Stochastic approach to square timing estimation ... - Semantic Scholar

Stochastic Approach to Square Timing Estimation with Frequency Uncertainty Jos´e A. L´opez-Salcedo and Gregori V´azquez Department of Signal Theory and Communications, Universitat Polit`ecnica de Catalunya. Email: {jlopez,gregori}@gps.tsc.upc.es

Abstract—This paper addresses the problem of Non-Data-Aided (NDA) symbol timing error estimation in presence of carrier frequency errors. For that purpose, a novel stochastic Maximum Likelihood (ML) estimator is derived following a Bayesian approach by considering the carrier frequency error an unknown deterministic parameter constrained within a certain range. Further on, it will be exploited the cyclostationary nature of the received data for obtaining an ML symbol timing estimator. In this way, it is found that the weighted cyclic autocorrelation function of the received data is a sufficient statistic of the problem and thus, an optimal ML symbol timing estimator is derived for low signal-to-noise ratios (SNR). As a consequence of the study, the well-known Oerder and Meyr (”Square Timing”) method becomes a particular case of this new general solution.

III, a Bayesian approach is introduced for obtaining a symbol timing error estimator in presence of frequency uncertainty. Next, section IV presents an analysis on the cyclostationarity of the Likelihood function, which enables us to derive an optimal timing estimation method based on the cyclic autocorrelation function of the received data. In section V, it is shown how the classical Square Timing recovery method is found to be a particular case of the general WCM, and how both algorithms are found to converge for maximum frequency uncertainty. Simulation results are discussed in section VI, and finally, conclusions are drawn in section VII.

Index Terms—Synchronization, Timing Estimation, Non-Data-Aided (NDA), low-SNR, Maximum Likelihood (ML), Bayesian, Carrier Frequency Error Uncertainty, Cyclic Autocorrelation Function (CAF).

II. D ISCRETE -T IME S IGNAL M ODEL AND S TOCHASTIC ML A PPROACH

I. I NTRODUCTION

The complex envelope model that will be used for a linearly modulated received signal immersed in noise is given by:

In this paper1 we are addressing a basic problem in digital communications such as the NDA symbol timing estimation, which constitutes one of the fundamental tasks of a digital receiver. There are a wide range of different approaches to attempt the NDA symbol timing estimation problem, but most of them are found to be based on some heuristic or ad-hoc reasoning [1]. However, it is interesting to notice that among all the NDA timing synchronizers, the well-known and efficient Oerder and Meyr method [2] (Square Timing recovery) is mostly adopted in many applications due to its simple implementation and its robust performance [3]. Following on this topic, the paper shows that is possible to follow a systematic Maximum Likelihood approach to derive an optimal symbol timing estimator for low SNR. In addition, the paper also extends the problem to a more general case by considering the presence of some carrier frequency uncertainty in the received data. By following a Bayesian approach, the new WCM timing estimator (Weighted Cyclic autocorrelation Method) is presented, which constitutes the main contribution of this paper to the synchronization field. By taking into consideration the cyclostationary properties of the received signal, the paper shows that the Cyclic Autocorrelation Function (CAF) [4] of the received data is a sufficient statistic for the problem, and thus an optimal ML symbol timing estimator is derived for low SNR scenarios. A second relevant result is related to the inclusion of the classical Square Timing recovery method by Oerder and Meyr as a particular case of the general WCM, which supplies a new stochastic interpretation of this important and popular algorithm. The structure of the paper is the following: the signal model and the stochastic ML approach are presented in section II. In section 1 This work was partially funded by the following research projects of the Spanish/Catalan Science and Technology Commissions (CICYT/CIRIT): TIC2002-04594-, TIC2001-2356-, TIC2000-1025-C02-01 and 2001SGR00268.

r (k) =

+∞ X

n=−∞

xn g (kTs − nT − τ ) ejωkTs + w (kTs )

(1)

where xn are the transmitted information symbols, g(kTs ) is the sampled transmission pulse shape, Ts is the sampling period, T = Nss Ts is the symbol period with Nss the number of samples per symbol and w(kTs ) is the complex additive white Gaussian noise (AWGN). Regarding the synchronization parameters, the signal model in (1) includes the symbol timing error τ , constrained within a symbol interval [−T /2, +T /2), and the carrier frequency error ω constrained to the Nyquist bandwidth [−π/Ts , +π/Ts ) that will be further on normal. ω T. ized to the symbol rate, that is, ν = 2π In a more convenient vectorial notation, the received signal can be expressed by means of a linear transfer matrix A (Θ) as follows: r = A (Θ) x + w

(2)

. A (Θ) = [a−K (Θ) , a−K+1 (Θ) , ..., a+K (Θ)] h 2π . an (Θ) = g (−nT − M Ts − τ ) e−j N ss Mν ,

