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Carrier Phase and Frequency Estimation for Pilot-Symbol Assisted

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					> T-SP-01823-2003 <                                                                                                                     1




            Carrier Phase and Frequency Estimation for
            Pilot-Symbol Assisted Transmission: Bounds
                          and Algorithms
           N. Noels, Student Member, H. Steendam, Member, M. Moeneclaey, Fellow IEEE, and H. Bruneel


                                                                           known pilot symbols (PS), while Non-Data-Aided (NDA) and
   Abstract—In this contribution we consider the Cramer-Rao                Decision-Directed (DD) estimators operate on modulated data
lower Bound (CRB) for the joint estimation of the carrier phase            symbols (DS). DD estimators are similar to DA estimators,
and the frequency offset from a noisy linearly modulated burst             but use, instead of PS, hard or soft decisions regarding the DS
signal containing random data symbols (DS) as well as known
                                                                           that are provided by the detector; NDA estimators apply a
pilot-symbols (PS). We point out that the CRB depends on the
location of the PS in the burst, the number of PS, the number of           non-linearity to the received signal to remove the data
DS, the signal-to-noise ratio (SNR) and the data modulation                modulation.
scheme. Distributing the PS symmetrically about the center of                 Assuming that the parameter estimate is unbiased, the
the burst and estimating the carrier phase in the center of the            variance of the estimation error is often used as a performance
burst interval decouples the frequency and phase estimation,               measure. The Cramer-Rao lower bound (CRB) is a
making the CRB for phase estimation independent of the specific
                                                                           fundamental lower bound on the variance of any unbiased
location of the PS. At low and moderate SNR, the CRBs for both
phase and frequency estimation decrease as the fraction of the PS          estimate [1], and is also known to be asymptotically
in the burst increases. In addition, the CRB for frequency                 achievable for a large enough number of observations, under
estimation decreases as the PS are separated with more DS.                 mild regularity conditions. The CRBPS(Np) for phase and/or
Numerical evaluation of the CRB indicates that the carrier phase           frequency estimation from Np known PS have been derived in
and frequency of a ‘hybrid’ burst (i.e., containing PS and DS)             [2] and [3]. The CRBDS(Nd) related to joint carrier phase and
can be estimated more accurately when exploiting both the
                                                                           frequency estimation from Nd random DS have been
presence of the DS and the a priori knowledge about the DS,
instead of using only the knowledge about the PS (and ignoring             addressed in [4-6]. In the latter case, the statistics of the
the DS), or considering all the received symbols (PS and DS) as            observation depend not only on the vector parameter to be
unknown (and ignoring the knowledge about the PS).                         estimated, but also on a nuisance vector parameter (i.e. the
Comparison of the CRB with the performance of existing carrier             unknown DS) we do not want to estimate. In order to avoid
synchronizers shows that the iterative soft-Decision-Directed              the computational complexity caused by the nuisance
(sDD) estimator with Data-Aided (DA) initialization performs               parameters, a modified CRB (MCRB) has been derived in [7]
very closely to the CRB, and provides a large improvement over
the classical Non-Data-Aided (NDA) estimator at lower SNR.
                                                                           and [8]. The MCRB is much easier to evaluate than the CRB,
                                                                           but is in general looser (i.e. lower) than the true CRB,
  Index Terms—Cramer-Rao Bound, Carrier Synchronization,                   especially at lower signal-to-noise ratio (SNR). In [9], the
Frequency estimation, Phase Estimation                                     high-SNR limit of the CRBDS(Nd) has been obtained
                                                                           analytically, and has been shown to coincide with the
                                                                           MCRBDS(Nd).
                         I. INTRODUCTION                                      Very often it may be beneficial for carrier synchronizers to

I  N burst digital transmission with coherent detection, the
   recovery of the carrier phase and the frequency offset is a
key aspect. We assume that phase coherence over successive
                                                                           utilize information on both PS and DS in the estimation
                                                                           process. In [10], it has been shown that a frequency estimator
                                                                           that utilizes both PS and DS may provide the combined
bursts cannot be maintained, so that the carrier phase and                 advantages of DA estimators and NDA estimators, and allow
frequency offset have to be recovered on a burst-by-burst                  more accurate synchronization at lower SNR. A similar
basis.                                                                     observation holds for DA estimators and DD estimators. The
   Most classical synchronizers belong to one of the following             proper operation of DD estimators requires an accurate
types: Data-Aided (DA) synchronization algorithms use                      initialization, which, at low SNR, can only be provided by a
                                                                           DA estimator using known PS. At the same time, exploiting
   Manuscript received November, 7, 2003. This work was supported by the   the DS guarantees a good performance at high SNR. Note that
Interuniversity attraction Poles Program P11/5 – Belgian Science Policy.   the PS also allow to resolve the ambiguity of the NDA and
   N. Noels, H. Steendam and M. Moeneclaey are with the Department of      DD phase estimates caused by the rotational symmetry of the
Telecommunications and Information Processing (TELIN) of the Ghent
University (UGent), Gent, B-9000 Belgium (phone: ++32-9-264-34-26; fax:
                                                                           constellation.
++32-9-264-42-95; e-mail: {nnoels,hs,mm}@ telin.UGent.be).                    In this contribution we derive the true CRBPS-DS(Np,Nd) for
> T-SP-01823-2003 <                                                                                                                                                   2

joint phase and frequency estimation from the observation of a     the Fisher Information Matrix (FIM) [1]. The (i,j)-th element
‘hybrid’ burst that contains Np pilot symbols as well as Nd        of J(u) is given by
data symbols. These CRBs can be viewed as a generalization                             ∂                   ∂                
of the CRBs derived in [2-6]. Numerical results are reported           J ij (u) = E r       ln( p (r; u))      ln( p (r; u))      (3)
for a QPSK constellation, indicating that ‘hybrid’ algorithms                          ∂u i
                                                                                                          ∂u j              
                                                                                                                             
