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					                            Time-Frequency Signal Processing:                                                                            
                                A Statistical Perspective
                                                                    Franz Hlawatsch and Gerald Matz
                     Institute of Communications and Radio-Frequency Engineering, Vienna University of Technology
                                             Gusshausstrasse 25/389, A-1040 Wien, Austria
                            phone: +43 1 58801 38916, fax: +43 1 5870583, email: fhlawats@email.tuwien.ac.at
                                           web: http://www.nt.tuwien.ac.at/dspgroup/time.html

                                                                                                 ¨ ¤
                                                                                                 6eS
   Abstract—Time-frequency methods are capable of analyzing and/or pro-                   with          the system’s impulse response, completely and
cessing nonstationary signals and systems in an intuitively appealing and                 uniquely describes the system’s frequency-domain character-
physically meaningful manner. This tutorial paper presents an overview of
some time-frequency methods for the analysis and processing of nonstation-                istics [2]. However, for a time-varying linear system, the
ary random signals, with emphasis placed on time-varying power spectra                    frequency-domain characteristics will be time-dependent and
                                                                                                        P
and techniques for signal estimation and detection.                                       we thus must look for a time-dependent frequency response of
   We discuss two major definitions of time-dependent power spectra—                                          ¨ ¦4
                                                                                                             £0I@ ¤
the generalized Wigner-Ville spectrum and the generalized evolutionary
                                                                                          the form            , which can again be interpreted as a time-
spectrum—and show their approximate equivalence for underspread ran-                      frequency representation of the system.
dom processes. Time-dependent power spectra are then applied to nonsta-                      In this tutorial paper, we will discuss time-dependent power
tionary signal estimation and detection. Specifically, simple expressions and              spectra and how they can be used for signal estimation and sig-
designs of signal estimators (Wiener filters) and signal detectors in the sta-
tionary case are extended to underspread nonstationary processes. This re-                nal detection in nonstationary environments. In Section II, we
sults in time-frequency techniques for nonstationary signal estimation and                consider two different approaches to defining a time-dependent
detection which are intuitively meaningful as well as efficient and stable.                power spectrum for nonstationary random processes. Section
                                                                                          III discusses the concept of underspread processes and shows
                              I. I NTRODUCTION                                            that the two time-dependent power spectra of Section II become
   It is an interesting fact that most papers on time-frequency                           effectively equivalent in the underspread case. The estimation
analysis consider deterministic signals whereas a large and im-                           (optimal filtering) and detection of nonstationary random pro-
portant group of applications require signals to be modeled as                            cesses will be considered in Sections IV and V, respectively.
random processes. As long as these random processes are sta-
tionary, a need for time-frequency methods does not arise since                                             II. T IME -D EPENDENT P OWER S PECTRA
the power spectral density1                                    £
                                                               3                             Apart from the physical spectrum, which can be interpreted
                     20(&$" £§¥£¡
                     1 )'% # ! ¨ ¤ ¢    © ¨ ¦¤ ¢                 5
                                                                    4                     as the expectation of the spectrogram [3–5], there are two fun-
                                                                                (1)       damentally different approaches to defining a time-dependent                       ¨
                                                                                                                                                                            F@ ¤
          76¤ ¢ 
          © ¨            ¨  A@
                          CB6¤            G¨@
                                           HF6¤                                           power spectrum for a nonstationary random process        .                    9
with           E     8
                     9           , provides a complete and unique
                                      D
                                      E9
description of the process’ second-order statistics and spectral                          A. Generalized Wigner-Ville Spectrum
properties [1]. In particular, due to stationarity the power spec-
tral density does not change with time.                                                     The first group of time-dependent spectra, called generalized
   When the process under analysis is nonstationary, it is intu-                          Wigner-Ville spectrum [4–7], is a formally simple extension of
                                                                                                                                                    ¨ ¦¤ ¢
                                                                                                                                                    £§¥U¡
itively clear that its spectral properties change with time and,                          the stationary power spectral density      in (1):
hence, a meaningful representation of these spectral properties
                                                                                                                                                               3
                                                                                                                                                               £
must depend on a time variable. Thus, we look for a time-                                                    2(($"FI@ ¤ 2tg ¢   s£0I@ ¤ 2phf
                                                                                                             1 )'% # ! ¨ 4 q i     r ¨ ¦ 4 q ig¢                  
                                                             £HI6¤ ¢ ¡
                                                             ¨ ¦4@                                                                                                                 (3)
dependent power spectrum of the form                , which can be
                                                                                          with
interpreted as a time-frequency representation of the process’
second-order statistics.                                                                      A ex &‡
                                                                                                €     @ 4       € x A w  r ¨ 4 q i
                                                                                                                ey£@ 7¢ vuFI@ ¤ $pg ¢                                 4 ‰
                                                                                                                                                                        "
                                                                                                   ‚     † „ ‚
                                                                                                          e…ƒ                                                  ˆ„
                                                                                                                                                                †                  (4)
   This situation is somewhat analogous (and, as we will see
presently, closely related) to the frequency-domain analysis of                                    © ¨‘@ 4 @¤ ¢
                                                                                                   ’tBFI6¥                   ¨
                                                                                                                               F@ ¤         G ¨‘@
                                                                                                                                            HtB6¤
                                                                                          where               E        9
                                                                                                                       “8    . The parameter
                                                                                                                                      E9
                                                                                                                                      D                is ar-– ”
                                                                                                                                                              —•„
linear systems. As long as the system is time-invariant, the fre-
                                                                                          bitrary a priori. Special cases are the Wigner-Ville spectrum
quency response       P                                       3
                                                              £                              ˜
                                                                                              ™©                                                                   ©
                                                                                          (
                                                                                          „        ) [3–8] and the Rihaczek spectrum (
                                                                                                                      ˜ f©                        ) [4,5,9].     d
                                                                                                                                                                 e€   „
                          © ¨ ¦
                          R£Q¤             20(&$" UT 
                                           1 )'% # ! ¨ ¤ S          5
                                                                    4                     The case          has certain advantages over other choices of ;
                                                                                                                           „                                                       „
                                                                                (2)                       4
                                                                                                      ¨£¦0I@ ¤ q0gag ¢ f
                                                                                          in particular,            is always real-valued (however, it is not
      V                                                                                   guaranteed to be everywhere nonnegative although it is approx-
  WFunding by FWF grant P11904-TEC.
   Throughout this paper,   denotes a deterministic signal or a random pro-
                             b aX
                                 `Y                                                       imately nonnegative for the practically important underspread
cess and integrals are from  to .
                               ec
                               d             d                                            processes—see Section III).

                                                                                      1
                                                                                                                       ¨ ¦4 q i
                                                                                                                       £0I@ ¤ 2pg ¢ f                                                                                                                                                                                                                         ©                    
   Under appropriate conditions,              can be interpreted                                                                                                                                                                                                                         evolutionary spectrum (            ) [25–28] and the transitory
                                                                                                                                                                                                                                                                                                                                                 © „                           d€
as the expected value of the so-called generalized Wigner dis-                                                                                                                                                                                                                           evolutionary spectrum (              ) [13,29]; furthermore, the
                                                                                                                                                                                                                                                                                                                                             „                                 d
                                                                                                                                                                                                                                                                                                                                                                            © e€ ‚                      ˜                              ¡
                                                                                                                                                                  ¨ ¦4@ q i
                                                                                                                                                                  £0I6¤ $pg ¢ f                                                                                                          Weyl spectrum is obtained with             and      chosen as the                       „
tribution [10–12] (cf. (19)). For any ,               reduces to                                           ¨ ¦¤
                                                                                                           £Q¢ ¡                                     „                                                                                                            ¨@
                                                                                                                                                                                                                                                                   F6¤                                                                                                                                          ¢
the power spectral density         in (1) if the process       is                                                                                                                                                                                      9                                 positive (semi-)definite square root of       (this choice has cer-
                                                                                                                                                                                                                                                                                                                                                                                                                           ¡
(wide-sense) stationary.                                                                                                                                                                                                                                                                 tain advantages over other choices of and ) [13]. Note that                                       „
                                                                                                                                                                                                                                                                                               £0I@ ¤ 2tg ¢
                                                                                                                                                                                                                                                                                               ¨ ¦4 q i            I
                                                                                                                                                                                                                                                                                                                   H    ˜
                                                                                                                                                                                                                                                                                         GES               .                                                                                                ¨@
                                                                                                                                                                                                                                                                                                                                                                                                            F6¤
B. Generalized Evolutionary Spectrum                                                                                                                                                                                                                                                        For a wide-sense stationary process      , the innovations sys-                                    9
                                                                                                                                                                                                                                                                                               ¡
                                                                                                                                                                                                                                                                                         tem      can always be chosen to be time-invariant, in which     P
   The second group of time-dependent spectra, called general-                                                                                                                                                                                                                                                                                                                   $       ¨ ¦4@ q i
                                                                                                                                                                                                                                                                                                                                                                                         £HI6¤ $pg %
                                                                                                                                                                                                                                                                                         case the generalized Weyl symbol                reduces to the sys-           ¨ ¦
                                                                                                                                                                                                                                                                                                                                                                       £Q¤
ized evolutionary spectrum [7,13], is based on an innovations                                                                                                                                                                                   ¨
                                                                                                                                                                                                                                                F@ ¤                                     tem’s frequency response         . Hence, the generalized evo-
                                                                                                                                                                                                                                                                                                                            P
system representation of the nonstationary process
                                               F@ ¤
                                               ¨               [14].                                                                                                                                                                9                                                    lutionary spectrum here reduces to the power spectral density:
That is,     is represented as the output of a linear, time-varying
                                       9                                                                                                                                                                                                                                                       T£0I@ ¤ 2tg ¢
                                                                                                                                                                                                                                                                                               © ¨ ¦4 q i              
                                                                                                                                                                                                                                                                                                                       P        £§¤ ¢ ¡ © ' Q£Q¤
                                                                                                                                                                                                                                                                                                                                ¨ ¦          ¨ ¦
                                                                                                    ¡                                                                                                                                                                                    GES                                .
system (linear operator) whose input is stationary white noise
                 ¨@
                 F6¤
¢
     with normalized power spectral density:
                                                                                                                                                                                                3                                                                                                    III. U NDERSPREAD S YSTEMS                                                                        AND                 P ROCESSES
                                                                      ¨
                                                                    © F@ ¤      ¤       ¡   ¢            ‡F6¤ ¨
                                                                                                          © ¨@                          ¨ ‘ (I6eS
                                                                                                                                            @4 @¤                   ¢       4‘@ ¨‘@¤                                                                                                        For the results obtained with the two classes of time-
                                                        9                                                              ¤
                                                                                                                       ¥£
                                                                                                                                                                                                                                                                                         dependent spectra described above, the time-frequency dis-
                                                                                                                                                                                                                                                                                                                                                                                                                                           ¡
                                  @ "©© tBFI@ ¤ §
                                    ¤ ¨    ¨‘@ 4 ¦                                          ¨ B@
                                                                                              ‘                   ¨‘@ 4 @¤
                                                                                                                  tB(I6eS                                                                                                                                                                placements caused by the innovations system         and the time-
with                          and        P denoting the impulse re-                 ‚                                                                                                                                                                                                                                                                                                                                                                                ¨@
                                                                                                                                                                                                                                                                                                                                                                                                                                                                     F6¤
                                     ¡                                                                                                                                          ¡                                                                                                        frequency correlation structure of the resulting process     play                                                                                                   9
sponse (kernel) of . (In the stationary case, is time-invariant
                             ' ¨£§¤ © £Q¤ ¢ ¡
                                       ¦       ¨ ¦                                                                                                                                                         ¡                                                                            an important role. A limitation of these time-frequency dis-
and                     .) The innovations system is obtained
                                                                                                                                                                   ¡
                                                                                                                                                                    ¡                                              ©
                                                                                                                                                                                                                    
