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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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