that exploit both PS and DS (in some intelligent way) are          Note that J(u) is a symmetrical matrix. When the element
potentially more accurate to estimate the carrier phase and        Jij(u)=0, the parameters ui and uj are said to be decoupled. The
frequency from a hybrid burst than algorithms that only use        expectation Er[.] in (3) is with respect to p(r;u). The
the PS (and ignore the DS) or algorithms that use all burst        probability density p(r;u) of r, corresponding to a given value
symbols (PS and DS) but ignore the knowledge of the PS.            of u, is called the likelihood function of u, ln(p(r;u)) is the
Comparing the obtained CRBs to the performance of the              log-likelihood function of u. When the observation r depends
hybrid estimation algorithm from [10] it is concluded that         not only on the parameter u to be estimated but also on a
more efficient hybrid algorithms may exist that perform more       nuisance vector parameter v, the likelihood function of u is
closely to the CRBs. We show that the iterative soft-DD            obtained by averaging the likelihood function p(r|v;u) of the
(sDD) estimator with DA initialization yields a close              vector (u,v) over the a priori distribution of the nuisance
agreement between the simulated performance and the new            parameter: p (r; u) = E v [ p (r | v; u )] . We refer to p(r|v;u) as
CRBs.                                                              the joint likelihood function, as p(r|v;u) is relevant to the joint
                                                                   maximum likelihood (ML) estimation of u and v.
                                                                      Considering the joint estimation of the carrier phase θ and
                    II. PROBLEM FORMULATION
                                                                   the frequency offset F from the observation vector r={rk}
   Consider the transmission, of a signal with digital linear      from (1), we take u=(u1,u2)=(θ,F). The nuisance parameter
modulation, over an AWGN channel with unknown carrier              vector v={ak: k∈Id} consists of the unknown DS. Within a
phase and frequency offset. Assuming ideal timing recovery,        factor not depending on F, θ and a, the joint likelihood
the matched filter output samples are given by
                                                                   function p(r|a;F,θ) is given by
   rk = a k e jθk + wk ,   k ∈ I = {K1, K1+1, …, K2}         (1)      p(r | a; F , θ ) = ∏ F (a k , ~k )
                                                                                                    r                              (4)
In (1), {ak: k ∈ I} is a sequence of L=K2-K1+1 transmitted                                    k ∈I

PSK, QAM or PAM symbols. We assume ak belongs to the               where
symbol alphabet {α0, α1, ..., αΜ−1}, with M denoting the
                                                                                         Es
                                                                                              (2 Re( a ~ ) −|a | )
                                                                                                       r  *             2


                                                                     F (a k , ~k ) = e N
                                                                                                          k k       k

number of constellation points and E[|ak|2]=1. The symbol ak                  r           0
                                                                                                                                   (5)
denotes a known PS for k belonging to the set of indices                ~ = r e − j ( 2πkFT +θ ) . Averaging (4) over the data symbols
                                                                   and rk       k
Ip={k0, k1, …, kNp-1} ⊆ I, where Np denotes the number of PS.      yields the likelihood function p(r;F,θ). For the log-likelihood
For k ∈ Id={I \ Ip}, ak denotes an unknown DS. The Nd (=L-         function ln(p(r;F,θ)) we obtain, within a term that does not
Np) DS are assumed to be statistically independent and             depend on (F,θ)
uniformly distributed over the constellation, i.e., the
                                                                                                      Es                    
transmitted DS can take any value from the symbol alphabet            ln p (r; F , θ ) = 2               Re e − jθ ∑ a k ~k  + ∑ ln I (~l )
                                                                                                                         *
                                                                                                                           r             r                           (6)
with equal probability. The sequence {wk: k ∈ I} consists of                                          N0                     l ∈I
                                                                                                                   k ∈I           p               d

zero-mean complex Gaussian noise variables, with
                                                                   where
independent real and imaginary parts each having a variance                       M −1
of N0/2Es. The quantities Es and N0 denote the symbol energy          I (~k ) =
                                                                         r        ∑ F (α          i   , ~k )
                                                                                                        r                                                            (7)
and the noise power spectral density (SNR=Es/N0),                                 i=0

respectively. The quantity θk is defined as (θ + 2πkFT), where     and {α0, α1, ...,αΜ−1} denotes the set of constellation points.
θ represents the carrier phase at k=0, F is the frequency offset      It follows from (2) that the error variance regarding the
and T is the symbol duration. Both θ and F are unknown but         estimation of θ and F is lower bounded by the Cramer-Rao
deterministic parameters.                                          Bound (CRB):
   Let us denote by p(r;u) the probability density function             r
                                                                            ˆ
                                                                       E [(θ − θ ) 2 ] ≥ CRBθ  ( N , N ) = J −1
                                                                                                         PS − DS            p   d (8)   (          )
                                                                                                                                            (θ , F ) 11
(pdf) of the observation vector r, where u is an unknown
deterministic vector parameter. Suppose one is able to produce
                                                                             ˆ
                                                                      E r [( F − F ) 2 ] ≥ CRB PS − DS ( N p , N d ) = J (−θ1, F )
                                                                                               FT
                                                                                                                                            (             ) 22
                                                                                                                                                                     (9)
                               ˆ
from r an unbiased estimate u of the parameter u. Then the                   -1
                                                                   where J denotes the inverse of the FIM. Similarly, (2) yields
estimation error covariance matrix R u −u = E[(u − u)(u − u) T ]
                                          ˆ        ˆ    ˆ          a lower bound on the variance of the estimation error on the
satisfies                                                          instantaneous phase:
                                                                             ˆ                      θ
                                                                      E r [(θ k − θ k ) 2 ] ≥ CRBPS − DS ( N p , N d )
                                                                                                                k
   R u −u − J −1 (u) ≥ 0
      ˆ                                                      (2)                                                                                 (10)
where the notation A ≥ 0 indicates that A is a positive semi-                     (           )                         (       )
                                                                               = J (−θ1, F ) 11 + 4πkT J (−θ1, F ) 12 + 4(πkT ) 2 J (−θ1, F ) 22        (        )
definite matrix (i.e., xTAx ≥ 0, irrespective of x), and J(u) is   The presence of the nuisance vector parameter v={ak: k∈Id}
> T-SP-01823-2003 <                                                                                                                                                      3

makes the analytical computation of the FIM J(θ,F) very hard.                                    be interpreted as the center of gravity of the sequence {βk}.
In order to avoid the computational complexity caused by the                                     We obtain J 12 ≠ 0 , unless kG=0, which is achieved if both PS
nuisance parameters, a simpler lower bound, called modified
                                                                                                 and DS are each located symmetrically about zero, and the PS
CRB (MCRB), has been derived in [7] and [8], i.e.,
                                                                                                 satisfy |ak|=|a-k|. For kG≠0, the parameters θ and F are coupled,
 E[( x − x ) 2 ] ≥ CRBPS − DS ( N p , N d ) ≥ MCRB PS − DS ( N p , N d ) ,
     ˆ                x                            x
                                                                                                 meaning that the inaccuracy in the carrier phase estimate has
           x
where MCRB PS − DS ( N p , N d ) is defined in the same way as                                   an impact on the frequency offset estimation and vice versa.
    x
                                                                                                 Note that the FIM does not depend on θ or F. Substituting
CRB PS − DS ( N p , N d ) in (8)-(10) but with the FIM J(θ,F)                                    (15) into (8)-(10) we obtain
replaced with the Modified FIM (MFIM) JM(θ,F) given by                                                                                           1       k2 
                                                                                                        ˆ                 θ
                                                                                                    E[(θ − θ ) 2 ] ≥ CRBPS − DS ( N p , N d ) =      1 + G      (18)
         2E s  ∑ k
                       γ    2πT ∑ kγ k                                                                                                        J 11  σ G 2
                                                                                                                                                            