                                                                                                                                                                                                                                           ¢                                            placements or time-frequency correlations leads to the impor-
as a solution to the factorization problem                   , where
                ¢                                                                                                                                                                            ¨ F@ ¤                                                       ¡
                                                                                                                                                                                                                                                                                        tant concepts of underspread systems or processes, respectively.
     is the correlation operator2 of the process         and        is                                                                                                                             ¢ 9
                                                                    ¡                           ¡                                                                                                                                                                                        Broadly speaking, an intuitively pleasing and meaningful inter-
the adjoint of . Thus, is a “square root” of          . This square                                                                                                                                                                    ©
                                                                                                                                                                                                                                        !
                                                                                                                                                                                                                                        "                                ¡               pretation of time-dependent spectra is only possible for under-
root is unique only up to a factor satisfying                  : if                                                                                                                               
                                                                                                                                                          ¡
                                                                                                                                                           ¡                        ©  
                                                                                                                                                                                        #                               ¢                                                                spread processes.
is a valid innovations system satisfying                 and sat-
                                                            ©
                                                            "
                                                            !                                                                                                                       ©U‘                                                                    
                                                                                                                                                                       ¡                               ¡
isfies            , then it is easily seen that
                                                         is a valid                                                                                                                                                           
                                                                                                                                                                                                                                                                                         A. Underspread Systems
innovations system as well.                                                                                                                                                         ¡
   A “time-dependent frequency response” of that extends the                                                                                                                                                                                                                                The time-frequency shifts caused by a linear time-varying
                                                                                                                                                                                                                                                                                                      ¡
frequency response of time-invariant systems in (2) is given by                                                                                                                                                                                                                          system are characterized by the generalized spreading func-
the generalized Weyl symbol [15–17]                                                                                                                                                           3                                                                                         tion [15–17]                                                                                                                          3
                                                                        20(&# "(I6¤ $pg S   s£0I@ ¤ 2pg %
                                                                        1 )'% ! ¨ 4@ q i      r ¨ ¦4 q i                                                                                                                                                                                                     ((# "FI@ ¤ 2pg S  r &S&6¤ $pg % ¡
                                                                                                                                                                                                                                                                                                               )'% ! ¨ 4 q i         ¨ R4  q i                                                                     £ U
                                                                                                                                                                                                                                                                                                                                                                                                                       T                   E@
                                                                                                                                                                                                                                                                                                                                                                                                                                           4
                                                                $
                                                                                                                                                                                                                                                                         (5)                                                                          £
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                (7)
with                                                                                                                                                                                                                                                                                                                         ¨ 4 q i
                                                                                                                                                                                                                                                                                                                             FI@ ¤ 2pg S
                                                                                                                                                                                                                                                                                         with             as defined in (6). It can be shown [15,16] that
                                     u(I6¤ $pg S
                                     r ¨ 4@ q i                               A € x &7
                                                                                      @ 4       € x A @ w S                                                                                                                           ‰
                                                                                                                                                                                                                                    4 "                                                                   ¨ R4 q i
                                                                                                                                                                                                                                                                                                            V& ¤ 2pg % ¡
                                                                                   ‚     † …’
                                                                                            „ ‚                                                                                                                 † „                                                      (6)             the magnitude of              is independent of , so that we may                                                                          „
                                                                                                                                                                                                                                                                                                   ¨&RV46¤           © ¨ R4 q i
                                                                                                                                                                                                                                                                                                             % ¡ W¥QV& ¤ 2pg % ¡ 
                                                                                                                                                                                                                                                                                                                                                                                                                                              ¨ R4 q i
                                                                                                                                                                                                                                                                                                                                                                                                                                               &V& ¤ 2tg % ¡
                                                                                                                                                                                                                                                                    ©            ˜       write                             . Furthermore,              is the
where          . Special cases are the Weyl symbol for
                                      – ” „                                                                                                                                                                                                        „                         ©           2-D Fourier transform of the generalized Weyl symbol in (5).
[16–21], Zadeh’s time-varying frequency response for                                                                                                                                                                                                           „                                                                                                                                                       ¨ R4  q i
                                                                                                                                                                                                                                                                                                                                                                                                                        &S&6¤ $pg % ¡
                                                                                                                                                                                                                                                                                           The generalized spreading function                   is the co-
     [16,17,20,22,23], and the Kohn-Nirenberg symbol (equiv-
        d€                                                                                                                                                                                                                                                                               efficient function of an expansion of             into elementary
                                                                                                                                                                                                                                                                                                                                                                                                                                  ¡
                                                                                                                                                                                                                                                                                                                                                                                                                                      
alently, Bello’s frequency-dependent modulation function) for
                         ©                                                                                                                ©                   ˜                                                                                                                          time-frequency shifts         , where
                                                                                                                                                                                                                                                                                                                                                     $YTq pg
                                                                                                                                                                                                                                                                                                                                                           i                                                         © F6¤ $pg
                                                                                                                                                                                                                                                                                                                                                                                                                         ¨@       q i
                                                                                                                                                                                                                                                                                                                                                                                                                               c9 T Y                                 9
                                                                                                                                                                                                                                                                                                                                                                                                                                                                           6¤
                                                                                                                                                                                                                                                                                                                                                                                                                                                                           @
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 ‚
                                                                                                                                                                                                                                                                                                                                        X                                                             X
                                                                                                                                                                                                                                                                                                                                                                                                       a`                    b
    „        [18,23,24]; the case    d€ ‚  has again certain ad-                                                                „                                                                                                                                                                                                T                       'fd i )' T )' ! ¨
                                                                                                                                                                                                                                                                                                                                                       q 0ge# pg 0(% ! £ U0(% "
                                                                                                                                                                                                                                                                                                                   [15–18,20,21,23,30,31]. Hence, for a
vantages over other choices of [17,18,20]. Note that the gen-                                                     „                                                                                                                                                                                                                             Q¨&S&6¤
                                                                                                                                                                                                                                                                                                                                                     R 4         ¨ R4 
                                                                                                                                                                                                                                                                                                                                                               ¡  &V6¤
                                                                                                                                                                                                                                                                                         given       ,             indicates how much the time-frequency
                                                                                                                                                                                                                                                                                                                                                            %
eralized Wigner-Ville spectrum in (3) is the generalized Weyl                                                                                 ¢                                                                                                                                            T q '0fidh# tg )((%
                                                                                                                                                                                                                                                                                                       i '                 !     @
                                                                                                                                                                                                                                                                                                               ! £ TU)0'(% "¨ 6¤ ©‡¨F@ ¤ q$ipg
symbol of the correlation operator    , i.e.,
                                                                                                                                    
