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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@ ¤ $pg ¢ 4 " e (4) This situation is somewhat analogous (and, as we will see presently, closely related) to the frequency-domain analysis of © ¨@ 4 @¤ ¢ tBFI6¥ ¨ F@ ¤ G ¨@ HtB6¤ where E 9 8 . The parameter E9 D is ar- linear systems. As long as the system is time-invariant, the fre- bitrary a priori. Special cases are the Wigner-Ville spectrum quency response P 3 £ © © ( ) [3–8] and the Rihaczek spectrum ( f© ) [4,5,9]. d e © ¨ ¦ R£Q¤ 20(&$" 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 £HI6¤ Ftg ¢ ¨ ¦4@ q i £HI6¤ 0pg ¢ ) 0 GES GES have been discussed in [32–35]). This underspread property is ' 3£0I@ ¤ 2pg satisﬁed by many nonstationary processes occurring in practice. © £HI6¤ $pg ¢ ¨ ¦4@ q i $1 1 ¨ ¦ 4 q i% ¨ ¦4@ q i £HI6¤ $pg ¢ f © With GES and 1 1 We emphasize that underspread should not be confused with $ ¨ ¦ 4 @ qI ¨ ¦ 4 q' i £0I6¤ !2tg % $ © £0I@ ¤ 2tg & i & & ¤ ¢ ¡ ¨ 4 V R % , and using an underspread innova- quasistationarity which only means a limitation of R ¡ 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 £0I@ ¤ 2pg ¢ ¨ ¦ 4 @ q ig¢ £HI6¤ $phf ) 0 & ¤ ¢ ¡ ¨ 4 Q V R GES (extension of with respect to ) is too long, whereas a £0I@ ¤ 2tg ¢ ¨ ¦4 q i 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 ¢ ¡ ¨ R4 q i 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 ¦ ¦ ¦ ¦ R @ @ @ @ (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. ¦ ¦ ¦ ¦ R @ @ @ @ (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 ¡ © ¨@ F6¤ ¤ ¡ © F6¤ ¨ ¨@ @ ¨ @ ¤ ¨ FI@ US @4 ¤ ) process whose expected generalized ambiguity function is not 9 ¤ £ 9 ¨ R4 &V6¤ 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- £ ¢ ¡ © ¤ ¤ ¦¤ ¥ A ¦ 4 e# ¨ d (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 ¨ ¦¤ ¤ £§e§¡ 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¤ ¨@ © ¨@ F6¤ A F6¤ ¨@ F6¤ ¨@ We may ask whether a similarly simple formulation as in the ary random signal from an observation 9 , ¢ stationary case can be obtained for the time-varying Wiener ﬁl- 4 ter £ ¢ ¡ by introducing in (11) an explicit time dependence, i.e., P 6 " ) A92Gb G 76 a`Y ¤2¢ 45 3 6 @ 8 9 @ ¢ £Q¤ ¨ ¦ 04 £§¤ ¤ ¡ ¡ ¨ ¦ e ¦ ¨ ¦ £Q¤ by substituting for and for suitably de- ﬁned time-dependent frequency responses and time-dependent F@ ¤ ¨ ' ) &0" ) power spectra. Indeed, for jointly underspread3 processes F6¤ ¨@ and it can be shown [47,48] that the time-varying Wiener ¢ ' ) 2" ) 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 @ system’s performance (mean-square error) and can hence be dis- ` ` regarded. (a) (b) The underspread part, denoted as in what follows, allows ¢¡ ¡ Figure 3. Time-frequency interpretation of the time-varying ¢ £ ¡ the approximate time-frequency formulation Wiener ﬁlter for jointly underspread signal and noise pro- cesses: (a) Effective time-frequency support regions of signal £0I@ ¤ $pg ¤ f ¨ ¦4 q i and noise, (b) time-frequency pass, stop, and transition regions $ £0I@ ¤ $pg % ¨ ¦ 4 q ¤£ ¢i 4 4 g¦ ¨ ¦ 4 @ q ig¤ ¨£¦0I@ ¤ q2ithf A £HI6¤ $phf (12) of the time-varying Wiener ﬁlter. ¨ ¦ 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 ¢ £I ¡ © its generalized Weyl symbol equal to the right-hand side of (12) nonstationary (underspread) case. For (recall that ¨ ¦4@ q £HI6¤ Hg g f [47,48]: is real-valued), (12) allows a simple and intuitively ¥ appealing time-frequency interpretation of (the underspread ¨ ¦ 4 @ q ig¤ £HI6¤ $phf $ r ¨ ¦ 4 q¤ £0I@ ¤ $pPi g ) component of) the time-varying Wiener ﬁlter (see Fig. 3). Let % ¨£¦0I@ ¤ q2itg ¦ f A £HI6¤ $pg ¤ f 4 ¨ ¦4@ q i (13) ¤ ¦ and ¦ denote the