   JM =           k ∈I           k ∈I
                                                      (11)
         N 0 2πT ∑ kγ k (2πT ) ∑ k γ k                                                                                                                  1
                                2       2
                                                                                                         ˆ
                                                                                                    E[( FT − FT ) 2 ] ≥ CRB PS − DS ( N p , N d ) =
                                                                                                                              FT
                                                                                                                                                                  (19)
              
                    k ∈I          k ∈I   
                                                                                                                                                     4π σ G J 11
                                                                                                                                                         2 2

where
                                                                                                       ˆ                                            1  (k − k G ) 2 
         a 2,
                                                                                                                           θ
                                                                                                   E[(θ k − θ k ) 2 ] ≥ CRBPS − DS ( N p , N d ) =
                                                                                                                              k                         1 +          (20)
                                              k ∈ Ip                                                                                              J 11     σG
                                                                                                                                                              2      
   γk =  k                                                                                                                                                         
        1,
                                              k ∈ Id
                                                                                                                                ˆ
                                                                                                 The lower bound on E[(θ − θ ) ] from (20) is quadratic in
                                                                                                                                            2
                                                                                                                                  k   k

           III. TRUE CRBPS-DS: ANALYTICAL RESULTS                                                k. Its minimum value is achieved at k=kG and is equal to 1/J11,
                                                                                                 which is the CRB for the estimation of the carrier phase when
   Partial differentiation of the log-likelihood function (6) with                               the frequency offset is a priori known. Note from (15)-(16)
respect to the carrier phase θ and the frequency offset F yields                                 that 1/J11 depends on the number (Np) of PS, the number (Nd)
    ∂                     E                                                                    of DS, and the particular pilot sequence that was selected, but
       ln p(r; F , θ ) = 2 s  ∑ Im(a k ~k ) + ∑ M (~l ) 
                                        *
                                          r          r        (12)
   ∂θ                     N 0  k ∈I                                                            not on the specific position of the PS in the burst. Let ∆1=|K1-
                                              l ∈I  p                           d
                                                                                                 kG| and ∆2=|K2-kG| represent the distance (in symbol intervals)
   ∂                       E                                                                   between the position of the minimum value of the CRB (20)
      ln p(r; F , θ ) = 4πT s  ∑ k Im(ak ~ ) + ∑ lM (~) 
                                        *
                                          rk          rl                                  (13)
  ∂F                           k∈I
                           N0                                                                  and the edges of the burst interval I. The bound (20) achieves
                                                l∈I      
                                                         p                            d
                                                                                                 its maximum value at
where
                                                                                                             K ,     ∆1 ≥ ∆ 2
                   M −1
                                                                                                     k max =  1
   M (~ ) = ∑ F (αi , ~ ) Im( i*~ ) / I (~ )
      rk              rk    α rk         rk                                               (14)               K 2 ,   ∆1 ≤ ∆ 2
                   i =0

F(.,.) and I(.) are defined as in (5) and (7) respectively and                                   i.e. at one of the edges of the burst interval I (or at both edges
                                                                                                 if kG=(K2+K1)/2). The difference between the minimum and
{α0, α1, ...,αΜ−1} denotes the set of constellation points.
                                                                                                 the maximum value of (20) over the burst amounts to
Substituting (12) and (13) into (3) yields
                                                                                                 ∆24π2 CRB PS − DS (Np,Nd), where ∆=max(∆1,∆2). Hence, for
                                                                                                             FT

        2E s  ∑ k
                    β       2πT ∑ kβ k 
   J=            k∈I              k∈I                                                                                            FT
                                                                                                 given values of 1/J11 and CRB PS − DS (Np,Nd), the detection of
         N 0 2πT ∑ kβ k (2πT ) ∑ k β k 
                                 2      2

              
                  k∈I             k∈I     
                                                         (15)                                   symbols located near the edge kmax suffers from a larger
                                                                                                 instantaneous phase error variance as ∆ increases.
            1                             2πTk G
    = J 11                                                                                         Let us define by J∞ and J0 the high-SNR and low-SNR
           2πTk G                (2πT ) (k + σ )
                                           2     2
                                                 
                                                 G
                                                                 2
                                                                 G                               asymptotic FIM, that are obtained as the limit of the FIM from
where                                                                                            (15) for Es/N0 → ∞ and Es/N0 → 0, respectively. It can be
         | a k | 2 ,                                                k∈Ip                        verified that J0 equals the FIM for estimation from the PS
                                                                                                only; it has been shown in [3] that this FIM is given by (15) in
   β k =  2Es         2
          N E r M (r ) ,     [        ]                             k ∈ Id
                                                                                          (16)
                                                                                                 which the summation over I is replaced with a summation
          0                                                                                     over Ip only. This indicates that at very low SNR, DA
and                                                                                              estimation techniques may perform close to optimal. The high
          ∑ kβ            k                  ∑ (k − k                G    )2 βk                  SNR asymptotic FIM J∞ equals the MFIM from (11).
   kG   =  I
                                   σ   2
                                           =     I
                                                                                          (17)      Note that, for a PSK type modulation, |ak|=1, some further
          ∑β                                     ∑β
                                       G
                          k                                           k                          simplification and interpretation of the above results is
               I                                             I
                                                                                                 possible
In (16), Er[.] denotes the average over r = a+n, where a is a                                                           θkG                        θkG
random variable that takes any value from the symbol                                             •    The ratio CRB PS − DS ( N p , N d ) / MCRB PS − DS ( N p , N d ) only
alphabet with equal probability and n is complex zero-mean                                       depends on the SNR and on the ratio Np/Nd.
Gaussian noise with independent real and imaginary parts                                         • For very low SNR, kG converges to the center of the pilot
each having a variance equal to N0/2Es. The quantity kG can                                      sequence. For very high SNR, kG converges to the center of
> T-SP-01823-2003 <                                                                                                                                                                                     4