                                                                                                                                                                                                                                                                                         shifted input signal                   ‚ 9       b 9 TY     X `
                                                                                                                                                                                                                                                                                         contributes to the output signal. It follows that the time-
                                                                              © ¨ ¦4@ q i
                                                                              £HI6¤ $pg ¢ f                            $               £HI6¤ $pg &
                                                                                                                                        ¨ ¦ 4 @ q(
                                                                                                                                                 '
                                                                                                                                                   i                        0
                                                                                                                                                                            )
                                                                                                                                                                                                                                                                                         frequency shifts caused by a linear time-varying system are                                                                               ¨ 4
                                                                                                                                                                                                                                                                                                                                                                                                                                   & & ¤ % ¡                        R
                                                                                                                                                                                                                                                                                                                                                                                                                                                                      V
                                                                                                                                                                                                                                                                                         crudely characterized by the effective support of
                                                                                                                                                                                                                                                                                                                   ¡
                                                                                                                                                                                                                                                                                                                                                         .                                                                            ¨ R 4 
                                                                                                                                                                                                                                                                                                                                                                                                                                     Q&S&6¤       ¡
  The generalized evolutionary spectrum is now defined as the                                                                                                                                                                                                                                A system is now called underspread if                    is con-                   &S&6¤
                                                                                                                                                                                                                                                                                                                                                                               ¨ R4 
                                                                                                                                                                                                                                                                                                                                                                                                                                                %
                                                                                                                                                                                                                                                                                                                                                                                                                                                         ¡
squared magnitude of the generalized Weyl symbol of the inno-                                                                                                                                                                                                                            centrated about the origin of the        -plane, so that causes
                                                                         ¡
vations system [13]:                                                                                                                                                                                                                                                                     only small time-frequency shifts (mathematically precise defi-
                                                                                          ¨ ¦4@ q i
                                                                                        r £HI6¤ $pg ¢             $2 1                   ' 3£0I@ ¤ 2pg
                                                                                                                                             1 ¨ ¦ 4 q i%                                 )                                                                                              nitions of underspread systems can be found in [16,17,30,31]).
                                                                             GES                                   1                       1
                                                                                                                                                                                                                                                                                         It should be noted that underspread is not equivalent to slowly
                                                                                                                                                                                                                                                                                         time-varying, since slow time variation only means a limita-
                                                                                                                                                                                                                                                                                                          ¡       ¨ 4 
                                                                                                                                                                                                                                                                                                                   & &6¤
Note that this definition contains a twofold ambiguity related                                                                                                                                                                                                                            tion of               %
                                                                                                                                                                                                                                                                                                                        R
                                                                                                                                                                                                                                                                                                                       Q S
                                                                                                                                                                                                                                                                                                              with respect to . In contrast, underspread
                                                                                                                                                                                                                                                                                                                                                                                     R

to the choice of        and the choice of the innovations sys-                 ”
                                                                             – h„                                                                                                                                                                                                                                                                                                                                                             R
                             ¡                                                                                                                           ¢                                                                                                                              means limitations with respect to both and ; however, the
tem for given correlation operator      . Special cases are the                                                                                                                                                                                                                          extents of these limitations can be exchanged for one another.
             4
  The correlation operator        is the positive (semi-)definite linear operator            75
                                                                                            6                                                                                                                                                                                            Hence, a slowly time-varying system will not be underspread              ¡          & & ¤
                                                                                                                                                                                                                                                                                                                                                                              ¨ 4                                                                                
whose kernel equals the correlation              E              .                                           6 8   b § aY
                                                                                                                    ` `
                                                                                                                      C A 9
                                                                                                                      D B @
                                                                                                                                                                  V X `Y X
                                                                                                                                                              A `
                                                                                                                                                          FGb BaY b p(E                                                                                                                if its memory (extension of               with respect to ) is too             %
                                                                                                                                                                                                                                                                                                                                                                                      V
                                                                                                                                                                                                                                                                                                                                                                                      R




                                                                                                                                                                                                                                                                                     2
long, whereas a fast time-varying system will be underspread if                                                                                                            lence of all above-mentioned time-dependent spectra in the case
its memory is sufficiently short.                                                                                                                                           of underspread processes. This equivalence is based on the fol-
                                                                                                                                                                           lowing two approximations valid for the generalized Weyl sym-
B. Underspread Processes                                                                                                                                                   bol of underspread systems (bounds on the associated approxi-
  Quasi-stationary processes have limited spectral correlation,                                                                                                            mation errors are provided in [16,17,30]):
                                                                                                                                                                                                                                                                                                       ¡
while quasi-white processes have limited temporal correlation.                                                                                                             1. The generalized Weyl symbol of an underspread system                                                                              is
These two situations are generalized by the concept of under-                                                                                                              approximately independent of , i.e.,                               „
spread processes. We first define the expected generalized am-                                                                                            ¨@
                                                                                                                                                        F6¤                                                 $     ¨ ¦4@ q i
                                                                                                                                                                                                                 £HI6¤ Ftg %                        $        ¨ ¦4@ q i
                                                                                                                                                                                                                                                               £HI6¤ (pg %  )
                                                                                                                                                                                                                                                                             0
biguity function [6,16] of a nonstationary process   as                                                       3                                               9                                                                                                                                            (8)
                                                                                                                                                         
        0(&# "(I6¤ $pg ¢   r &S&6¤ $pg ¢   ¡
        )'% ! ¨ 4@ q i         ¨ R4  q i                                                          £ U
                                                                                                      T               © @                4                  4              2. For an underspread system
                                                                                                                                                                                                                                          ¡
                                                                                                                                                                                                                                                    , there is
                                                   £
                                                                                                                                     E     §¨¥¦9 q$YT ipg X (¤¢ 9£
                                                                                                                                                                                                       $        £0I6¤ $pg %
                                                                                                                                                                                                                 ¨ ¦ 4 @  qI i                          $1    ' 1 £HI6¤ $pg
                                                                                                                                                                                                                                                                     ¨ ¦4 @ q i%   )
                 FI@ ¤ $pg ¢ 
                 ¨ 4 q i                                                                                                                                                                                                    %                            1        1                                       (9)
with            as in (4). The expected generalized ambiguity
function is the generalized spreading function (see (7)) of the       ¢                                                                                                     With these approximations, we now obtain the following
correlation operator    , i.e.,                                                                                                                                                                                                                                                                 ¨
                                                                                                                                                                                                                                                                                                F@ ¤
                                                                                                                                                                           equivalence results valid for an underspread process :                                                           9
                                         &S&6¤ $pg & © &V6¤ $pg ¢   ¡
                                         © ¨ R 4  q( ¡
                                                      i  ¨ R4  q i                                                                                                                         ¨ ¦ 4 q ig¢
                                                                                                                                                                                          © £0I@ ¤ 2pyf                       ¨ ¦ 4 @ q(
                                                                                                                                                                                                                              £HI6¤ $pg &
                                                    '
                                                                                                                                                                             With
                                                                                                                                                                                                                      $
                                                                                                                                                                                                           , and since for an underspread
                                                                                                                                                                                                                                       '
                                                                                                                                                                                                                                         i
                                                                                                                                                                                                     ¢ 
it is furthermore the 2-D Fourier transform of the generalized                                                                                                             process     is an underspread system, it follows from (8) that
Wigner-Ville spectrum in (3). It can be shown [6,16] that the                                                                                                              the generalized Wigner-Ville spectrum is approximately inde-
                                                                                                                                &S&6¤ 2pg ¢   ¡
                                                                                                                                ¨ R4  q i                                 pendent of , i.e., [7,16]
                                                                                                                                                                                             „
expected generalized ambiguity function                      describes
the average correlation of all time-frequency locations separated
                                                          R
                                                                                                                                                                                                            ¨ ¦ 4 @ q  ig¢
                                                                                                                                                                                                           £HI6¤ Fthf                                   ¨ ¦ 4 @ q  ig¢
                                                                                                                                                                                                                                                          £0I6¤ 0¨phf          )
                                                                                                                                                                                                                                                                               e
by in time and by in frequency.                                                           ¨@
                                                                                          F6¤
   A nonstationary process           is now called underspread if                     9                                                                                                          £0I@ ¤ 2pg ¢
                                                                                                                                                                                                 © ¨ ¦4 q i                    $1   ' 1 £HI6¤ $pg
                                                                                                                                                                                                                                         ¨ ¦4 @ q i%
     & 6¤ $pg ¢   ¡ 
    ¨ 4  q i         V
                      R      ©                  & & ¢   ¡ 
                                                 ¨ 4 ¤                      R
                                                                            V                                                                                                With GES                           , and since for an under-
                                                                                                                                                                                                                               1       1

       ¨ R4
       &V& ¤                  is concentrated about the origin of                                                        F6¤
                                                                                                                          ¨@                                               spread process we can always find an underspread innovations
the        -plane, so that components of          that are sufficiently                                                9                                                               ¡
                                                                                                                                                                           system , it further follows from (8) that the generalized evolu-
separated in the time-frequency plane will be nearly uncorre-                                                                                                              tionary spectrum (based on an underspread innovations system
lated (mathematically precise definitions of underspread pro-                                                                                                               ¡
                                                                                                                                                                              ) is approximately independent of , i.e., [7,13,16]                                „
cesses can be found in [6,7,16]; furthermore, somewhat similar
concepts of processes with limited time-frequency correlation
                                                                                                                                                                                                                  ¨ ¦4@ q i
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have been discussed in [32–35]). This underspread property is                                                                                                                                                                                     ' 3£0I@ ¤ 2pg
satisfied by many nonstationary processes occurring in practice.
                                                                                                                                                                                  © £HI6¤ $pg ¢
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                                                                                                                                                                               With GES                              and             1              1
We emphasize that underspread should not be confused with                                                                                                                  $   ¨ ¦ 4 @  qI      ¨ ¦ 4 q' i
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quasistationarity which only means a limitation of
                                         R
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                                                                                                                                                                           tions system , it follows from (9) that the generalized evo-
with respect to . In contrast, underspread means limitations                     R                                                                                        lutionary spectrum is approximately equal to the generalized
with respect to both and , where again the extents of these                                                                                                                Wigner-Ville spectrum, i.e., [7,13,16]
limitations can be exchanged for one another. Hence, a quasi-
stationary process will not be underspread if its correlation time                                                                                                                                                   ¨ ¦4 q i
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(extension of              with respect to ) is too long, whereas a                                                                                                                          £0I@ ¤ 2tg ¢
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                                                                                                                                                                                                                H         ˜
fast nonstationary process will be underspread if its correlation                                                                                                          Since GES               , this also shows that the generalized
time is sufficiently short (quasiwhite process).                                                                                                                            Wigner-Ville spectrum of an underspread process is approxi-
   The concepts of underspread systems and underspread pro-                                                                                                                mately real-valued and nonnegative.
                                                                             &S&6¤ $pg ¢   ¡
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cesses are related since                 is the generalized spread-
                                                      ¢                                                                             ¨
                                                                                                                                     F@ ¤                                     These equivalence results simplify the spectral analysis of
ing function of       , and hence a process            is underspread                                                ¢          9                                         nonstationary processes a great deal: Even though there exist
if and only if its correlation operator       is an underspread sys-                                                                                                       infinitely many different time-dependent spectra (there are the
tem. Furthermore, the time-frequency shifts caused by the inno-
                                     ¡                                                                                                                                     two distinct classes of generalized Wigner-Ville spectrum and
vations system       are related to the time-frequency correlation                                       ¨@
                                                                                                         F6¤                                                               generalized evolutionary spectrum, plus the dependence on                                                                            „
structure of the associated process
             ¡
                                            . If the innovations sys-                               9                                          © ¢     ¡
                                                                                                                                                         ¡                within each class), all these spectra are approximately equiva-
tem is underspread, the correlation operator                        is                                                                                         F6¤
                                                                                                                                                               ¨@          lent for underspread processes.
an underspread system as well, and hence the process                is                                  ¨@
                                                                                                        F6¤                                                9                  This approximate equivalence is demonstrated by Fig. 1
itself underspread. Conversely, if          is underspread, then not          ¡
                                                                                                9                                                                          which compares various time-dependent spectra of an under-
every innovations system is underspread but an underspread
¡                                                                                                                                                                          spread process. It is seen that all spectra are very similar and,
    can always be found.                                                                                                                                                   furthermore, that they are smooth time-frequency functions. In
                                                                                                                                                                           fact, the spectra of underspread processes must be smooth func-
C. Equivalence of Time-Dependent Spectra
                                                                                                                                                                           tions since the generalized Wigner-Ville spectrum is the 2-D
  The importance of the underspread property in the context of                                                                                                             Fourier transform of the expected generalized ambiguity func-
nonstationary spectral analysis is due to the approximate equiva-                                                                                                          tion (which is concentrated about the origin) and the generalized

