Sample Covariance Matrix Based Parameter Estimation for Digital Synchronization. Javier Villares and Gregori Vazquez. Department of Signal Theory and ...
Sample Covariance Matrix Based Parameter Estimation for Digital Synchronization Javier Villares and Gregori Vazquez Department of Signal Theory and Communications, Polytechnic University of Catalunya E-mail: havi,gregori}@gps.tsc.upc.es closed-loop estimators. Thus, the wanted parameter is modeled as a random variable with an a priori distribution @nor) containing the a priori knowledge ahout the parameter. In a covariance matrix. The estimator coefficients are optimized inorclosed-loop implementation, once the initial error has been acder to yield minimum mean squared error (MSE) estimates oftbe parameter. Some linear constraints are introdueed into the op- quired, the parameter error fluctuates around zero and, thus, a timization procesr allowing the designer to bave control over the very "narrow" prior can he deemed to design optimal parameestimator characteristie response. For those scenarios where bias ter discriminators in the steady-state. On the other hand, if an is forbidden, as it happens in ranging applicatiom, we provide the open-loop scheme is preferred with the aim of decreasing the optimal solutmo minimizingthe estimates bias within the range of the received parameter. The adopted approach is Bayesian as we initial acquisition time, the prior domain must be extended in treat the wanted parameter as a random variable with P known a order to cover the total initial uncertainty. When the estimator is intended for ranging applications, the p"o" probability distribution (prior). This modeling allows us to unify the design of both open- and closed-loop estimators. absence of bias at the estimator output is a desirable feature. The proposed formulation encompasses all the linear modula- However, as shown in [5], it is not possible to force the lintions as well as the binary Continuous Phase Modulation (CPM). The new approach supplies optimal estimation schemes without earity of the characteristic curve except in a limited number of the need of assuming a given statistics for the unknowo symbols, points (or near its origin). This limitation comes from the nonthat is, avoiding the common adoption of the gaussian assump- linear dependence of the received signal covariance matrix with tion, which does not apply in digital communications. Special at- the input parameter. In this paper, we recognize this obstacle tention is paid to those low-complexity implementationsfor which and we focus on the minimization of the overall open-loop esthe Maximum Likelihood efficiencyis not guaranteed. timator bias. At this point, the maximum bias cancellation is Indrx TermsSyochmnization, parameter estimation, cantinachieved by imposing some constrains on the optimization pmuous phase modulation, non-data-aided, carrier and timing recovcedure. ery. The structure of the paper is the following. The signal model and the problem statement are presented in Section 11. In sec1. INTRODUCTION tion Ill we introduce the estimator formulation based on the linear processing of the received sample covariance matrix. The In the synchronization field most techniques have been prosecond-order approach is founded on the Stochastic Maximum posed so far from an heuristic reasoning and conceived for a specific modulation format [I]. This fact precludes most times Likelihood which is discussed in the section. In section N we (or makes it really difficult) from adapting these techniques deduce the estimator mean squared m o r (MSE) and, in secto new transmission schemes or to translate them to solve the tion V, we obtain the estimator coefficients minimizing the bunch ofdifferent estimation problems that, nowadays, modem Bayesian Risk defined as the estimates MSE weighted by the digital communications are stating. In [2][3] the authors pre- parameter prior. In this section we also allow the introducsented a general framework allowing the formulation of any tion of some linear constraints on the estimator mean response. NDA synchronizer based on second order moments under a Next, in section VI we address the design of open-loop estimaMaximum Likelihood (ML) perspective. There are two basic tors. In this section we find the constraints minimizing the esreasons for limiting the analysis to quadratic synchronizers. timator bias within the parameter prior interval and the optimal It is shown [Z] that the NDA stochastic ML solution becomes minimum-bias open-loop estimator is deduced. Section VI1 dequadratic for low SNRs, whereas it is still unknown for moder- duces the closed-loopestimator that minimizes the parameter ate to high SNRs because the difficult treatment of the