we derive an asymptote-based lower bound which provides an interesting alternative between accuracy and computational cost. I. INTRODUCTION.
´ BAYESIAN CRAMER-RAO BOUND FOR DYNAMICAL PHASE OFFSET ESTIMATION S. Bay(1) , C. Herzet(2) , J.M. Brossier(3) , J.P. Barbot(1) , A. Renaux(4) and B. Geller(1) (1)
SATIE, ENS Cachan, 61 av du President Wilson, F-94230 Cachan, France Dept. of EECS, UC Berkeley, Wireless Foundations, 273 Cory Hall, Berkeley, CA 94720, USA (3) LIS, INPG, 961 rue de la Houille Blanche, BP 46, 38402 St. Martin d’Heres cedex, France Washington Univ St. Louis, Campus Box 1127, One Brookings Drive 63130 St. Louis, Missouri, USA (2)
(4)
ABSTRACT In this paper, we present a closed-form expression of a Bayesian Cram´er-Rao lower bound for the estimation of a dynamical phase offset for a random BPSK sequence in an AWGN channel. The received symbols are disturbed by carrier phase offset which follows a Wiener model. Considering a received observation sequence, we provide a first analytical expression of the Bayesian CRB. Then, we derive an asymptote-based lower bound which provides an interesting alternative between accuracy and computational cost. I. INTRODUCTION Synchronization has recently become one of the most challenging task of digital receivers. Due to the high data-rates requirements of the modern communication standards, phase synchronizers have to estimate rapidly time-varying phase parameters [1]. In order to evaluate the estimator performance, lower bounds on the Mean Square Error (MSE) are needed. One of the most used bound is the Cram´er-Rao Bound (CRB) [2]. Considering dynamical parameter estimation, the unknown parameters can no longer be regarded as ”deterministic” and the performance analysis requires the derivation of the Bayesian CRB (BCRB) [3]. This bound is sometimes difficult to derive, particularly in the case of nuisance parameters, and a Modified BCRB (MBCRB) easier to evaluate has been proposed in [4]. The problem of deriving BCRBs suited to time-varying parameters has been recently addressed. In [5], the authors propose a general framework for deriving analytical expression of the socalled on-line or filtering CRBs. In [6], the authors developed a numerical graph-based algorithm to evaluate the BCRB for timevarying scenarios. In this contribution, we focus on the original Van Tree’s BCRB [3]: we address the open problem of deriving an analytical expression of the off-line, also called smoothing BCRB for timevarying phase estimation in Non-Data-Aided (NDA) scenarios. Explicit expressions of the bound and its modified version are provided. This bound helps us to evaluate and also to predict the estimator performance without any particular assumption or simulation restriction. The asymptotic cases at low and high Signalto-Noise Ratio (SNR) are presented. In particular it is shown that the Modified BCRB is equal to the derived asymptote at high SNR. We finally propose an Asymptotic BCRB which is lower than the BCRB but more accurate than its classical modified version. This paper is organized as follows. In Section II we set the system model. In Section III we present the BCRB and its modified version. Then, we derive the off-line BCRB in Section IV. In Section V the asymptotic cases are considered and lead to the Asymptotic BCRB. Finally, the different results are illustrated and interpreted in Section VI and a conclusion is given Section VII. The notational convention adopted is as follows: italic indicates a scalar quantity, as in a; boldface indicates a vector quantity, as in a and capital boldface indicates a matrix quantity as in A. Entry (k, l) of matrix A is denoted [A]k,l . The matrix transpose operator
1-4244-0955-1/07/$25.00 © 2007 IEEE.