the complete burst, i.e. (K2+K1)/2.                                                         θ kG                                N0
                                                                                 CRB PS ( N p ) =                                                                                                    (21)
• Independent of the presence of the PS (number, value,                                                                        2N p Es
location), the MFIM (11) related to a burst containing Np                                                                                   θkG                                         θkG
pilot-symbols and Nd data symbols equals the MFIM for                         i.e.,    the               ratio                  CRBPS − DS ( N p , N d ) / MCRBPS − DS ( N p , N d )
transmitting a sequence of L=Np+Nd unknown DS, that has                                                                                                                                 θkG
                                                                              converges to L/Np. At very high SNR, the CRB PS − DS ( N p , N d )
been shown to coincide with the high-SNR limit of the FIM
for the estimation from L unknown DS in [9]. This implies                     converges to its high-SNR asymptote, i.e., the ratio
                                                                                      θkG                                                          θkG
that, at high SNR, estimation techniques that make no use of                  CRB PS − DS ( N p , N d ) / MCRB PS − DS ( N p , N d ) converges to 1.
PS may perform close to optimum.
                               ˆ
• The lower bound on E[( FT − FT ) 2 ] from (19) does not                     B. True CRB for frequency estimation
depend on the choice of the time origin.                                         Fig. 3 corresponds to the frequency estimation error. At low
                                                                              and intermediate SNR, increasing the number of PS (Np)
                                                                              decreases                              the                            ratio
                                                                                    FT                         FT
 IV. TRUE CRBPS-DS: NUMERICAL RESULTS AND DISCUSSION                          CRB PS − DS ( N p , N d ) / MCRB PS − DS ( N p , N d ) (Fig. 3a versus Fig.
   Numerical results were obtained for a QPSK constellation                   3b).                                             In                                  contrast                          with
and a symmetrical observation interval, i.e. I={-K, …,K}. In                          θkG
                                                                              CRB PS − DS ( N p , N d ) / MCRB PS − DS ( N p , N d ) ,
                                                                                                                                                   θkG
                                                                                                                                                                                            the      ratio
this case the MFIM from (11) becomes diagonal and the
                                                                                  FT                           FT
MCRBs reduce to [8]                                                           CRB PS − DS ( N p , N d ) / MCRB PS − DS ( N p , N d ) also depends on
         θ                           θ          N0                            the specific position of the PS within the burst. For a fixed
   MCRB PS − DS ( N p , N d ) = MCRB DS ( L) =
            kG


                                               2 LE s                         number              of         PS           (Np),         the       ratio
                                                                                   FT                           FT
                                                     3N 0                      CRB PS − DS ( N p , N d ) / MCRB PS − DS ( N p , N d ) decreases as the
   MCRBPS − DS ( N p , N d ) = MCRBDS ( L) =
       FT                          FT

                                                 2π L( L2 − 1) E s
                                                    2                         spacing s increases.                                                  At             very           low         SNR,    the
                                                                                   FT
We assume a burst of L=2K+1=Np+Nd symbols, containing                         CRB PS − DS ( N p , N d )
two parts of Np/2 PS spaced with s DS, as proposed in [10].
Two different burst structures are considered. They are shown
in Fig. 1, where the shaded areas indicate the location of the                        #1
PS. In burst structure #1 the PS are concentrated at the                                           Np/2                  s        Np/2

beginning of each burst, whereas burst structure #2 is                                #2
symmetric yielding kG=0, so that carrier phase and frequency                                                                                      Np/2         s       Np/2
estimation are decoupled (with θ k = θ ).
                                        G
                                                                              Fig. 1: burst structure, location of the PS
   Figs. 2 and 3 show the ratio CRBPS-DS(Np,Nd)/MCRBPS-
DS(Np,Nd) as a function of the SNR, for the reference phase
error in k=kG and for the frequency error, respectively. Results                                              10
are presented for L=321, Np/L equal to (approximately) 10%
and 20% (Np=32, 64 if s≠0 and Np=33, 65 if s=0), and for s=0,
(s-1)=Np/2, and (s-1)=3Np/2. For comparison, the lower bound
CRBDS(L)=CRBPS-DS(0,L) for the estimation from a burst
without PS is also displayed. The gray curves correspond to
the lower bounds CRBPS(Np)=CRBPS-DS(Np,0) for the
                                                                                                   CRB/MCRB




estimation from the PS only, which are the low-SNR
asymptotes of the CRBPS-DS(Np, Nd).                                                                                                    Np = 0
                                                                                                                                       Np = 33 (10%L)
                                                                                                                                       Np = 65 (20%L)
A. True CRB for the estimation of the reference phase in k=kG                                                                          low SNR asymptote

   Fig. 2 corresponds to the reference phase estimation error in
                            θkG                         θkG
k=kG. As the ratio CRB PS − DS ( N p , N d ) / MCRB PS − DS ( N p , N d )                                                                   L=321
                                                                                                                                            QPSK
                                                                                                                               reference phase estimate in k = kG
is determined only by Np/L and Es/N0, the curves for all burst
structures with the same ratio Np/L coincide. At low and
intermediate           SNR,              the             ratio                                                1
      θkG                         θkG
CRBPS − DS ( N p , N d ) / MCRBPS − DS ( N p , N d ) decreases as Np/L                                             -14   -12    -10    -8     -6    -4    -2       0    2     4    6    8     10

                                                                                                                                                         Es/No
                                                        θ
increases. At very low SNR, the                   CRB PSG− DS ( N p , N d )
                                                       k
                                                                              Fig. 2: CRB/MCRB for the reference phase estimate in k=kG

converges to its low-SNR asymptote that is given by
> T-SP-01823-2003 <                                                                                                                                                                                                               5

                                1000                                                                                              1000

                                                                                                                                                                                              Np = 0
                                                                                                                                                                                              #1, Np = 65, s = 0
                                                                                                                                                                                              #2, Np = 65, s = 0
                                                                                                                                                                                              #1, Np = 64, s = 33
                                                                                          Np = 0                                                                                              #2, Np = 64, s = 33
                                                                                          #1, Np = 33, s = 0                                                                                  #1, Np = 64, s = 97
                                                                                          #2, Np = 33, s = 0                                                                                  #2, Np = 64, s = 97
                                     100                                                  #1, Np = 32, s = 17                          100                                                    low SNR assymtote
                                                                                          #2, Np = 32, s = 17
                          CRB/MCRB                                                        #1, Np = 32, s = 49




                                                                                                                            CRB/MCRB
                                                                                          #2, Np = 32, s = 49
                                                                                          low SNR asymptote




                                      10                                                                                               10

                                                        L=321, Np=10%L
                                                             QPSK
                                                       frequency estimate
                                                                                                                                                         L=321, Np=20%L
                                                                                                                                                              QPSK
                                                                                                                                                         frequecy estimate