(3)

2π

g (−nT − (M − 1)Ts − τ ) e−j Nss (M−1)ν , ..., iT 2π 2π g (−nT + M Ts − τ ) ej Nss Mν e−j Nss n0 ν

(4)

Θ = [τ, ν]

(5)

where an observation window of 2K +1 symbols and a total of 2M +1 samples have been considered. Finally, n0 is a constant that reflects the arbitrary time origin. Synchronization methods derived under the Stochastic or Unconditional Maximum Likelihood approach are the most extent in literature,

0-7803-7802-4/03/$17.00 © 2003 IEEE

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and they are based on applying the ML principle assuming that the transmitted symbols x are all random. In this way, an NDA estimation can be performed by maximizing the marginal Likelihood function of the wanted parameters: b = arg max Λ (r|Θ) = arg max Ex [Λ (r|Θ; x)] Θ Θ

Θ

From equations (6) to (8), the Bayesian stochastic log-Likelihood function for low-SNR asymptotically becomes: L (r|τ ) = ln (Eν,x [Λ (r|τ, ν; x)]) ≈ ³ h i ´ C b T r Eν AAH R 4 σw

(6)

In the presence of AWGN, the parameter estimation problem for any linear modulation is easily formulated based on the Gaussian noise probability density function as follows: ¶ µ 1 (7) Λ (r|Θ; x) = C exp − 2 kr − A(Θ)xk2 σw However, the computation of the expectation Ex in (6) generally exhibits insuperable obstacles. To circumvent this limitation, further on this paper, a low SNR will be assumed. In [5] it is shown that for low SNR, the sample covariance matrix of the received data given b = rrH , becomes a sufficient statistic for the NDA parameter by R estimation problem, and thus, the stochastic ML function depends only on the second order moments of the received data. As in [5], for a long enough observation interval, the log-Likelihood function for low SNR becomes asymptotically: ³ ´ C . b L (r|Θ) = ln (Ex [Λ (r|Θ; x)]) ≈ 4 T r A(Θ)ΓAH (Θ)R σw (8) i h . Γ = E xxH (9)

where C is an irrelevant constant, ”T r” stands for the trace operator, and Γ is the autocorrelation matrix for the transmitted symbols, which will be assumed to be normalized to the transmitted mean power, that is, Γ = I. For simplicity, the dependence of the linear transfer matrix A on the parameter vector Θ further on will be omitted. III. T IMING E STIMATION WITH C ARRIER F REQUENCY U NCERTAINTY: A BAYESIAN A PPROACH

For attempting the expectation Eν in (10), it is convenient to further develop the outer product matrix AAH . Each of the (p, q) entries of the resulting matrix is given by: h i = lim AAH K→∞

∞ X

n=−∞

p,q

2π

g (pTs + nT − τ ) g∗ (qTs + nT − τ ) ej Nss ν(p−q)

(11)

Now, when including the prior distribution in equation (11) it leads to:

lim Eυ

K→∞

·h

+∞ X

n=−∞

AAH

i

p,q

¸

=

+N Zss /2

fν (ν) e

2π ν(p−q) jN ss

dν

−Nss /2

g (pTs + nT − τ ) g ∗ (qTs + nT − τ )

(12)

Thus, considering this Bayesian approach, the log-Likelihood function can be expressed as follows: ³ ´ C b T r M R 4 σw h³ ´ i M = Gτ GH ¯V τ

L (r|τ ) =

(13) (14)

where the ”¯” operator stands for the component-wise or SchurHadamard product, Gτ is the pulse shaping matrix with only dependence on the symbol timing error parameter τ , Gτ = A (τ, ν = 0)

In section II it has been introduced that our signal parameter vector Θ is containing both the symbol timing τ and the carrier frequency error ν. However, we will here expose the situation when we are only interested in obtaining an estimate for τ . In practice, both the symbol timing τ and the carrier frequency error ν are unknown deterministic parameters, and for the problem at hand, the most demanding goal would be the joint estimation of these two parameters. Here in, we will focus only on the problem of the marginal estimation of the symbol timing error τ . Under this condition, the carrier frequency error ν will be undetermined within a given interval ∆ν , that is, ν ∈ [−∆ν /2, +∆ν /2). There are two main cases of specific interest, such as the case where no carrier frequency error is present (that is, the carrier has been perfectly acquired before the timing error estimation), and the case where the carrier frequency error lies within the maximum frequency deviation interval given by the normalized Nyquist bandwidth = T /Ts = Nss . However, the study will be done for the gen∆max ν eral case, and it will be later particularized for these two cases. Despite of the deterministic nature of the carrier frequency error ν, following a Bayesian approach a given prior is considered for the unknown parameter. The lack of knowledge about the carrier frequency error is then reflected by adopting a uniform prior distribution, that is, fν (ν) = 1/∆ν .