is indicated by a superscript T as in AT . |.| denotes the determinant of a matrix. < {A} and = {A} are respectively the real and the imaginary part of A. The actual value of a parameter is indicated by the superscript ∗ as in A∗ . Ex [.] denotes the expectation operator with respect to the probability density function (pdf) of x. bK is equal to the integer part of K. δk,l is the Kronecker symbol. ∇θ and ∆θψ represent the first and second-order partial derivatives h iT operator i.e. ∇θ = ∂θ∂1 · · · ∂θ∂K and ∆θψ = ∇ψ ∇Tθ . The error R 2 x function is erf (x) , √2π 0 e−t dt. II. MODEL We consider the transmission of a BPSK modulated sequence a = [a1 · · · aK ]T over an Additive White Gaussian Noise (AWGN) channel affected by a carrier phase offset θk . Assuming that the received signal has been ideally filtered and sampled at the optimum sampling instant, the discrete-time baseband signal is given by yk = ak ejθk + nk
with k = 1 . . . K,
(1)
th
where ak is the k unknown transmitted BPSK symbol (ak = ±1) and nk is a zero-mean circular Gaussian noise with a known Es variance σn2 . Since the energy per symbol is normalized, N = 0 1 . We consider that the system operates in NDA synchronization 2 σn mode, i.e., the transmitted symbols are independent and identically distributed (i.i.d.) with p(ak = ±1) = 21 . We will use the Wiener phase-offset evolution model commonly used (see e.g. [7]) to describe the behavior of practical oscillators, θk = θk−1 + wk ,
(2)
where wk is an i.i.d. zero-mean Gaussian noise with a known 2 2 variance σw . In practice, σw ¿ 1. III. THE BCRB AND ITS MODIFIED VERSION In this section we first present the general expression of the BCRB [3]. Then we introduce the Modified BCRB (MBCRB) which is generally simpler. Van Trees [3] shows that any estimator ˆ θ(y) of a random parameters vector θ is bounded by the inverse of the Bayesian Information Matrix (BIM), say B, as follows £ ¤ ˆ ˆ Ey,θ (θ(y) − θ) (θ(y) − θ)T ≥ B−1 . (3) The BIM can be written with the expected value of the Fisher Information Matrix (FIM) F(θ) [2] with respect to the a priori pdf p (θ) £ ¤ £ ¤ B = Eθ F(θ) + Eθ −∆θθ log p(θ) , (4) £ ¤ θ F(θ) = Ey|θ −∆θ log p(y|θ) . (5) The first term of (4) can be interpreted as the average information with respect to θ brought by the observation y; on the other hand,
the last term can be regarded as the information available from the prior knowledge on θ, i.e., p(θ). This term actually accounts for the time dependence between phase offsets at different instants. The practical evaluation of the BIM can be quite tedious. In order to circumvent this problem a MBCRB has been considered in [4]. The MBCRB comes from the inversion of the following £ ¤ £ ¤ matrix C C = Eθ G(θ) + Eθ −∆θθ log p(θ) , (6) £ ¤ θ with G(θ) = Ea Ey|a,θ −∆θ log p(y|a, θ) , (7) where G(θ) corresponds to the modified FIM. IV. THE OFF-LINE BCRB AND ITS MODIFIED VERSION FOR DYNAMICAL PHASE ESTIMATION In this section, we derive an analytical expression of the BCRB associated to off-line carrier-phase-offset estimation. For this scenario, the receiver processes all the received observations to compute the estimates of the phase offsets. We first obtain analytical expressions of the two terms contributing to the BIM (4). Then, the expression of the diagonal elements of the BCRB is derived. In order to have a full characterization of the system performance, we propose an analytical expression of the MBCRB. IV-A. Computation of Eθ [F(θ)] The evaluation of Eθ [F(θ)] requires the computation of F(θ). Using the observation model defined in Section II, the loglikelihood function can be expanded X as log p(y|θ) = log p(y|a, θ) p(a). (8) a
Using the whiteness of the noise and the independence of the transmitted symbols, one then obtains that K X ∆θθ log p (y|θ) = ∆θθ log p (yk |θk ) . (9) k=1
It is important to note that each term of the sum (9) is a matrix with only one non-zero element, namely, h i ∂2 ∆θθ log p (yk |θk ) = log p (yk |θk ) . (10) ∂θk2 k,k As a direct consequence, ∆θθ log p (y|θ) is a diagonal matrix with the kth diagonal element given by equation (10). Moreover, because of the phase model and the Gaussian nature of the noise, p (yk |θk ) is invariant with respect to the time index k. Then, one has that Eθ [FK (θ)] = JD IK ,
(11)
where IK is the K ×K identity matrix and JD is defined as follows · ¸ ∂ 2 log p (yk |θk ) JD , Ey,θ − . (12) 2 ∂θk IV-B. Computation of Eθ [∆θθ log(p(θ))] Due to the Wiener structure of the phase model (2), ∆θθ log p (θ) can be expanded as K X ∆θθ log p (θ) = ∆θθ log p (θ1 ) + ∆θθ log p (θk |θk−1 ) . (13) k=2
The first hterm is a matrix one non-zero element which is i with only 2 p(θ1 ) equal to ∆θθ log p (θ1 ) = ∂ log . The other terms in (13) ∂θ 2 1,1
1
are matrices with only four non-zero elements, namely, (k − 1, k − 1), (k − 1, k), (k, k − 1) and (k, k). Due to the Gaussian nature of the noise, one finds that: h i h i −1 ∆θθ log p (θk |θk−1 ) = ∆θθ log p (θk |θk−1 ) = 2 σw k,k k−1,k−1 h i h i 1 θ θ ∆θ log p (θk |θk−1 ) = ∆θ log p (θk |θk−1 ) = 2 σw k,k−1 k−1,k
With these previous expressions, one finally obtains − Eθ [∆θθ log p(θ)] = i h 2 1 ∂ log p(θ ) 1 2 − E θ1 2 σ ∂θ1 w −1 2 σw 0 .. . 0
−1 2 σw
0
2 2 σw
−1 2 σw
..