unknown tracking variance in the steady-state. Simulations results and transmitted symbols. The second reason is that quadratic algo- their comments can be found in Section VllI and. finally, conrithms allow efficient digital implementations. clusions are drawn in Section IX. In this paper, we propose a different approach to the design of second-order NDA estimators without resorting to the com11. DISCRETE-TIME SiGNAL MODEL mon gaussian assumption about the statistical distribution of the nuisance parameters [4]. Fuqhermore, we have adopted a The noisy samples of the received complex envelope for any Bayesian approach in order to unify the design of open- and linearly modulated signal can be expressed as follows: Abstmcf-In this paper we develop a new, versatile framework for the design of optimal Non-Data-Aided (NDA) parameter estimators based on the exploitation of the received signal sample
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where g ( t ) is the shaping pulse, xi = cieje are the nuisance parameters (or pseudo-symbols) with ci the transmitted symbols of a given alphabet and 9 the camer-phase error, T is the symbol period, T, is the sampling period and N., = TJT, is an integer number of samples per symbol, such that WT. < 1 with W the two-sided bandwidth of the shaping pulse g ( t ) . Finally, the additive zero-mean noise term w(nT.) is assumed to be gaussian distributed with a known covariance sequence. In (I) we have also included the effect of the synchronism errors under consideration, i.e., the carrier frequency offset v , that is constrained to the Nyquist bandwidth [-1/2Ta,+1/2T,) and the timing error T that takes values in the interval [-TJ2, +T/2). Finally, let us recall that binary Continuous Phase Modulations (CPM) can be seen as the superposition of a set of linearly modulated signals (1) by resorting to the Laurent's expansion [ 6 ] . Therefore, the Laurent's decomposition allows to extend the model in ( I ) to any binav CPM modulation. For more details, the reader is referred to [2]. For convenience, we will adopt a vectorial notation. Let r stand for the column vector containing the N observed samples. Vector r can be expressed in a compact form as follows:
r = A A X +w
(2)
where the columns of the linear transfer matrix A Aare those T-seconds delayed versions of the shaping pulse g ( t ) involved into the current N-samples block and X is the wanted parameter to be estimated, parameterizing matrix A A ,that is, the carrier frequency error u,or the timing error r . Then, the expected value of the sample covariance matrix RA= rrH is given by:
,.
RA= E {RA} = A
A r A A H
+ C,
(3)
where ( . ) H stands for the matrix transpose conjugate operator, I7 = E {xx"} is the pseudo-symbols covariance matrix and C , = E {ww"} is the noise covariance matrix. Matrix r allows to easily incorporate into the model the pseudo-symbols correlation, which takes place in the case of correlative encoding modulations (e.g., duobinary transmission), or in the presence of a frequency-selective channel, as studied in [7]. 111. STOCHASTIC MAXIMUM LIKELIHOOD APPROACH
The classical stochastic approach to the NDA estimation problem consists in maximizing the marginal likelihood function f (./A) with respect to the argument X as indicated below:
showed that for the carrier frequency error or the timing error estimation problems of any linear modulation (including DS-CDMA and Multi-Camer modulations) and of any binary CPM modulation, the estimation scheme for low SNRr become quadratic on the received data vector r. Thus, we can state that any parameter estimator working in a low SNR scenario is kased on the linear processing of the sample covariance matrix RAand, in general, is given by:
i= ~ " M T= T ~ ( M R ~=)M ~ R ~( 5 ) for a given weighting matrix M and where the calligraphic notation will be used in the sequel denoting matrix vectorization, that is, = uec(RA) = uec(rr") and M = wec(M"). The main drawback when following a low SNR approach is that the design of the estimator will not take into account the contribution of the self-noise to the estimator variance. The effect of the self-noise term is specifically important when considering higher operating SNRs. In the counterpart, some quadratic solutions have been obtained to avoid the presence of self-noise for high SNRs at the expense of certain noise enhancement for low SNRs [81[91[lO]. Following the simple general quadratic expression given in ( 5 ) we will try to design the weighting matrix M such that a minimum mean squared error (MSE) performance is achieved for a given distribution of the unknown parameter. A trade-off between the noise and the self-noise contributions will be set by the estimator for a given working SNR.