                                       1                                                                                                 1
                                           -14   -12   -10   -8    -6    -4    -2     0    2    4    6     8     10                          -14   -12    -10   -8    -6     -4    -2     0   2     4     6        8   10
                                                                              Es/No
                                                                                                                      (a)                                                         Es/No
                                                                                                                                                                                                                            (b)
Fig. 3: CRB/MCRB for the frequency estimate

becomes     close to its low-SNR asymptote that is given by                                                                        use of the presence of the DS (of the knowledge about the PS)
[3],[10]                                                                                                                           in the estimation process.
                                                                  3N 0
   CRBPS ( N p ) =
      FT
                                                                                                                (22)               D. Effect of the burst structure
                       2π 2 E s [ N p ( N p − 1) + 3 N p s ( s + N p )]
                                                             2
                                                                                                                                      For a fixed Np and fixed s, burst structures #1 and #2 yield
i.e.,          assuming            Np,L>>1,                                               the      ratio                                                                   θkG
                                                                                                                                   the same CRB PS − DS ( N p , N d ) , while the asymmetric burst
      FT                          FT
 CRB PS − DS ( N p , N d ) / MCRB PS − DS ( N p , N d )                                   converges to                                                                  FT
                                                                                                                                   structure #1 yields the smallest CRB PS − DS ( N p , N d ) (at any
(1+3q+3q2)-1(L/Np)3, where q=s/Np. For fixed Np and fixed s,
                                  FT                                                                                               SNR). However, as the following example illustrates, we
the low SNR asymptote of the CRB PS − DS ( N p , N d ) is the same
                                                                                                                                   should be very careful when interpreting these results. Fig. 4
for burst structures #1 and #2, as (22) is not affected by a                                                                                       θ
                                                                                                                                   depicts the CRB PS − DS ( N p , N d ) for the reference phase error
                                                                                                                                                                              k

time-shift of the pilot sequence within the burst. At very high
                 FT                                                                                                               as a function of the symbol index k at Es/N0=2 dB for burst
SNR the CRB PS − DS ( N p , N d ) converges to its high-SNR
                                                                                                                                  structures #1 and #2 with Np=64 and s=33. The following
asymptote,                    i.e.,                 the           ratio                                                           observations can be made
    FT                              FT
CRB PS − DS ( N p , N d ) / MCRB PS − DS ( N p , N d ) converges to 1.                                                            • Although burst structure #1 yields the smallest
                                                                                                                                        FT
For both burst structures #1 and #2, it can be easily verified                                                                     CRBPS − DS ( N p , N d ) , its CRB on the reference phase error
that, assuming Np,L>>1 and for a fixed value of q=s/Np, the                                                                        variance at k=K is larger than for burst structure #2. This can
            FT                         FT
ratio CRB PS − DS ( N p , N d ) / MCRB PS − DS ( N p , N d ) is mainly                                                             be explained by noting that, at a value of SNR as low as 2dB,
determined by the ratio Np/L. Simulation results, not reported                                                                     the distance (∆) between the positions of the minimum and
                                                                                                                                                                   θ
here, indicate that the assumption Np,L>>1 holds even for the                                                                      maximum value of the CRB PS − DS ( N p , N d ) is significantly       k



relatively small values of Np and L from Fig. 3.                                                                                  larger for burst structure #1 than for burst structure #2.
C. CRBPS-DS(Np,Nd) versus CRBPS(Np) and CRBDS(Np+Nd)                                                                              • Although burst structure #2 results in the smallest
                                                                                                                                                        θ
                                                                                                                                  maximum for CRBPS − DS ( N p , N d ) over the burst, other than
                                                                                                                                                                                    k

   It follows from both Fig. 2 and Fig. 3 that the CRBPS-
DS(Np,Nd) is smaller than both CRBPS(Np) and CRBDS(Np+Nd).                                                                         for burst structure #1, this maximum value is reached near
This indicates that it is potentially more accurate to estimate                                                                    both edges of the burst interval. This implies that in burst
the carrier phase and frequency of a hybrid burst with a hybrid                                                                    structure #2 more symbols are affected by a large
algorithm that exploits both PS and DS (in some intelligent                                                                        instantaneous phase error variance than in burst structure #1.
way) than with an algorithm that only uses the PS (and                                                                                Hence, the ‘best’ burst structure depends strongly on the
ignores the DS) or with an algorithm that uses all received                                                                        operating SNR and on the maximum allowable phase error
symbols (PS+DS) but ignores the a priori knowledge about                                                                           variance for proper symbol detection.
the      PS.     The      ratio     CRBPS(Np)/CRBPS-DS(Np,Nd)
(CRBDS(Np+Nd)/CRBPS-DS(Np,Nd)) depends on the operating
SNR and on the burst structure, and indicates to what extent
synchronizer performance can be improved by making clever
> T-SP-01823-2003 <                                                                                                                                             6

                           1.8E-02                                                                              FT
                                                                                                  decreases CRB PS ( N p ) , thus improving the performance of
                                         SNR = 2 dB
                           1.6E-02
                                                                                                  the DA frequency estimate at high SNR. However, below a
                                                                                                  certain SNR threshold, the performance dramatically degrades
                           1.4E-02                                                                across a narrow SNR interval, resulting in a MSEE much
                                                                                                  larger than the CRBPS(Np). This so-called threshold
                           1.2E-02                                                                phenomenon results from the occurrence of estimates with
                                                      #1                                          large errors, i.e., outlier estimates [2]. The presence of
                                                      #2
                                                                                                  important secondary peaks in the likelihood function results in
              CRB_θk




                           1.0E-02

                                                                                                  a large probability of generating outlier frequency estimates at
                           8.0E-03                                                                lower SNR, because these secondary peaks can more easily
                                                                                                  exceed the central peak when noise is added. The SNR
                           6.0E-03                                                                threshold decreases with the number of available signal
                                                                                                  samples Np. For Np consecutive PS (as in burst structure #1),
                           4.0E-03                                                                the threshold is very low so that the DA estimator usually
                                                                                                  operates above threshold. However, the SNR threshold tends
                           2.0E-03                                                                to increase as the PS are separated by DS [3],[10]. The
                                  -160   -120   -80        -40   0   40   80   120   160
                                                                 k                                simulation results, reported in Fig. 5, illustrate this behavior
Fig. 4: CRB for the reference phase estimate at Es/N0=2 dB                                        and show that s≈Np/2 provides a good compromise between a
                                                                                                                         FT
                                                                                                  small value of CRB PS ( N p ) and a low SNR threshold. A
 V. PRACTICAL ESTIMATORS FOR HYBRID BURST STRUCTURES
                                                                                                  minimum of 105 trials have been run to ensure accuracy. Each
  In this section we consider some practical joint carrier
                                                                                                  trial a new phase θ and frequency offset FT is taken from a
phase
                                                                                                  random uniform distribution over [-π, π] and [-0.1; 0.1],
and frequency estimators for hybrid burst structures.
                                                                                                  respectively. The estimated reference phase error was
A. DA Synchronization                                                                             measured modulo 2π, i.e., in the interval [-π, π].
   The DA estimates are given by [2]                                                              B. NDA Synchronization
                                 ~                                                                 Assuming a QPSK constellation, the NDA estimates are
   F = arg max ∑ a k rk e − j 2πkFT 
    ˆ                *
                                                                                           (23)
           ~
           F   k ∈I                                                                             given by [2],[11]
                                    