                                                                                                                                                                       3
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                              @                           @                                                         @                                                                               @
           (a)                          (b)                                         (c)                                                                           (d)                                                    (e)
Figure 1. Time-dependent spectra of an underspread process: (a) Wigner-Ville spectrum, (b) real part of Rihaczek spectrum,
(c) Weyl spectrum, (d) (transitory) evolutionary spectrum using a positive semidefinite innovations system (here, the evolutionary
spectrum equals the transitory evolutionary spectrum since the positive semidefinite innovations system is used [13]), (e) magnitude
of expected ambiguity function (the rectangle shown has area 1 and thus permits to assess the underspread property of the process).
The process was generated by means of the time-frequency synthesis technique introduced in [36]. The signal length is 256 samples.

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                              @                           @                                                         @                                                                               @
           (a)                          (b)                                         (c)                                                                           (d)                                                    (e)
Figure 2. Time-dependent spectra of an overspread process: (a) Wigner-Ville spectrum, (b) real part of Rihaczek spectrum, (c) Weyl
spectrum, (d) (transitory) evolutionary spectrum, (e) magnitude of expected ambiguity function. The overspread characteristic of
this process is due to strong statistical correlations between the ‘T’ and ‘F’ components. Note that the expected ambiguity function
in (e) is widely spread out beyond the rectangle with area 1.

                                                                                                        ¨
                                                                                                        F@ ¤                                                                                                                              ¨@
                                                                                                                                                                                                                                          F6¤
evolutionary spectrum is similar to the generalized Wigner-Ville                        where     ¢
                                                                                                     is nonstationary noise uncorrelated with , by                                                          ¡
                                                                                                                                                                                                                                   
spectrum.                                                                               means of a linear, time-varying system . Hence, the signal
   In contrast, Fig. 2 shows that the various spectra yield dra-                        estimate is given by                                                                                                         3
matically different results for an “overspread” process (i.e., a                                               ¡        © ¨@
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process whose expected generalized ambiguity function is not                                                                                             9                          ¤ £
                                                                                                                                                                                                          9
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sufficiently concentrated about the origin of the         -plane, cf.
part (e)). Furthermore, the spectra are no longer smooth func-                          The system that minimizes the mean-square error is the time-
tions; they contain rapidly oscillating components (cross terms)                        varying Wiener filter [42–45]
that can be attributed to the strong statistical correlations exist-
                                                                                                                                             £
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                                                                                                                                                                                                                                                (10)
ing between widely separated time-frequency points [7,37].                                    ¤
                                                                                              ¥
                                                                                                                                   ¦
                                                                                        where      and     denote the correlation operators of signal and                                                                             ¡
                                                                                        noise, respectively. For stationary random processes,        sim-
         IV. N ONSTATIONARY S IGNAL E STIMATION                                         plifies to a time-invariant system whose frequency response is
   In the remainder of this paper, we shall show how time-                              given by [1,42–46]                   P
                                                                                                                                                                                ¨ ¦¤ ¤
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dependent spectra can be applied to nonstationary signal esti-                                                                           ¢       © ¨ ¦
                                                                                                                                                 £§¤                   4
                                                                                                                                                                          ¨£Q¤ e¡ A £§¤ ¤ ¡
                                                                                                                                                                            ¦ ¦        ¨ ¦                                                      (11)
mation and detection. We shall use the generalized Wigner-
Ville spectrum because it has a mathematically simple struc-                                          £Q¤ ¤ ¡
                                                                                                      ¨ ¦                                        ¨ ¦¤ ¦
                                                                                                                                                 £§£h¡
ture. However, for underspread processes the generalized evolu-                         where          and         denote the power spectral densities
tionary spectrum is approximately equivalent to the generalized                         of signal and noise, respectively. This frequency-domain ex-
Wigner-Ville spectrum (as explained above), and hence it can                            pression contains simple products and reciprocals of functions
be substituted for the generalized Wigner-Ville spectrum in the                         (instead of products and inverses of operators as in (10)) and
relevant equations. Other approaches to nonstationary signal es-                        hence allows a simple design and interpretation of time-invariant
timation are discussed in [34,35,38–41].                                                Wiener filters.
   The enhancement or estimation of signals corrupted by noise                          A. Time-Frequency Formulation of Time-Varying Wiener Filters
or interference is important in many signal processing applica-
tions. In this section, we consider the estimation of a nonstation-
                        F6¤
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ary random signal        from an observation   9                  ,       ¢
                                                                                        stationary case can be obtained for the time-varying Wiener fil-

                                                                                    4
ter
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        by introducing in (11) an explicit time dependence, i.e.,
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by substituting for         and for                 suitably de-
fined time-dependent frequency responses and time-dependent                                                                                                                           F@ ¤
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power spectra. Indeed, for jointly underspread3 processes
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and      it can be shown [47,48] that the time-varying Wiener
                     ¢
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                                                                                                                                                                                                                                                                                                                         1
filter
                         ¡           ¢
           can be decomposed into an underspread component
and an overspread component with the following properties:
                                                                                                                                                                                                                                                                                                              " ) 0¨' ) H H
  The overspread system component has negligible effect on the                                                                                                                                                              ' )                                                                                         FDb C6 a`Y ¤2¢ B45 3
                                                                                                                                                                                                                                                                                                                        E 8    9
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system’s performance (mean-square error) and can hence be dis-                                                                                                                                                                                                                                   `                                                                                  `
regarded.                                                                                                                                                                                                                                               (a)                                                                                           (b)
  The underspread part, denoted as    in what follows, allows
                                                                                                               
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                                                                                                                                                                                                Figure 3. Time-frequency interpretation of the time-varying
                                                                                                                                                                                                                                           ¢
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the approximate time-frequency formulation                                                                                                                                                      Wiener filter       for jointly underspread signal and noise pro-
                                                                                                                                                                                                cesses: (a) Effective time-frequency support regions of signal
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                                     ¨ ¦ 4 @ q ig¤
                                     £HI6¤ $phf                                              ¨ ¦ 4 @ q ig¦
                                                                                             £HI6¤ $phf                                                                                         B. Time-Frequency Design of Time-Varying Wiener Filters
where               and                denote the generalized
Wigner-Ville spectra of signal and noise, respectively.
                                                                                                                                                                                                   The time-frequency formulation (12) suggests a simple time-
  The time-frequency formulation in (12) provides the looked-                                                                                                                                   frequency design of nonstationary signal estimators. Let us de-
for extension of the frequency-domain formulation (11) to the                                                                                                                                   fine the “time-frequency pseudo-Wiener filter”          by setting
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nonstationary (underspread) case. For                (recall that                                                               „
    ¨ ¦4@ q
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            is real-valued), (12) allows a simple and intuitively
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appealing time-frequency interpretation of (the underspread                                                                                                                                                                                                                                          ¨ ¦ 4 @ q ig¤
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component of) the time-varying Wiener filter (see Fig. 3). Let                                                                                                                                                                                         %
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                                                                                                                                                                                                                                                                                            4                   ¨ ¦4@ q i                                                         (13)
¤ ¦ and
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and            , respectively. Regarding the action of the time-                                                                                                                                For jointly underspread processes   ,      where (12) is                                                                            ¢

varying Wiener filter, the following three time-frequency re-                                                                                                                                    a good approximation, combination of (12) and (13) yields
gions can be distinguished:
                                                                                                                                                                                                $      ¨ ¦ 4 @ q¤
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   Pass region. In the “signal only” time-frequency region                                                                                                                                      pseudo-Wiener filter         is a close approximation to (the un-
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energy is present, there is                 . Thus,      passes                                                             €
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distortion.                                                                                                                                                                                                                                                                                                                                                       ¢
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frequency regions where there is no signal.                                                                                                                                                                                      ¡      ¨
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region             where both signal energy and noise energy                                                                                                                                                             pB(I6¤ ¢ R S
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are present,            assumes values approximately between                                                                                                                                                        $      ¦ 4 @ q¤
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uation that depends on the relative signal and noise energy at                                                                                                                                      ¢ SI       ¨ ‘ FI@ ¤
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signal and noise energy, i.e., time-frequency points       with                                                                                                                                        ©               $        ¦ q ¤ £ # £ g $((% ! ‰ 04 ‘ @
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                                                                                         , there is
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centrated within the same region about the origin of the        -plane. For ex-                                                             b % #
                                                                                                                                            e&9 (Y                                                Compared to the Wiener filter             , the time-frequency
ample, a quasistationary process and a quasiwhite process may be individually                                                                                                                   pseudo-Wiener filter        possesses two practical advantages:
                                                                                                                                                                                                                                                                            ¡   ¢ I
underspread but not jointly underspread.