e~
-
IV. ESTIMATOR MEANSQUARED ERROR (MSE) Using the notation defined in the last section, any generic quadratic estimator of the parameter X can be written as a linear processing ofthe sample covariance matrix. The mean squared error (MSE) of the quadratic estimator in ( 5 ) is given by:
c* (A)
=E
{i,i - XI*}
=M
H Q ~ M - M H ~ ~ ~ - ~ ~ ~ M + ~ (6)
where the expectation is performed with respect to the noise w and the unknown pseudo-symbols x, matrix Q A = E {RA'@} is the correlation matrix of the sample covariance vectorRAand%A = E{X*RA}. The matrix QA in ( 5 ) is composed of the fourth-order moments of the received signal r. For any rotational symmetric constellation holding E{xilzil*"} = E{zflziI*") = 0 (Vi,n). QAadmits the following closed-form expression:
{ - 1+(B;8 B A ) K ( B ; @ B A ) ~
Q~ = E R& The main problem of this approach is the difficulty associated to the analytical formulation of the marginal distribution E, { f ( r J x ,A)] being only known for a few number of simple linear modulations. To circumvent this important limitation, a low SNR approximation of (4) was obtained in [2]. In this work, the authors
-R~R?+R:BR~+
(7)
where BA= A A r l l zwith P/2rH/2 = r, 0 stands for the Kronecker product and (.) * is the conjugate operator. In equation (7) we have introduced the modulation fourthorder cumulant matrix K which is given by:
K = E{vec(ssH)uecH(ssH)} -ZZH - I
464
(8)
where the inversion of Q is guaranteed if the noise covariance matrix C , is positive definite. On the other hand, the Moore-Penrose pseudo-inverse (.)# has been introduced in (12) in order to coverthose cases in which C H Q - ' C is rankdeficient. Actually, depending on whether the system of equations C"M = k is underdeteiminedor incompatible, equation (12) provides the minimum-norm or the least-squares solution to C H M = k,respectively. Next section addresses the problem of choosing C with the aim of controlling the open-loop estimator bias within the range of the unknown parameter.
where s is a whitened version of the pseudo-symbols vector x = l"I2s, I denotes the identity matrix and Z its vectorization. It is well-hown that K would vanish if the pseudosymbols were normally distributed. However, this does not happen in digital communications and matrix K provides the complete non-gaussian information about the discrete pseudosymbols that second-order estimators ( 5 ) are able to exploit. In the case of linear modulations, such as M-PSK, QAM or, in general, APSK, matrix K reduces to:
K = ( p - 2)diag(Z)
(9)
VI. OPEN-LOOPDESIGN In (1 2) we deduced the minimum BR estimator under certain linear constraints. If no constraints are imposed (a_ = 0), the estimator coefficients, and the the resulting estimator BR, are the following:
(Vi)is the fourthwhere the scalar p = to second-order moment ratio, which is specific of the modulation under consideration, and diag(.) converts a vector into a diagonal matrix. In the case of the binary CPM modulation, a general closed-form for K is not feasible because of the symbols dependence introduced at the CPM modulator. Anyway, K can be easily obtained element-by-element or, preferably, averaging a few independent outcomes of the random vectors.