                                     p


                                                                                                     ˆ 1 arg max
                                                                                                                                           ~
                                                                                                              ~ ∑ rk
                                                                                                                     2 j 4 arg{rk } − j 2πkFT
           
                             ˆ                                                                    F=               e            e                         (25)
   θ = arg ∑ a k rk e − j 2πkFT 
    ˆ             *
                                                           (24)                                         M     F
                                                                                                                 k∈I
            k ∈I
                                
                                                                                                                                            ˆ 
                       p
                                                                                                          1
                                                                                                    θˆ =     arg ∑ rk e j 4 arg{rk }e − j 8πkFT 
                                                                                                                       2
Of the L=Nd+Np received samples, only the Np known PS are                                                                                                (26)
                                                                                                          M       k∈I                           
                                     ˆ    ˆ
used. For FT << 1, the estimates F and θ from (23) and (24)
                                                                                                  All (L) received samples are taken into account, but the
are unbiased [2]. The mean square estimated errors (MSEE) of
                                                                                                  knowledge of the PS is disregarded. For FT<<1 and θ∈[-π/4,
(23) and (24) are lower bounded by the CRBPS(Np) (with
                                                                                                                       ˆ          ˆ
                                                                                                  π/4], the estimates F and θ from (25) and (26) are unbiased
CRBPS(Np)≥CRBPS-DS(Np,Nd)). This implies that DA
synchronization is intrinsically suboptimal, especially at high                                   [2]. The resulting MSEE converges to CRBDS(L) at high SNR.
SNR where CRBPS(Np)>>CRBPS-DS(Np,Nd) for Nd>>1. Still, it                                         Simulation results indicate, however, that the value of SNR at
is interesting to understand the behavior of the DA algorithm                                     which the MSEE becomes close to the CRBDS(L) may be quite
since multi-stage synchronization procedures are often                                            large. The SNR threshold for the NDA estimator is much
initialized with a DA estimate (see subsections V.B and V.C).                                     higher than for the DA estimator, as the non-linearity
                            ˆ      ˆ   ˆ    ˆ                                                     increases the noise level. To cope with this problem, a two-
   The DA estimates F and θ k = θ + 2πkG FT , resulting from G
                                                                                                  stage coarse-fine DA-NDA estimator has been proposed in
(23) and (24), is not affected by a time-shift of the pilot                                       [10]. A ML DA estimator is used to coarsely locate the
sequence within the burst. This implies that, for a given value                                   frequency offset, and then the more accurate NDA estimator
of Np and s, the corresponding MSEEs are the same for burst                                       attempts to improve the estimate within the uncertainty of the
structures #1 and #2. It is well known that, at high SNR, the                                     coarse estimator. In fact, the search range of the NDA
MSEE reaches the CRBPS(Np). Considering the burst                                                 estimator is restricted to the neighborhood of the peak of the
                              θ                    FT
structures from Fig.1, CRB PS ( N p ) and CRB PS ( N p ) are
                                                           kG
                                                                                                  DA based likelihood function. This excludes a large
given by (21) and (22). Increasing the number of PS (Np)                                          percentage of secondary peaks from the search range of the
              θ                                                                                   NDA estimator, and thus considerably reduces the probability
decreases CRB PS ( N p ) , thus improving the performance of
                              kG

                                                                                                  to estimate an outlier frequency. Assuming the MSEE of the
the DA reference phase estimate at k=kG at high SNR.                                                                                    FT
                                                                                                  initial DA estimate equals the CRB PS ( N p ) , this uncertainty
Increasing the number of PS (Np) and/or the spacing (s)
                                                                                                  range can be determined as
> T-SP-01823-2003 <                                                                                                                                                                                                          7

                                          100                                                                                                10000

                                                                                                                                                                                                 QPSK, L=321
                                                                                                                                                                                                frequency error


                                                                                                 QPSK, L=321
                                                                                         reference phase error in k = kG

                                                                                                                                              1000




                                                                                                                                 MSEE/MCRB
                                   MSEE/MCRB



                                               10                                                                                             100




                                                                   #1-#2, Np=65, s=0
                                                                                                                                                               #1-#2, Np=65, s=0
                                                                   #1-#2, Np=64, s=33                                                           10             #1-#2, Np=64, s=33
                                                                   #1-#2, Np=64, s=97                                                                          #1-#2, Np=64, s=97
                                                                   #1-#2, Np=33, s=0                                                                           #1-#2, Np=33, s=0
                                                                   #1-#2, Np=32, s=17                                                                          #1-#2, Np=32, s=17
                                                                   #1-#2, Np=32, s=49                                                                          #1-#2, Np=32, s=49
                                                                   CRB_PS (Np)                                                                                 CRB_PS(Np)

                                                 1                                                                                              1
                                                     -6       -4    -2        0        2      4         6      8      10                             -6   -4        -2      0         2     4         6       8   10
                                                                                      Es/No                                                                                         Es/No
                                                                                                                           (a)                                                                                         (b)
Fig.5: The ratio MSEE/MCRB for DA synchronization