effective support regions of ¦ £0I@ ¤ 0g g ¤ f ¨ ¦4 q ¨ ¦4@ q £0I6¤ Hg g ¦ f ¨@ F6¤ ¨@ F6¤ 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¤ £0I6¤ 2tPi g $ ¨ ¦ 4 @ q ¤Q £HI6¤ 2pg % ¢i % , and hence the time-frequency Pass region. In the “signal only” time-frequency region pseudo-Wiener ﬁlter is a close approximation to (the un- ¢ £I ¡ ¦ ¨¤ ¦ § ¦ , i.e., in the time-frequency region where only signal derspread part of) the optimal Wiener ﬁlter ; furthermore, ¡ ¢ $ £HI6¤ ¤0ag % ¨ ¦ 4 @ q£ ¢g ¡ ¢¡ ¢ £I ¡ will then be nearly independent of the value of used in energy is present, there is . Thus, passes all “noise-free” observation components without attenuation or (13). For processes that are not jointly underspread, however, ¡ ¢ I must be expected to perform poorly. distortion. ¢ £I ¡ While the time-frequency pseudo-Wiener ﬁlter is de- Stop region. In the “noise only” time-frequency region signed in the time-frequency domain, the actual calculation of ¤ ¦§ ¦ ¦ where only noise energy is present, there is $ £0I@ ¤ ¤Hg g % ¨ ¦ 4 q£ ¢ the signal estimate can be performed directly in the time domain , i.e., ¢¡ ¡ suppresses all observation components in time- according to 3 frequency regions where there is no signal. ¡ ¨ © F@ ¤ £I ¢ ¡ ¨@ © F6¤ ¢ RS 4 @ ¨ 6¤ ¨ FI6¤ @ @4@ ` 9 b 9 Transition region. In the “signal plus noise” time-frequency ¦ ¤ ¦ © ¦ ¤ ¥£ region where both signal energy and noise energy pB(I6¤ ¢ R S ¨ @ 4 @ ¢ £I ¡ $ ¨ ¦ 4 @ q£ £HI6¤ ¤0ag % ¢g where , the impulse response of , can be obtained are present, assumes values approximately between $ ¦ 4 @ q¤ ¨£HI6¤ $pPi g and . Here, performs a time-frequency varying atten- ¡ ¢ from % as [15–17] uation that depends on the relative signal and noise energy at ¢ SI ¨ FI@ ¤ @4 the respective time-frequency point. In particular, for equal £0I6¤ ¨ ¦4@ 3 signal and noise energy, i.e., time-frequency points with © $ ¦ q ¤ £ # £ g $((% ! 04 @ 1 )' ¦ x A @ A x w $pPi g q¤ % ) £HI6¤ Hg g ¦ f T£HI6¤ Hg g ¤ f ¨ ¦4@ q © ¨ ¦4@ q £HI6¤ ¤0ag % ¨ ¦ 4 @ q£ (14) , there is $ ¢g d . 1 An efﬁcient implementation of the time-frequency pseudo- Wiener ﬁlter ¡ ¢ I that is based on the multiwindow short-time The processes and b pI are said to be jointly underspread if their ex- `Y b a¥ `Y pected generalized ambiguity functions, and , are con- ! " b % # e&9 $Y ! ' b% # e&9 $Y Fourier transform is discussed in [48,49]. ¡ ¢ 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. ¤ ¦ ¢ £I ¡ © ¨ ¦4 q £0I@ ¤ 0g P g 8 ¤ 4 4E¨£¦HI@6¤ (& (10)), whereas the design of requires knowledge of the gen- $ %% 7 ¤ ¦¤ A @¤ 9 £HI6¤ $pg ¤ f ¨ ¦4@ q i ¨ ¦4@ q i £HI6¤ $pg ¦ f ¢ eralized Wigner-Ville spectra and (cf. %& ( d ( %& (13)). Although correlation operators and generalized Wigner- ¨£¦HI@6¤ 4 ¤ ¦ 4 ©T¨£¦HI@6¤ Ville spectra are mathematically equivalent due to the one-to- where is the indicator function of9 (i.e., 9 for ¨ ¦4@ £HI6¤ and 0 otherwise). Note that is ¤ ¦ $ £0I6¤ Hg P g ¨ ¦4@ q 7% 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- ¤ A¦ 8 G 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 . ' ¢ ¡ 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 ¢ £I ¡ C. Robust Variations ¨ F@ ¤ ¨ F@ ¤ 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 0.12 1 1 (a) 1.5 (c) (e) (g) p 0.09 © ¦ 0.98 1 d 0.5 f 0.5 0.06 0.96 0.03 0 0 ¢ £ £ ¤ £ 1 ¡ £ ¢ £ £ ¤ £ 1 ¡ £ 0 0 10 20 ¢ 30 40 0.94 0.1 0.2 0.4 0.6 0.8 1 q g ¥ § ¨¦ 0.12 1 1.5 (b) 1.5 (d) (f) (h) p 0.09 © ¦ 0.98 1 1 d f 0.5 0.5 0.06 0.96 0.03 0 0 ¢ £ £ ¤ £ 1 ¡ £ ¢ £ £ ¤ £ 1 ¡ £ 0 0 10 20 ¢ 30 40 0.94 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 1 [11] A. J. E. M. 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