(10)

(15)

and finally, the entries of the Doppler shaping (or spreading) matrix V are given by: µ ¶ ∆ (p − q) (16) [V]p,q = sinc Nss Due to the fact that both the outer product Gτ GH τ and V are related by means of a Schur-Hadamard product, but also because both matrices exhibit a symmetric structure, the Doppler shaping matrix can be seen to mold the Likelihood function according to the prior distribution for the carrier frequency error. IV. A NALYSIS ON THE C YCLOSTATIONARY S IGNAL S TRUCTURE In this section, we will show that the cyclic autocorrelation function (CAF) of the received data is a sufficient statistic for the NDA timing estimation problem. As a consequence of the structural analysis of matrix Gτ GH τ , the log-Likelihood cost function is shown to be expressed as a weighted sum based on the received samples and the known pulse CAF, depending on the prior selected for modelling the carrier frequency uncertainty. Assume the signal model presented in (1) in the absence of timing nor carrier frequency errors, to be given by r(k) = s(k) + w(k) with

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P s(k) = +∞ n=−∞ xn g (kTs − nT ). Considering independent uncorrelated symbols, the second-order moment for the cyclostationary signal s(k) can be expressed by means of its time-varying autocorrelation as: Rs (k; m) = E [s(k)s∗ (k + m)] = σx2 Rg (k; m) and a periodic behavior is observed, as Rs (k; m) = Rs (k + lNss ; m) with l ∈ Z. This effect can be translated to the log-Likelihood cost function in (13), which retains the cyclostationary nature of the transmitted signal. In particular, the diagonal and sub-diagonal entries of the outer-product matrix Gτ GH τ in (14) can be found to be given by: i h = Gτ GH τ k,k+m

+∞ X

n=−∞

h

g (kTs + nT − τ ) g ∗ ((k + m) Ts + nT − τ )

Gτ GH τ

i

k,k+m

h i = Gτ GH τ

(17) (18)

(k+lNss ),(k+lNss )+m

with l ∈ Z and T = Nss Ts , which can be identified with the signal time-varying autocorrelation evaluated at the m-th time lag: ¤ £ Gτ GH τ k,k+m = Rg (k; m). Due to the periodic behavior previously mentioned, each of these diagonal entries can be expanded in terms of its Fourier Series (FS) expansion in k: i h X α = Rg (k − τ ; m) = Rg (m)ejα(k−τ ) (19) Gτ GH τ k,k+m

α∈A

with A = {−π ≤ α ≤ π}, and Rsα (m) given by the cyclicautocorrelation function evaluated at the m-th time-lag and α cyclefrequency: bα R g (m) =

M X 1 g (k) g ∗ (k + m) e−jαk 2M + 1

(20)

k=−M

The key question is that the FS expansion in (19) can be further simplified by noting that for linear modulations, Rα g (m) 6= 0 when l, l ≤ integer (1 + β), and β the pulse shape roll-off factor |α| = N2π ss [6]. In particular, the DC term (l = 0) does not provide any information on the timing parameter τ and hence, it can be directly discarded. Even harmonics can be found to be all equal to zero due to the pulse shape even symmetry, and finally, all odd harmonics except from the first one (l = ±1) are also found to be zero by recalling that we are dealing with a bandlimited signal, constrained to ±r = ±1/T , and that no aliasing is incurred. In this way, the diagonal and sub-diagonal entries of matrix M can be expressed, for estimation purposes, as pure tones that are shaped by matrix Gτ GH τ and the Doppler shaping matrix V, whose pattern depends on the prior considered for the carrier frequency uncertainty. · ¸ 2π 2π j 2π k (21) [M]k,k+m ≈ [V]k,k+m Re e−j T τ RgNss (m)e Nss [V]k,k+m = sinc

µ

¶ ∆ν . m = dV (m) Nss

(22)

Keeping these results in mind, the log-Likelihood cost function in (13) can be further manipulated in order to obtain a more intuitive expression. Representing matrix M as a Fourier series expansion of each of its periodic diagonal and sub-diagonal entries, it is found that: L (r|τ ) =

³ ´ C b = T r M R 4 σw

à " +M 2π X 2π C −j 2π τ T dV (0) RgNss (0) = 4 2 Re e |r(k)|2 ej Nss k + σw σx k=−M

¯ 2π ¯ +M−m X ¯ N ¯ X ∗ j 2π (k+ m ) ss ¯ 2 + dV (m) ¯Rg (m)¯¯ r (k)r(k + m)e Nss

m>0

k=−M

!# ¯ 2π ¯ +M−m X ¯ N ¯ X 2π (k+ m ) j ∗ ss 2 dV (m) ¯¯Rg (m)¯¯ r(k)r (k − m)e Nss

m