-
M = Q-'R
{
EAE I
= % - EHQ-'%
(13)
where the negative term gHQ-'7? is the maximum BR reduction any second-orderestimator can yield with respect to the original uncertainty X1. The above open-loop estimator will yield biased estimates whenever this bias contributes to minimize the overall ER. In some applications (such as positioning) bias is not tolerated and, thus, the objective is to minimize the estimates bias. The estimator squared bias is given by the following expression:
V. ESTIMATOR OPTIMIZATION. BAYESIAN RISK (BR) MINIMIZATION Returning to equation (9,the design of an open-loop estimator for the parameter acquisition or a closed-loop discriminator for the parameter tracking can be unified if we follow a Bayesian approach, that is, we model the unknown parameter as a random variable with an a priori distribution (prior) ~ A ( X ) . We restrict the present study to even priors, that is, ~A(-A) = ~ A ( Xwhich ) is a common assumption in digital synchronization. The Bayesian Risk (ER) is then formulated by weighting the estimates MSE in (10) by the parameter prior, that is:
5 2 = ~ h { ~ 2 ( ~ ) } = ~ H ~ ~ -
i - XI'}
{ I M ~ -Rxiz}~ = M ~ Q M- M % - i i H+~ R (14)
where we have introduced the matrix Q = EA{ R A R ~ with } RA= vec(RA) the vectorization of the signal covariance matrix. The estimator coefficients minimizing the estimator bias (IO) ~ n E - E H ~ + R (14) are those holding the following equation:
for Q = EA( Q A }= EAE{%A%F}, % = EA{EA}and
QM=%
-
(15)
X2 = EA{ IXlz} the aprion parameter variance. It is found that equatcn (15) is underdetermined because Then, our goal is to find the estimator coefficients in M that belongs to the span of Q which is rank-deficient. The minimum minimize the estimator Bayesian Risk in (IO). Additionally, we BR estimator holding (1 5 ) can be obtained by replacing C = can introduce some constraints into the optimization procedure Q and k = 3 into (12) and, thus, we have that: in order to define the estimator mean response, that is, E{ i). The solution to this optimization problem requires the miniM = Q-'%+P QQ-') (16) mization of the following cost function with respect to the estimator coefficients: where we have made use of the hermitian symmetry of Q and
(I-
C ( M ) = M H Q M - M H E + (kH - M H C )a
(11)
where is the vector of Lagrange multipliers imposing the set of linear constraints C H M = k. The minimization of (1 I ) yields the following general expression for the parameter estimator:
I?.
introduced the matrix P = Q-'Q (QQ-'Q)# for the sake of clarity. Finally, if we substitute (16) into the bias expression in equation (14), we can compute the minimum (squared) bias corresponding to any estimator holding equation ( I 5 ) :
EA{IMHR*- XI2} = R -
eHP%
(17)
where-we have taken into account that QPQ = and Q P H R = R. Notice that matrix P in (17) can he replaced by any other matrix satisfying these two conditions, e.g., Q#.
465
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4 Estimator (squand) bias as a function of the parameter range AA (uniform prior). The cowrained OL plot is that obtained imposing the closedloop consminfs in (18) to the minimum MSEopen-Imp estimator in (12). Fig. 1.
VII. CLOSED-LOOP DESIGN
In this section the estimator is required to detect the parameter fluctuations omund X=O with minimum variance and no bias. The optimal parameter discriminator can be obtained from the results in section V if we adopt a impulsive prior at the origin and impose three constraints about the estimates bias around X=O as follows: f A ( N = J(A)
C = [ % So
701
; k H = [ O1 0 1
(18)
where S(.) is the Dirac delta function and R o is the (stacked) mean covariance matrix for a null value of A. Likewise, SO and 70 stand for the (vectorized) first and second derivative of the covariance matrix with respect to the parameter, that is, SA= &RAand h = &RA , evaluated at X = 0. The impulsive prior allows to simplify (12) taking into account that % = 0 and hence:
M = Q;'C (CHQ;'C)-' k
(19)
The closed-loop constraints in (18) force the S-curve linearity, E { x } = A, in the neighborhood of X = 0. In fact, the S-curve slope (second constraint) controls the loop bandwidth and has been set to 1with the aim of comparison. 'It can be shown that the first and third constraints, M H % = M H Z = 0, are always fined due to the discriminator symmetry and, thus, the closed-loop solution reduces to:
and, if (20) is plugged into (IO) with x'i = 0, the discriminator tracking error variance can be written as:
Fig. 2. Estimatar MSE as a function of the input parameter range AAconsidcnng an uniform prior.