 ± m CRB PS ( N p ) , where m should be carefully chosen.
         FT                                                                                                                              normal operating SNR of the DD estimators is situated above
                                                                                                                                         threshold (large number of available samples (L), no noise
When the parameter m increases, the search region increases,                                                                                                                           ˆ       ˆ
                                                                                                                                         enhancement). The required initial estimate (θ ( 0) , F ( 0) ) can
as well as the probability of comprising outlier peaks, which
may result in a degradation of the performance at low SNR                                                                                be obtained from the NDA method; however, the performance
(outlier effect). However, if m decreases, the search region                                                                             below the NDA threshold rapidly degrades, because of an
decreases, as well as the probability of comprising the                                                                                  inaccurate initial estimate. If PS are available, it is better to
(correct) central peak, which in turn may result in a                                                                                    use DA initialization. We will further refer to these schemes
degradation of the performance at high SNR. After frequency                                                                              as DA-hDD and DA-sDD. After phase and frequency
and phase correction, the samples for k∈Ip are compared to                                                                               correction, the samples for k∈Ip are compared to the original
the                                                                                                                                      PS and, if necessary, an extra multiple of π/2 is compensated
original PS and, if necessary, an extra multiple of π/2 is                                                                               for.
compensated for.                                                                                                                         The DD estimators take advantage of both the (Np) PS and the
   A major disadvantage of this DA-NDA algorithm is that it                                                                                                                              ˆ
                                                                                                                                         (Nd) DS. At high SNR, the DD estimates F and θˆ are
does not exploit the knowledge of the PS in the NDA fine                                                                                 unbiased and their MSEE equals the CRBPS-DS(Np,Nd).
estimation step. Therefore, its MSEE is lower bounded by the                                                                             However, the DD estimates become biased at low SNR,
CRBDS(Np+Nd) (with CRBDS(Np+Nd)≥CRBPS-DS(Np,Nd)). This                                                                                   implying that the CRB is no longer a valid lower bound on the
implies that the DA-NDA algorithm is intrinsically suboptimal                                                                            estimators’ performance in this region. Fig. 6 illustrates this
(especially at low and intermediate SNR) in the sense that                                                                               behavior. The presented results are for burst structure #2 with
under no circumstances its performance may meet the CRBPS-                                                                               Np=64, s=33 and L=321. Note that F and θ are decoupled
DS. Some other estimator may yield a MSEE between CRBDS                                                                                  (with θ k =θ). The initial phase error θ was set to 0.2 rad and
                                                                                                                                                          G
and CRBPS-DS, but it should fully exploit the knowledge of the
                                                                                                                                         the initial frequency offset FT was set to 10-4. The mean
PS.
                                                                                                                                         estimate is plotted versus the SNR. We observe that increasing
C. Iterative DD Synchronization                                                                                                          the number of DD iterations enlarges the SNR range for which
   DD estimators extend the sum over Ip in (23)-(24) with                                                                                the DD estimates are unbiased. The hDD algorithm converges
terms over Id in which the quantities ak are replaced by hard                                                                            somewhat faster to an unbiased estimate at intermediate SNR,
(hDD) or soft (sDD) decisions, that are based upon a previous                                                                            but only the sDD algorithm yields unbiased estimates up to
estimate of (θ, F). For QPSK {1,j,-1,-j}, the soft decisions are                                                                         values of Es/N0 as low as 3dB.
given by [4]                                                                                                                                It can be easily shown that the sDD algorithm proposed in

    ak =
    ˆ
                    [
          sinh N Re(~k ) + j sinh N Im(~k )
               2E
                     r    s

                          0
                                    2E    ( n −1)
                                            r             ] (27)
                                                                          [       s

                                                                                  0
                                                                                              ( n −1)
                                                                                                        ]                                [4] for carrier phase estimation and extended here to joint
                                                                                                                                         carrier phase and frequency estimation involves a practical
               2E
                    [
          cosh N Re(~k ) + cosh N Im(~k )
                     r    0
                              s     2E
                                            r
                                               ( n −1)
                                                          ]               [       s

                                                                                  0
                                                                                              ( n −1)
                                                                                                        ]                                implementation of the ML estimator by means of the
                                                                                                                                         expectation-maximization (EM) algorithm. This algorithm
In (27), ~k                                                                                                                              converges iteratively to the ML estimate provided that the
              ( n −1 )                    ˆ ( n −1) + 2πkF ( n −1)T )
                                                         ˆ
         r               = rk e − j (θ                     ˆ
                                 . The hard decisions a k are
                                                                                                                                         initial estimate is sufficiently accurate [12].
determined as the constellation points closest to ~k( n −1) . The
                                                  r
> T-SP-01823-2003 <                                                                                                                                                                                                           8

                                                     0.22                                                                                             1.E-04



                                                                                                                                                      1.E-04
                                                      0.2


                                                                                                                                                      9.E-05


                                                     0.18
                                                                                                                                                      8.E-05
                         mean phase estimate (rad)




                                                                                                                            mean frequency estimate
                                                     0.16                                                                                             7.E-05


                                                                                               1 it hDD
                                                                                               2 it hDD                                                                                          1 it hDD
                                                                                               4 it hDD                                               6.E-05                                     2 it hDD
                                                                                               8 it, 16 it hDD                                                                                   4 it hDD
                                                     0.14                                                                                                                                        8 it, 16 it hDD
                                                                                               1 it sDD
                                                                                               2 it sDD                                                                                          1 it sDD
                                                                                               4 it sDD                                                                                          2 it sDD
                                                                                                                                                      5.E-05                                     4 it sDD
                                                                                               8 it sDD
                                                                                               16 it sDD                                                                                         8 it sDD
                                                                                                                                                                                                 16 it sDD
                                                     0.12
                                                                                                                                                      4.E-05
                                                                              FT = 0.0001                                                                                       FT = 0.0001
                                                                             phase = 0.2 rad                                                                                   phase = 0.2 rad


                                                      0.1                                                                                             3.E-05
                                                                                                                                                               3   4   5   6        7        8         9           10
                                                            3   4   5    6       7        8          9           10
                                                                        Es/No (dB)                                                                                         Es/No
                                                                                                                      (a)                                                                                               (b)
Fig. 6: Mean estimate after n iterations of the sDD and hDD algorithms