                                                                                                                                                                                            5
      Modified a priori knowledge. The calculation (design) of
                                                                                                                                                                                        ¡       ¢               filter is obtained as [54,55]
    requires knowledge of the correlation operators      and     (cf.
                                                                                                                                   ¤                                              ¦

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    eralized Wigner-Ville spectra               and              (cf.                                                                                                                                                                                                       %&
                                                                                                                                                                                                                                                                            (                                                                                                    d
                                                                                                                                                                                                                                                                                                                                                                                                                                                                        (
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    (13)). Although correlation operators and generalized Wigner-                                                                                                                                                                                                  ¨£¦HI@6¤
                                                                                                                                                                                                                                                                       4                                                                                                                                                           ¤ ¦                             4
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    Ville spectra are mathematically equivalent due to the one-to-                                                                                                                                              where              is the indicator function of9   (i.e.,                                                                                                                                                                               9
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                                                                                                                                                                                                                                        and 0 otherwise). Note that              is
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                                                                                                                                                                                                                                                                                                                                                                                                                                                            ¨ ¦4@ q
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    one mapping (3), the generalized Wigner-Ville spectra are much                                                                                                                                                                                                                  ”
    easier and more intuitive to handle than the correlation opera-                                                                                                                                             piecewise constant, expressing constant time-frequency weight-                                                                                                                             ¤
                                                                                                                                                                                                                                                                                                                                                                                                           ¥¦
    tors or correlation functions. For example, an approximate or                                                                                                                                               ing in a given time-frequency region . It can be shown [54,55]
    partial knowledge of the Wigner-Ville spectra will often suffice                                                                                                                                             that the performance of the robust time-frequency Wiener filter
    for a reasonable filter design. This fact is especially important
                                                                                                                                                                                                                    ¡       5 I
                                                                                                                                                                                                                     is approximately independent of the actual operating condi-
    for practical applications where the a priori knowledge has to                                                                                                                                              tions as long as they are within , . An intuitive and compu-
                                                                                                                                                                                                                                                                                                                                                                                          4 !
    be estimated from the available data [49].                                                                                                                                                                  tationally efficient approximate time-frequency implementation
                                                                                                                                                           ¡                ¢                                   of
                                                                                                                                                                                                                                      ¡       5 I
                                                                                                                                                                                                                         in terms of the multi-window Gabor transform [56,57]
       Reduced computation. The calculation (design) of           re-                                                                                                                                                                                                                                                          ¤
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    quires a computationally intensive and potentially unstable op-                                                                                                                                             exists if the partition        corresponds to a uniform rectangular
    erator inversion (or, in a discrete-time setting, a matrix inver-                                                                                                                                           tiling of the time-frequency plane [55].
    sion). In the time-frequency design (13), this operator inver-
    sion is replaced by simple and easily controllable pointwise di-                                                                                                                                            D. Simulation Results
    visions of functions. Assuming discrete-time signals of length
       , the computational cost of the design of      grows with    ,
                                                                                                     ¡           ¢                                                                          ¡
                                                                                                                                                                                            ¢                      Fig. 4 shows the Wigner-Ville spectra and expected ambiguity                                ˜ ©                                                                                                                                                         ¨
                                                                                                                                                                                                                                                                                                                                                                                                                                                                           F@ ¤
    whereas that of
                                       ¡   ¢ I
                         (using divisions and FFTs) grows only with
                                                                                                                                                                                                                functions (with  ¨@
                                                                                                                                                                                                                                 F6¤      ) of signal and noise processes      and                       „                                                                                                                                                          
          '
         log .
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                                                                                                                                                                                                                      as well as the Weyl symbols of the resulting Wiener filter
                                                                                                                                                                                                                                  ¢                                                                                                                           
                                                                                                                                                                                                                                                                                                                                                            ¢¡
                                                                                                                                                                                                                                                                                                                                                            ¡
                                                                                                                                                                                                                     , its underspread part       , and the time-frequency pseudo-
                                                                                                                                                                                                                Wiener filter        . From the expected ambiguity functions in
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                                                                                                                                                                                                                parts (c) and (d), it is seen that the processes       and      are                                                                                                                                                                                    ¢

       If the actual correlations    and
                                                                 ¤
                                             deviate from the nominal
                                                                                 ¦
                                                                                                                                                                                                                jointly underspread. From the Weyl symbols in parts (e)–(g),
    correlations for which the Wiener filter         was designed, the
                                                                                         ¡       ¢                                                                                                              the time-frequency pass, stop, and transition regions (cf. Fig. 3)
    filter’s performance may degrade significantly. This sensitivity                                                                                                                                              of the filters       ,      , and       are easily recognized. It is
                                                                                                                                                                                                                                                                                            ¡        ¢                 ¡               ¢                                             ¡     ¢
    of the performance of the Wiener filter (and also of the time-                                                                                                                                               verified that the Weyl symbol of           closely approximates that
                                                                                                                                                                                                                                                                                                                                                                                         £I I
                                                                                                                                                                                                                                                                                                                                                                                         ¢    ¡

    frequency pseudo-Wiener filter) to variations of the second-                                                                                                                                                 of
                                                                                                                                                                                                                                          ¡           ¢ 
                                                                                                                                                                                                                         . The mean SNR improvement achieved was 6.14 dB
    order statistics motivates the use of minimax robust Wiener fil-                                                                                                                                             for the Wiener filter        , 6.10 dB for its underspread part    ,
                                                                                                                                                                                                                                                                                                                           ¡                ¢                                                                                                                                     ¡¢¡
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     
    ters that optimize the worst-case performance within specified                                                                                                                                               and 6.11 dB for the time-frequency pseudo-Wiener filter            .
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  ¢£I ¡

    uncertainty classes of operating conditions.                                                                                                                                                                Hence, the performance of the time-frequency pseudo-Wiener
       Recently, the minimax robust time-invariant Wiener filters                                                                                                                                                filter is seen to be very close to that of the Wiener filter.
    introduced in [50–53] for stationary processes were extended                                                                                                                                                   To illustrate the application and performance of the robust
    to the nonstationary case, and time-frequency designs of ro-                                                                                                                                                time-frequency Wiener filter described in Subsection IV-C, we
    bust time-varying Wiener filters were proposed [54,55]. Par-                                                                                                                                                 defined -point uncertainty classes and             based on
                                                                                                                                                                                                                                                                    £                                                                                                                                   !                  4                                               B
                                                                                                                                                                                                                                                                                                                                                                                                                                                                           C© 0
    ticularly simple and intuitively appealing results were obtained                                                                                                                                            rectangular time-frequency regions . The regional energies
                                                                                                                                                                                                                                                                                                                                                                                                      ¤ ¦                                                                  ¤ 
    for so-called -point uncertainty classes. Let
                                   £                               be                                             ' ©¨©¨©¨ ' © ¤§¦¤ ¦ G ¥¦¤ ¦ 8
                                                                                                                                          ¤      Y d               Y            Y                                                                 ¢        ¤
                                                                                                                                                                                                                and used for the definition of and were derived from the
                                                                                                                                                                                                                                                                                                                                                                                          !                     4
                                                                                                                                                                                                                                                                                                                                                                                                       ¨ ¦4 @ q g¤
                                                                                                                                                                                                                                                                                                                                                                                                       £0I6¤ Hg yf
    a partition of the time-frequency plane, i.e.,               and                                                       –    d                                                                              nominal Wigner-Ville spectra                  and            shown
                                                                                                                                                                                                                                                                                                                                                                                  3 £30I6¤ 0g yf
                                                                                                                                                                                                                                                                                                                                                                                    ¨ ¦4 @ q g¦
     © % ¦ © ¦
               ¤     for
                                                  