which constitutes a new lower-hound for the variance of any unbiased estimator based on the second-order statistics of the received signal. It can he shown that the S-curve odd symmetry, which made possible the odd constraints removal in (18), is due to the discriminator hermitian symmetry in ( 5 ) in the case of camer recovery, whereas it is caused by the shaping pulse even symmetry for the timing synchronization case. Finally, it is worth noting that the minimum bias condition stated in (15) reduces to the closed-loop constraints given in (1 8) when the parameter range A A tends to zero. VIII. SIMULATION RESULTS The simulations have been carried out for the MSK (Minimum Shift Keying) modulation as a particular case of the binary Continuous Phase Modulations CPM. This transmission scheme is adopted because it allows a simple extension to linear modulations as well as multiple access modulations [I I]. Recall that the Laurent expansion [6] [ I ] allows the formulation of binary CPM signals in terms of the model presented in Section 11. The simulations have been done for additive white gaussian noise (AWGN) and two samples per symbol ( N s s = 2). The signal-to-noise ratio (SNR)has been set to 10 dB and the observation window has been limited to N = 4 samples. The parameter prior for open-loop estimators is uniform within the given parameter range A A .We have focused ou the carrier frequency offset estimation problem as an illustrative case. - Figure 1 and 2: Fig. 2 shows the MSE degradation when the estimator proposed in (16) is forced to minimize its overall bias (Fig. 1). On the other hand, Fig. 2 shows the convergence of the estimator in (16) to the closed-loop solution (20) when the parameter range A A approaches zero. - Figure 3 and 4: Fig. 3 shows clearly how the minimum bias solution in (16) cannot force the linearity of the estimator
466
Fig. 3. Characteristic CYWS of the minimum bias open-loop estimatm deduced in (16) for different d u e s of the parameter m g e 4.The closed-loop BS well BS the vncanswined (minimum MXE) open-loop sol~liona x ploncd for comparison. Notice that we have opted to only plot the positive semi-axis becaws of the c h c t s r i r t i c curves odd symmcny
Characteristic curve when the initial uncertainty is a hit greater than 0.5/T. In those cases the minimum bias solution (16) distributes the bias throughout the whole prior range in order to achieve the overall bias minimization for a given observation length (see Fig. 3). The unconstrained open-loop estimator formulated in ( I 3) is shown to yield a better MSE performance within the given parameter range because it is not forced to deliver unbiased estimates. Actually, the open-loop estimator in (13) makes a trade-off between variance and bias in order to minimize the global USE. Finally, Fig. 4 deserves two more comments. Firstly, notice that the USE open-loop curves are not flat within the uniform parameter range, as expected, because of the short sample ( N 4 ) [ 5 ] . Secondly, Fig. 4 manifests the minor improvement that closed-loop schemes offer with respect to open-loop implementations when used in low-SNR scenarios (51, hence motivating the open-loop study.
(Bayesian Risk). Simulations showed that the unconstrained solution outperforms the constrained one because the former is able to make a trade-off between bias and noise variance. For the closed-loop case, an optimal (unbiased) discriminator is obtained whose tracking variance is a lower bound for NDA quadratic unbiased estimators even for short samples.
IX. CONCLUSIONS This paper presented a new, versatile approach for designing both open- and closed-loop optimal estimators following a Bayesian approach. The paper focuses on blind parameter estimation hut the extension to assisted schemes is straightforward. The estimator is demanded to minimize the Bayesian Risk, that is, the expected value of the mean squared error with respect to the apriori dishihution of tbe parameter. The formulation is rather general and encompasses most modulation formats without having to adopt any assumption on the unknown pseudo-symbols statistics. For the open-loop case, we obtained the optimal estimator of the parameter having minimum bias within the input parameter interval by imposing some constraints upon the cost function
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