       VI. COMPARING PERFORMANCE WITH TRUE CRBPS-DS                                                                                         • Above its SNR threshold (at about 4 dB), the NDA
   The true ML estimator is known to be asymptotically                                                                                      estimator performs more or less closely to the CRBDS(L).
optimal in the sense that it achieves the performance predicted                                                                             At high SNR, the performance of the DA-NDA estimator
by the CRB for large data records. However, the performance                                                                                 matches that of the NDA estimator, but the performance below
for finite signal durations cannot be determined analytically.                                                                              the SNR threshold degrades less rapidly and is still adequate
In this section we compare the simulated MSEE of the                                                                                        for reliable receiver operation.
different estimators listed in section V to the CRBPS-DS(Np,Nd)                                                                             • At low SNR the DA-hDD estimator performs worse than
derived in section III.                                                                                                                     the DA-NDA estimator (Fig. 7b for frequency estimation). At
   Numerical results pertaining to the different algorithms are                                                                             (very) low SNR the initial DA estimates become even more
obtained in Fig. 7. We assume a burst with L=641 QPSK                                                                                       accurate than the steady-state hDD estimates. Hence, hard
symbols, Np=128 and s=65. The PS are organized as in burst                                                                                  decisions are not useful at low SNR.
structure #2 from Fig.1. Note that F and θ are decoupled (with                                                                              • The DA-sDD estimator outperforms by far the DA-hDD
θ k =θ. A minimum of 105 simulations has been run to ensure                                                                                 estimator, especially at low SNR, and provides a considerable
   G
                                                                                                                                            improvement over the DA-NDA estimator. For large L, the
accuracy. Each simulation a different phase and frequency                                                                                   DA-sDD estimator performs very closely to the CRBPS-DS.
offset was randomly generated from [-π,π]and [-0.1;0.1],
respectively. We have plotted the ratio MSEE/MCRB for the
estimation of F and θ as a function of the SNR. The phase                                                                                                                       VII. CONCLUSION
error is measured modulo 2π and supported in the interval
                                                                                                                                                    In this contribution, we have investigated the joint phase
[−π, π], except for the NDA estimator. The phase error of the
                                                                                                                                                 and frequency estimation from the observation of a ‘hybrid
NDA estimator was estimated modulo π/2, i.e. in the interval                                                                                     burst’ that contains PS as well as DS. We have compared the
[−π/4, π/4], as this estimator gives a 4-fold phase ambiguity.                                                                                   CRBPS-DS with the performance of existing carrier
For the DA-NDA estimation we chose m=3. The performance                                                                                          synchronizers. Numerical evaluation of the CRB shows how
of the DA estimator is not displayed: as Np<<L, the MSEE of                                                                                      much can be gained in estimator performance by using a
the DA estimates is much larger than the MSEE resulting                                                                                          ‘hybrid algorithm’ that exploits both PS and DS (in some
from the other estimators. The MSEE resulting from the DA-                                                                                       intelligent way), rather than an algorithm that only uses the PS
hDD estimator reaches a steady state after about five                                                                                            (and ignores the DS) or an algorithm that uses all received
iterations. The MSEE resulting from the DA-sDD estimator                                                                                         symbols but ignores the a priori knowledge about the PS. We
reaches a steady state after 10 to 15 iterations. We note that                                                                                   have pointed out that the hybrid DA-NDA estimator proposed
using a combined DA-NDA initialization instead of a DA                                                                                           in [10] is suboptimal, because it does not fully exploit the
initialization the same steady state performance as for DA                                                                                       knowledge about the PS. Further, we have shown that the
initialization is obtained after considerably less (no more than                                                                                 iterative sDD estimator with DA initialization outperforms the
5) iterations, which indicates the importance of an accurate                                                                                     DA-NDA estimator and operates very closely to the CRBPS-DS.
initial estimate to speed up convergence. Further, our results
show that:
> T-SP-01823-2003 <                                                                                                                                                                                       9

                                                10                                                                                   10


                                                              phase estimate                NDA
                                                                                                                                                                                NDA
                                                              L=641, N=128                  DA-NDA
                                                             burst structure #2                                                                                                 DA-NDA
                                                                                            DA-sDD 15it
                                                                                                                                                                                DA-sDD 15it
                                                                                            DA-hDD 5it                                                frequency estimate        DA-hDD 5it
                                                                                            CRB_DS                                                       L=641, N=128           CRB_DS
                                                                                            CRB_PS-DS                                                  burst structure #2       CRB_PS-DS




                          CRB/MCRB, MSEE/MCRB




                                                                                                               CRB/MCRB, MSEE/MCRB
                                                 1                                                                                    1
                                                     3   4    5         6           7   8      9          10                              3   4   5       6        7        8   9       10
                                                                            Es/No                    Es/No
                                                                          (a)                                                                                                                 (b)
Fig. 7: MSEE performance of some algorithms using PS and DS. Burst structure #2, L=641, Np=128, s=65


                                                     REFERENCES                                                                   signal,” in Proc. IEEE Globecom 2002, Taipe, Taiwan, paper CTS-04-2,
                                                                                                                                  Nov. 2002
[1]   H.L. Van Trees, Detection, Estimation and Modulation Theory. New                                                       [7] A.N. D’Andrea, U. Mengali and R. Reggiannini, “The modified Cramer-
      York: Wiley, 1968                                                                                                           Rao bound and its applications to synchronization problems,” IEEE
[2]   D.C, Rife and R.R. Boorstyn, “Single-tone parameter estimation from                                                         Trans. Comm., vol COM-24, pp. 1391-1399, Feb.,Mar.,Apr. 1994.
      discrete-time observations,” IEEE Trans. Inf. Theory, vol. IT-20, No. 5,                                               [8] F. Gini, R. Reggiannini and U. Mengali, "The modified Cramer-Rao
      pp. 591-597, Sept. 1974                                                                                                     bound in vector parameter estimation," IEEE Trans. Commun., vol.
[3]   J.A. Gansman, J.V. Krogmeier and M.P. Fitz, “Single Frequency                                                               CON-46, pp. 52-60, Jan. 1998
      Estimation with Non-Uniform Sampling,” in Proc. of the 13th Asilomar                                                   [9] M. Moeneclaey, “On the true and the modified Cramer-Rao bounds for
      Conference on Signals, Systems and Computers, Pacific Grove, CA,                                                            the estimation of a scalar parameter in the presence of nuisance
      pp.878-882, Nov. 1996                                                                                                       parameters,” IEEE Trans. Commun., vol. COM-46, pp. 1536-1544, Nov.
[4]   W.G. Cowley, “Phase and frequency estimation for PSK packets:                                                               1998
      Bounds and algorithms,” IEEE Trans. Commun., vol. COM-44, pp. 26-                                                      [10] B. Beahan, “Frequency Estimation of Partitioned Reference Symbol
      28, Jan. 1996                                                                                                               Sequences,” Master Thesis, University of South Australia, April 2001,
[5]   F. Rice, B. Cowley, B. Moran, M. Rice, “Cramer-Rao lower bounds for                                                         available from www.itr.unisa.edu.au/rd/pubs/thesis/theses.html
      QAM phase and frequency estimation,” IEEE Trans. Commun., vol. 49,                                                     [11] A.J. Viterbi and A.M. Viterbi, “Nonlinear estimation of PSK-modulated
      pp 1582-1591, Sep. 2001                                                                                                     carrier phase with application to burst digital transmission,” IEEE Trans.
[6]   N. Noels, H. Steendam and M. Moeneclaey, “The true Cramer-Rao                                                               Inform. Theory, vol. IT-29, pp. 543-551, July 1983
      bound for phase-independent carrier frequency estimation from a PSK                                                    [12] R. A. Boyles, “On the convergence of the EM algorithm,” J.R. Statist.
                                                                                                                                  Soc. B, 45, No. 1, pp. 47-50, 1983

				
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