                                                   ©
                               . Extending the stationary case defini-                                                                                                                                                                                                                                                                                                            ¦ @ £HI6¤ Hg hf )& FE© ¤
                                                                                                                                                                                                                                                                                                                                                                                        ¨ ¦ 4 @ q g¤ ( D D
                                                                                                                                                                                                                in Fig. 4(a), (b) according to                                       3 3                                                                                                                                                                                                      and
    tion in [51,53], -point uncertainty classes can be defined for
                                           £                                                                                                                                                                                 ¤                ©                    ( & GD
                                                                                                                                                                                                                                                                       D                 ¨ ¦4@ q
                                                                                                                                                                                                                                                                                    ¦ @ £0I6¤ Hg g ¦ f
    Wigner-Ville spectra as [54]
                                                                                                                                                                                                                ¢
                                                                                                                                                                                                                                                                                                                                                                        . The Weyl symbol of the robust
                                                                              3             3
                                                                                                                                                                                                                time-frequency Wiener filter         obtained for these uncertainty
                                                                                                                                                                                                                                                                                                                                                                             56I
                                                                                                                                                                                                                                                                                                                                                                             ¡
        $ ¨ ¦4 q " ©
        %£0I@ ¤ Hg g ¤ f #—!                                     (&
                                                  £0I6¤ Hg g ¤ f )' 
                                                  ¨ ¦4@ q                             © ¦ @
                                                                                                                       ¤ 
                                                                                                                       4                 ©
                                                                                                                                                       €
                                                                                                                                                               4
                                                                                                                                                               ) ) 
                                                                                                                                                                   )            2 0
                                                                                                                                                                                31p4                            classes is shown in Fig. 4(h). The rectangular time-frequency re-
                                                                              3           3                                                                                                                     gions
                                                                                                                                                                                                                                                               ¤ ¦
                                                                                                                                                                                                                           underlying the uncertainty classes are clearly visible.
    &   %£0I@ ¤ Hg g f #© 4
          $ ¨ ¦4 q "       ¦                            £0I6¤ Hg g ¦ f (
                                                        ¨ ¦4@ q                        © ¦ @                 ¢       © 4¤
                                                                                                                                                  €
                                                                                                                                                           4 2 04 )
                                                                                                                                                           U31p) ) 4                                           Fig. 5 compares the performance (output SNR vs. input SNR) of
                                                                                                                                                                                                                                                                                                                                                        ¡           ¢
                                                                                                                                                                                                                the ordinary Wiener filter        designed for the nominal Wigner-
                                                                                                                                                                                                                Ville spectra, the robust time-frequency Wiener filter       , and a
                                                                                                                                                                                                                                                                                                                                                                                                                                                             ¡   5 I
    i.e., as the sets which contain all Wigner-Ville spectra                                                                                                                                                    trivial filter
                                                                                                                                                                                                                                                                                H
                                                                                                                                                                                                                                                                                6
                                                                                                                                                                                                                                                                                ¡
                                                                                                                                                                                                                                  that suppresses (passes) all signals in the case
        ¨ ¦ 4 @ q g¤
        £HI6¤ Hg hf
                and
                                           £0I6¤ 0g yf
                                           ¨ ¦4 @ q g¦
                                that have prescribed energies
                                                                                                                                                                           ¤                H           ˜
                                                                                                                                                                                                                of negative (positive) input SNR. It is seen that at nominal op-
    and       ¢   ¤   H        ˜
                in prescribed time-frequency regions .
                                                                                                                                ¥¦
                                                                                                                                ¤                                                                               erating conditions        performs only slightly better than
                                                                                                                                                                                                                                                                                                                   ¡               ¢                                                                                                                                                      ¡   5 I
       The minimax robust time-frequency Wiener filter         is now
                                                                                                                                             5
                                                                                                                                             6I
                                                                                                                                             ¡
                                                                                                                                                                                                                but at its worst-case operating conditions         performs much
                                                                                                                                                                                                                                                                                                                                                                                                                               ¡     ¢
    defined as the linear, time-varying system whose Weyl symbol                                                                                                                                                 worse than       or even
                                                                                                                                                                                                                                                                            ¡   5 I
                                                                                                                                                                                                                                              . Hence, in this example, the robust
                                                                                                                                                                                                                                                                                                                                                ¡               H
    minimizes a time-frequency expression of the mean-square error                                                                                                                                              time-frequency Wiener filter        achieves a drastic performance
                                                                                                                                                                                                                                                                                                                                                                         5
                                                                                                                                                                                                                                                                                                                                                                         6I
                                                                                                                                                                                                                                                                                                                                                                         ¡


    for the worst-case choice of                   and
                                                                  £0I@ ¤ 0g g ¤ f
                                                                  ¨ ¦4 q
                                                                                             ”
                                                                                                         !                              ¨ ¦ 4 q gg¦
                                                                                                                                        £0I@ ¤ 0ahf
                                                                                                                                                                                                    ”           improvement over          at worst-case operating conditions with
                                                                                                                                                                                                                                                                                                               ¡               ¢
4       [55]. The Weyl symbol of this robust time-frequency Wiener                                                                                                                                              only a slight performance loss at nominal operating conditions.

                                                                                                                                                                                                            6
     1                                                    1                                                                                                                                                                                             at nominal correlations
                                                                                                                                                                                30                                            ¡ ¢                       at worst-case correlations
                                                                                                                                                                                                                             ¡ £                       at any         ,                    5         ¨¦"
                                                                                                                                                                                                                                                                                                     § ¥                      5           ©'
                                                                                                                                                                                                                                                                                                                                          ¥
                                                                                                                                                                                25                                          ¤¢                                                                      ' "




                                                                                                                                                       output SNR [dB]
    0.5                                                 0.5                                                                                                                                                                     
                                                                                                                                                                                                                                                      at any     ,                    5                        5

                                                                                                                                                                                20


     0                                                    0                                                                                                                     15

                                                                                                                                                                               10
           t                                f                         t                                           f
                                                                                                                                                                                    5
                                 (a)                                                 (b)
                                                                                                                                                                                    0
                                        R                                                  R

                                                                                                                                                                                −5
                                                                                                                                                                                         −30                                  −20                                                  −10                                        0                                 10                           20                       30
                                                                                                                                                                                                                                                                                                                   input SNR [dB]
                                                                                                                 
                                                                                                                              Figure 5. Performance of the ordinary Wiener filter      , the ro-
                                                                                                                                                                                                                                                                                                                                                                                                                          ¡     ¢
                                                                                                                              bust time-frequency Wiener filter    , and the trivial filter     .
                                                                                                                                                                                                                                                                                                                                              5
                                                                                                                                                                                                                                                                                                                                              6I
                                                                                                                                                                                                                                                                                                                                              ¡                                                                                                     ¡         H
                                 (c)                                                 (d)                                      The input SNR was varied by scaling     .
                                                                                                                                                                                                                                                                                                                                                                       ¤
     1                                                    1
                                                                                                                              Ville spectrum as time-dependent power spectrum, while keep-
                                                                                                                              ing in mind that, in the underspread case considered, the gener-
    0.5                                                 0.5                                                                   alized evolutionary spectrum can approximately be substituted
                                                                                                                              in the relevant equations.
                                                                                                                                 We consider the detection of a nonstationary, Gaussian ran-                           ¨@
                                                                                                                                                                                                                       F6¤                                                                                                                                                                                                                                  ¨@
                                                                                                                                                                                                                                                                                                                                                                                                                                                            F6¤
     0                                                    0                                                                   dom signal        from a noise-contaminated observation        .                                                                                                                                                                                                                                                  9
                                                                                                                              The hypotheses are                                                                                              P
           t                                f                         t                                           f                                                                                                                           P                                $                    ¨
                                                                                                                                                                                                                                                                                                © F@ ¤                                    ¢                 F6¤
                                                                                                                                                                                                                                                                                                                                                            ¨@
                                                                                                                                                                                                                                                           g                                           9
                                 (e)                                                 (f)                                                                                                                                                                       d
                                                                                                                                                                                                                                                                               $               © F@ ¤
                                                                                                                                                                                                                                                                                               9
                                                                                                                                                                                                                                                                                                  ¨
                                                                                                                                                                                                                                                                                                                                                            ¨@
                                                                                                                                                                                                                                                                                                                                                          AUF6¤             ¢           4 ¨@
                                                                                                                                                                                                                                                                                                                                                                                        £F6¤
                                                         1
                                                                                                                                                                                         ¨@
                                                                                                                                                                                         F6¤
     1                                                  0.8                                                                   where   ¨@
                                                                                                                                      F6¤is nonstationary, Gaussian noise uncorrelated with
                                                                                                                                                                                ¢

                                                                                                                                  . The optimal detector (likelihood ratio detector) [42–44]                                                                                                                                                                                                                ¨@
                                                                                                                                                                                                                                                                                                                                                                                                            F6¤
                                                        0.6
                                                                                                                              forms a quadratic form of the observed signal    ,                                                                                                                                                                                                                    9
    0.5                                                 0.4                                                                                                                                                                                                                                                                                                                                                       3 3
                                                        0.2                                                                                            ¨ ¤
                                                                                                                                                          9                              !          
                                                                                                                                                                                         "9 4 9  ¡ ©                                                                     ©                                                ¨ ‘ 6¤ ¨ ‘ (I6¤  S
                                                                                                                                                                                                                                                                                                                                  @
                                                                                                                                                                                                                                                                                                                                     9
                                                                                                                                                                                                                                                                                                                                         @4@
                                                                                                                                                                                                                                                                                                                                                                                              D 9
                                                                                                                                                                                                                                                                                                                                                                                                            4 ‘ @ @ F6¤
                                                                                                                                                                                                                                                                                                                                                                                                                      ¨@
                                                                                                                                                                                                                                                                                                                                                                                                                                                        (15)
                                                                                                                                                                                                                                                                                                        £               ¤ £
     0                                                   0


                                                                                                                              with the operator
                                                                                                                                                                                                                                                  ¡                       given by
           t                                f                         t                                           f
                                 (g)                                                 (h)
                                                                                                                                  ¡                                    ©                     #
                                                                                                                                                                                               h¦
                                                                                                                                                                                               d
                                                                                                                                                                                                                   ‚
                                                                                                                                                                                                                                  ¤                   ¤               A                          # &¦
                                                                                                                                                                                                                                                                                                     ¨          d
                                                                                                                                                                                                                                                                                                                e                 ©                        #
                                                                                                                                                                                                                                                                                                                                                             e¦
                                                                                                                                                                                                                                                                                                                                                              d                     ¤ ¤¤               A                    4 e# &¦
                                                                                                                                                                                                                                                                                                                                                                                                                                d  ¨
                                                                                                                                                                                                                                                                                                                                                                                                                                                        (16)
Figure 4. Time-frequency representations of signal and noise
                                                                                                                              or
                                                                                                                                               ¡
                                                                                                                                                 , where
                                                                                                                                                                           ©            
                                                                                                                                                                                  is the time-
                                                                                                                                                                                                   #
                                                                                                                                                                                                   ¡ h¦
                                                                                                                                                                                                     d                                ¢                                                                     ¡       ¢                 ©
                                                                                                                                                                                                                                                                                                                                      
                                                                                                                                                                                                                                                                                                                                      #                     ¤ ¤¤                   A                    ¨
                                                                                                                                                                                                                                                                                                                                                                                                        # "¦  d
                                                                                                                                                                                                                                                                                                                                                                                                              e
                                                                                                                                                                                                                                                                                                                                                                                                                                           P
statistics and of various Wiener-type filters: (a) Wigner-Ville
                                 F@ ¤
                                 ¨                                                                     F6¤
                                                                                                       ¨@                     varying Wiener filter considered in Section IV. The test statistic
                                                                                                                              ¨ ¤ P
spectrum of       , (b) Wigner-Ville spectrum of
                                                        , (c) mag-                             ¢
                                                                               F6¤
                                                                               ¨@                                               9    is then compared to a threshold to decide whether       or                                                                                                                                                                                                                                                         g
nitude of expected ambiguity function of       , (d) magnitude of                                                                                                                                                                                                                                                                                                                      ¨ ¤
                                                          ¨
                                                          F@ ¤                                                                    is in force. For stationary processes,
                                                                                                                                       d                                     can be expressed                                                                                                                                                                                             9
expected ambiguity function of       , (e) Weyl symbol of Wiener
                                                    ¢

filter
           ¡   ¢
           , (f) Weyl symbol of underspread part           of     ,
                                                                                                   ¡          ¢ £ ¢   ¡       in the frequency domain as
                                                                                                                                                                                                                                                                                                                                                                                                                      3
(g) Weyl symbol of time-frequency pseudo-Wiener filter             ,
                                                                                                             ¢ I      ¡
                                                                                                                                                                                   © ¨ ¤                                                           4 ¦ ' ¨£§¤& 
                                                                                                                                                                                                                                                                ¦
                                                                                                                                                                                                                                                                                    £Q¤ §¡
                                                                                                                                                                                                                                                                                   ¨ ¦ ¤
(h) Weyl symbol of robust time-frequency Wiener filter         . The
                                                                                                              ¡  6I
                                                                                                                 5                                                                       9                                                                             $%¨£§¤£¦e¡ AU£Q¤ ¤ ¡ ¨£§£e¡
                                                                                                                                                                                                                                                                           ¦         ¨ ¦ # ¨ ¦¤ ¦                                                                                                                                                       (17)
                                                                                                                                                                                                                              1
rectangles in parts (c) and (d) have area 1 and thus permit to       ¨@
                                                                     F6¤              ¨@
                                                                                      F6¤
assess the underspread property of       and       . The processes
                                                                                 ¢                                                                                              £§&
                                                                                                                                                                                ¨ ¦¤                                                                                                                                                                                                                                                    F6¤
                                                                                                                                                                                                                                                                                                                                                                                                                                        ¨@
     F6¤
     ¨@                F6¤
                       ¨@                                                                                                     where
                                                                                                                               £Q¤ §¡
                                                                                                                              ¨ ¦ ¤         is the Fourier transform of the observation    and         ¡       ¦            ¨ ¦
                                                                                                                                                                                                                            £Q¤                                                                                                                                                                                                         F6¤ 9
                                                                                                                                                                                                                                                                                                                                                                                                                                        ¨@
     and       were generated using the time-frequency synthesis
                   ¢
                                                                                                                                     and           are the power spectral densities of     and                                                                                                                                                                                                                                       
technique introduced in [36]. The signal length is 128 samples.
                                                                                                                                 ¨@
                                                                                                                                 F6¤ ¢
                                                                                                                                  , respectively. This frequency-domain expression involves
                                                                                                                              simple products and reciprocals of functions (instead of prod-
                       V. N ONSTATIONARY S IGNAL D ETECTION                                                                   ucts and inverses of operators as in (16)) and hence allows a
                                                                                                                              simple interpretation and design of optimal detectors in the sta-
   Next, we discuss time-frequency methods for nonstationary                                                                  tionary case.
signal detection. We shall again use the generalized Wigner-
                                                                                                                          7
                                                                                                                                                                                                                                                                                                                                                                          3 3
A. Time-Frequency Formulation of Nonstationary Detectors                                                                                                       R               £¦ I6¤ D $pg ¢ f £¦ I@ ¤ 2pg ¤ f
                                                                                                                                                                                ¨ 4@ q i         ¨ 4 q i
                                                                                                                                                                                                                            © ¨ ¤                                                                                                                                      ¦ @        )
                                                                                                                                                                                                                                                                                                                                                                                    c
                                                                                                                                                                    ¤ ¨£¦ I6¤ q$pg ¦ f e£¦ I6¤ $pg ¤ f £ £¦ I@ ¤ 2tg ¦ f 1 £
                                                                                                                                                                          4@ i           A ¨ 4@ q i          ¨ 4 q i              9
  It is known [58–60] that the quadratic test statistic in (15) can
be rewritten as an inner product in the time-frequency domain,                                                                                                                                                                                                                                           ¨@
                                                                                                                                                                                                                                                                                                         F6¤                 ¨@
                                                                                                                                                                                                                                                                                                                             F6¤
                                                                                      3 3                                                               For jointly underspread processes           ,      where (20) is                                                                                             ¢

                © ¨ ¤         q i ¢ f 4 2pg % $ £
                                         q¡              ©
                                                                                    ¦ @ £0I@ ¤ $pg ¢ f £HI6¤ $pg %
                                                                                          ¨ ¦4 q i      ¨ ¦ 4 @ q¢                                      a good approximation, combination of (20) and (21) yields
                     9       ¥ $pg         i                                $
                                                                                                D                 i                           c
                                                                                                                                              )
                                                                                                                                                        $    £0I@ ¤ 2tPi g
                                                                                                                                                              ¨ ¦4 q
                                                                                                                                                                                 . Thus,
                                                                                                                                                                                                            $           ¨ ¦ 4 q£
                                                                                                                                                                                              is a close approximation to
                                                                                                                                                                                                                         £0I@ ¤ $pg %
                                                                                                                                                                                                                                 ¢ i                                           I¡
                                                              £     1                                                                                        
                                                                                                                                                            ¢%                                                                                                                                                                        ¡          
                                                                                                                                             (18)       (the underspread part of) the optimal operator ; furthermore,
Here,                                                                                                                                                    I¡will then be nearly independent of the value of used in                                                                                                                                          „
¨ ¦4@ q i
£HI6¤ $pg ¢ f                                                                                                                                           (21). Hence, for jointly underspread processes the performance
                                                                                                                        3                                                                                                                                                                                     R   ¨ ¤
                                                                                                                                                        of the time-frequency designed detector          will be similar to                                             ¨ ¤                                          9
     A ex "w
       €    @    ‰       € A @    
                          ex £w  fr                                                               20(&# ! ‰ 
                                                                                                    1 )'%                                
                                                                                                                                                        that of the optimal detector         and approximately indepen-
          ‚ D 9     † …‚ 
                       „         9                                                            † „                                            (19)                                                                                                                          9
                                                                                                                                                        dent of . For processes that are not jointly underspread, how-
                                                                                                                                                                         R            ¨ „¤
is the generalized Wigner distribution [10–12] of the observed                                                                                          ever,        must be expected to perform poorly.
                                                                                                                                                                                         9                                                   R       ¨ ¤
signal     . Thus,
                      ¨ F@ ¤
                        can be interpreted as a weighted integral
                             9
                                                 ¨ ¤
                                                    9                                                                                                      While the detector       is designed in the time-frequency do-                               9
                 £0I6¤ D $pg ¢ f
                 ¨ ¦4@ q i                                                                                                                              main, it can be implemented directly in the time domain accord-
of             , where the time-frequency weighting function is
                                                                                                    ¡                                                  ing to (cf. (15))
the generalized Weyl symbol of the operator .                                                                                                                                                                                                                                                                                                              3 3
   In analogy to Subsection IV-A, a simplified approximate                      ¨ ¤
                                                                                                                                                                    R             © ¨ ¤                         I¡                 4                ©                        ¨ ‘ 6¤ ¨ ‘ FI6¤  R S
                                                                                                                                                                                                                                                                                      @      @4@                                                     4 ‘ @ @ F@ ¤
                                                                                                                                                                                                                                                                                                                                                               ¨
time-frequency formulation of          exists for jointly under-                  9                                                                                                    9                                             9 9          !          £           ¤
                                                                                                                                                                                                                                                                         ¥£
                                                                                                                                                                                                                                                                                         9                                                D 9
spread processes       and      . Here, the operator
                                                 ¨@
                                                 F6¤     ¢   F@ ¤
                                                             ¨
                                                           can be
                                                                                                                    ¡        
                                                                                                                                                                                         pB(I6¤  R S
                                                                                                                                                                                         ¨‘ @ 4 @                                                                                                                    I¡
decomposed into an overspread component whose effect is neg-                                                                                            where                                                           , the impulse response of    , can be obtained
ligible and an underspread component, denoted as         , whose
                                                                                                                 
                                                                                                                A
                                                                                                                ¡
                                                                                                                                                        from
                                                                                                                                                                              $        ¨£HI6¤ $pPi g
                                                                                                                                                                                        
                                                                                                                                                                                       ¢%
                                                                                                                                                                                          ¦4@ q
                                                                                                                                                                                                                          by an inverse Weyl transformation (cf. (14)).
generalized Weyl symbol can be approximated as [60]
                                                                                                                                                        Efficient implementations of the time-frequency detector
                                                                                                                                                                                                                                                                                                                                                                               R   ¨ ¤
                                                                                                                                                                                                                                                                                                                                                                                      9
                                                         £0I6¤ $pg ¤ f
                                                        ¨ ¦4@ q i                                                                                       that are based on the multiwindow short-time Fourier transform
                   £HI6¤ 2pg %
                    ¨ ¦ 4 @ qQ
     $
                              ¢ i      ¥¨£¦HI6¤ q$pghf AU£0I@ ¤ 2phf £ £0I6¤ $pihf
                                       ¤ 4@ i ¦           ¨ ¦ 4 q ig¤ ¨ ¦ 4 @ q g¦
                                                                                                                                     )
                                                                                                                                             (20)       or the multiwindow spectrogram are discussed in [61].                                                                                                ¨ ¤
                                                                                                                                                           Compared to the optimal detector         , the time-frequency             R    ¨ ¤                                                                  9
                                                                                                                                                        designed detector        is practically advantageous because the                     9
Inserting (20) in (18), we obtain the following approximate                                                                                             statistical a priori knowledge used in its design is formulated
time-frequency formulation of our test statistic,                                                                                                       in the intuitively accessible time-frequency domain, and be-
                                                                                                                                  3 3                   cause its design is computationally less intensive and numer-
                                                ¨ 4 q ig¤
                               £¦ I6¤ D $pg ¢ f £¦ I@ ¤ 2phf
                               ¨ 4@ q i
                                                                    f ¨ ¤
                                                                                                                                ¦ @         c
                                                                                                                                             )          ically more stable (since operator inversions are replaced by
                   ¤ ¨£¦ I6¤ q$phf e£¦ I6¤ $phf £ £¦ I@ ¤ 2thf 1 £
                         4 @ ig¦        A ¨ 4 @ q ig¤     ¨ 4 q ig¦        9                                                                            pointwise divisions of functions). These advantages are anal-
                                                                                                                                                        ogous to the advantages of the time-frequency pseudo-Wiener
                                                                                                                                                        filter discussed in Subsection IV-B.
This extends the frequency-domain expression (17) to the non-
                                                                                        ˜ ©                         ¨ ¦4 q
                                                                                                                    £0I@ ¤ 0g g f
stationary (underspread) case. For          (note that                           „                                       ¥                              C. Simulation Results
                   £0I6¤ Hg g ¢ f
                   ¨ ¦4@ q
and               are real-valued), the above approximation al-
lows a simple and intuitively appealing time-frequency inter-                                                                                              Fig. 6 compares the performance of the optimal likelihood
                                                                                                                                                                                                                       ¨ ¤
pretation that is analogous to the one given in Subsection IV-A                                                                                         ratio detector        with that of the time-frequency designed de-
                                                                                                                                                                                  R        ¨ ¤                           9
in the context of the approximation (12). In essence, the test                                                                                          tector       for jointly underspread signal and noise processes.
                                                                                                                                                                                              9
statistic
                            ¨ ¤
               picks up energy of the observation
                               9                         in time-                                           9
                                                                                                                ¨@
                                                                                                                F6¤                                     It is seen that the time-frequency designed detector closely ap-
frequency regions where there is large mean signal energy but                                                                                           proximates the optimal detector.
little mean noise energy, and tends to reject observation compo-                                                                                           In the previous example, the noise contained a strong white                                                                              ¦                           ¤        A            ¦
nents in time-frequency regions where there is little mean signal                                                                                       component and hence the operators          and           (that have                                          ¡        
energy and large mean noise energy.                                                                                                                     to be inverted for calculating        according to (16)) were non-
                                                                                                                                                        singular. In practice, this need not be the case. Our next exam-                                                                                                                                                ¨ ¤
                                                                                                                                                        ple considers the application of the optimal detector           and                                                                              R                                                                9
B. Time-Frequency Design of Nonstationary Detectors                                                                                                     the time-frequency designed detector            to the detection of
                                                                                                                                                                                                                                                                                                               ¨ ¤
                                                                                                                                                                                                                                                                                                                 9
                                                                                                                                                        knocking combustions in car engines (see [61] for background
   The time-frequency formulation (20) suggests a simple time-                                                                                                                                      P
                                                                                                                                                        and details). The hypotheses are
frequency design of nonstationary detectors. In analogy to Sub-
section IV-B, we define the system     by setting its generalized
                                                                                  I¡                                                                                                               P                        $             © ¨@
                                                                                                                                                                                                                                           F6¤        © F6¤
                                                                                                                                                                                                                                                          ¨@                                               A ¨
                                                                                                                                                                                                                                                                                                           UF@ ¤              ¢                   F6¤
                                                                                                                                                                                                                                                                                                                                                  ¨@
                                                                                                                                                                                                                g                                g 9            9                                                  g
Weyl symbol equal to the right-hand side of (20) [60]:                                                                                                                                                                       $               ¨@
                                                                                                                                                                                                                                          ©F6¤ d © F6¤   ¨@                                              AUF@ ¤ d
                                                                                                                                                                                                                                                                                                               ¨              ¢                4E¨F6¤@
                                                            ¨ ¦ 4 q ig¤
                                                            £0I@ ¤ 2phf
                                                                                                                                                                                                                    d                                9        9                                   
         $         £0I6¤ $pPi g
                  r ¨ ¦4@ q                                                                                                      )                                                           ¨
                                                                                                                                                                                             F@ ¤                                                            ¨@
                                                                                                                                                                                                                                                             F6¤
                            ¡%
                                       ¤ ¨£¦0I6¤ q2ipg ¦ f AU£0I@ ¤ 2tg ¤ f £ £0I@ ¤ 2tg ¦ f
                                             4@                ¨ ¦4 q i       ¨ ¦4 q i                                                       (21)       where      and       denote respectively the non-knocking and
                                                                                                                                                                                        g                                        d
                                                                                                                                                                                                                                                       ¨@
                                                                                                                                                                                                                                                       F6¤ ¢
                                                                                                                                                        knocking signal and      is stationary white Gaussian noise.
                                                                                                                                                        The format of these hypotheses is different from that of our                                                                                                                                                    ¨ ¤
Inserting in (18) yields the time-frequency designed test statistic                                                                                     previous hypotheses; however, the optimal detector        and                                                                                                                                                      9

                                                                                                                                                    8
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                                                                                                                                                                                                                             0.1        0.2             0.4     0.6       0.8   1

                                                                                                                                                                                     q g ¥                                                     § ¦

Figure 6. Comparison of optimal detector           and time-frequency designed detector
                                                                                                      ¨ ¤
                                                                                              : (a) Wigner-Ville spectrum of      , (b)
                                                                                                                                                                                               R    ¨ ¤                                                                                ¨@
                                                                                                                                                                                                                                                                                        F6¤
                                                         F6¤
                                                         ¨@
                                                                                                         9
                                                                                                                                                   ¡                                                  9                                                                                        I¡
Wigner-Ville spectrum of      , (c) Weyl symbol of optimal operator , (d) Weyl symbol of time-frequency designed operator ,
                                                     ¢
                                                                                                                                                                      ¨ ¤
(e) conditional probability density functions (pdf’s) of optimal test statistic
                                                                              R     under either hypothesis, (f) conditional pdf’s of
                                                                                             ¨ ¤                                                                         9                                                                                                 ¨ ¤
time-frequency designed test statistic      , (g) receiver operator characteristics (ROC) [43,44] of optimal detector
                                                                                         ¨ ¤ 9R                             , and (h)                                                                                                                                        9
ROC of time-frequency designed detector          . The performance results in (e)–(h) were obtained by Monte Carlo simulation. The
                                                                                           9
signal length is 128 samples.
                                                                                   R     ¨ ¤
the time-frequency designed detector          can be extended to                            9
this more general type of hypotheses [61]. In this example,                                                                                                                                                 (a)                                                                     (b)
the second-order statistics (correlations or Wigner-Ville spec-                                                                                         20                                                                         20
tra) were estimated from a set of labeled training data, the opti-
                                                                                                                                                        15                                                                         15
mal and time-frequency designed detectors were constructed us-
ing these estimated second-order statistics, and the performance                                                                                        10                                                                         10
of the detectors was evaluated by applying them to a different
set of labeled data. The calculation of the optimal detector was                                                                                              5                                                                     5
made difficult by the poor conditioning of the estimated correla-
                                             ¨@
                                             F6¤                       ¨
                                                                       F@ ¤                                                                                   0                                                                     0
tion operators of        and       (hence, pseudo-inverses were
                                       g 9               9     d                                                                                                        30                          60               90                           30                      60                  90
used for implementing the necessary inversions). In contrast,
                                  R   ¨ ¤
the design of          using the estimated Wigner-Ville spectra
                                         9                                                                                                                                                                  (c)                                                                     (d)
merely involves divisions of functions that are easily stabilized.                                                                                     20                                                                        20
                                                                                                                          ¨@
                                                                                                                          F6¤
Fig. 7 shows the estimated Wigner-Ville spectra of            and                                                        g 9
        ¨@
        F6¤                                                                                                                                            15                                                                        15
9   d  as well as the resulting time-frequency weighting func-
              $   ¨ ¦ 4 @ q¢
                   £HI6¤ Hg g %                                                           ¨ ¤               $   ¨£¦H4I@6¤ Hg P g
                                                                                                                           q  %
tions            for the optimal detector
                                               and             for                          9                                                          10                                                                        10
                                                                                   R         ¨ ¤
the time-frequency designed detector            (see (18)). The re-                             9                                                         5                                                                        5
ceiver operating characteristics of the two detectors are com-
pared in Fig. 8. It is seen that, due to its more stable design, the                                                                                      0                                                                        0
                                                                                                                                                                       30                         60                90                            30                      60                 90
time-frequency designed detector performs significantly better
than the theoretically optimal detector.                                                                                                          Figure 7. Estimated Wigner-Ville spectra of the observed sig-
                                                                                                                                                  nal and time-frequency weighting functions of the two detec-                                                        ¨@
                                                                                                                                                                                                                                                                      F6¤
                                                   VI. C ONCLUSION                                                                                tors: (a) Estimated Wigner-Ville spectrum of           calculated                                           g 9
                                                                                                                                                  from non-knocking training data, (b) estimated Wigner-Ville
                                                                                                                                                                                      ¨@
                                                                                                                                                                                      F6¤
   We have shown that the time-frequency domain allows an ex-                                                                                     spectrum of         calculated from knocking training data, (c)
                                                                                                                                                                        9        d
                                                                                                                                                                                                                                                                                              ¨ ¤
tension of the spectral representation of random processes and                                                                                    time-frequency weighting function of the optimal detector       ,                                                                              9
the frequency-domain formulation of statistical signal process-                                                                                   (d) time-frequency weighting function of the time-frequency de-
                                                                                                                                                                                      R    ¨ ¤
ing techniques to the nonstationary case. However, it is im-                                                                                      signed detector       . Horizontal axis: crank angle (in degrees)
                                                                                                                                                                                              9
portant to be aware that this extension is possible only if the                                                                                   which is proportional to time, vertical axis: frequency (in kHz).
processes involved are underspread. In this paper, we have                                                                                        The signal length is 186 samples.
emphasized techniques for signal estimation and signal detec-

                                                                                                                                          9
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