..
. .
...
..
...
.
−1 2 σw
..
.
..
.
2 2 σw −1 2 σw
0
0 .. .
(14) 0 −1 2 σw 1 2 σw
In the sequel, we assume a classical non-informative i prior on θ1 h 2 p(θ1 ) (see e.g., [5]). As a consequence, Eθ1 ∂ log = 0. 2 ∂θ 1
IV-C. Analytical Expression of the Off-Line BCRB In this subsection, an analytical expression of the diagonal elements of the BCRB is displayed. These elements lower-bound the MSE achievable by any off-line estimator of the time-varying phase offset θk ’s.From (11) and (14), the BIM can be written as A+1 1 1 A 1 . . . , .. .. .. BK = b (15) 1 A 1 1 A+1 2 2 where b , −1/σw and A , −σw JD − 2. Based on the symmetric sparse structure of BK , one finds in Appendix I that the kth diagonal element of B−1 K can be expressed as h £ −1 ¤ 1 BK k,k = ρ21 (b + r1 )2 r1K−3 + ρ22 (b + r2 )2 r2K−3 |BK | i b2 − (r1k−2 r2K−k−1 + r1K−k−1 r2k−2 ) , (16) A−2 where ´ ´ p p b³ b³ r1 , A + A2 − 4 , r 2 , A − A2 − 4 , (17) 2 2 √ √ A2 − 4 + A A2 − 4 − A √ √ , ρ2 , , (18) ρ1 , 2 A2 − 4 2 A2 − 4 h i |BK | = (A + 2) b ρ1 (r1 )K−1 + ρ2 (r2 )K−1 . (19)
Note that the practical evaluation of (16) requires the calculus of JD which is detailed in Section V. IV-D. The Off-line MBCRB We now consider the MBCRB, (see equation (6)). The second term in the side of (6) is given by equation (14). The first £ right-hand ¤ term Eθ G(θ) requires the evaluation of G(θ) which corresponds to the modified FIM defined in (7). Using the observation model, one has that ∆θθ log p(y|a, θ) is a diagonal matrix where ∂2 [∆θθ log p(y|a, θ)]k,k = log p(yk |ak , θk ). (20) ∂θk2 Due to the gaussian distribution of the noise, one further has ´ ∂2 2 ³ ∗ −jθk log p(y |a , θ ) = < −a y e . (21) k k k k k ∂θk2 σn2 2 (22) [G(θ)]k,k = 2 . Then, it follows that σn £ ¤ Straightforwardly, we have Eθ G(θ) = 2/σn2 IK . Hence, the modified BIM is obtained exactly like the BIM previously derived in subsection IV-C with JM , σ22 playing the role of JD in (15). n Note, however, that the MBCRB is usually looser than the BCRB.
V. EVALUATION OF JD AND ASYMPTOTIC CASES In this section we first show that, in the general case, the evaluation of JD implies to resort to numerical integration. We then derive high-SNR and low-SNR asymptotes of the BCRB, and emphasize that their evaluation is straightforward. Finally, we show that these asymptotes are themselves lower bounds on the MSE. In particular, we illustrate that the combination of low and high SNR asymptotes leads to an alternative and tight lower bound, whose evaluation is straightforward. V-A. Evaluation of JD In this part, we calculate an expression of JD defined in (12). First, using the Gaussian nature of the noise and the equiprobability of the data symbols, one finds that µ ¶ 2 k| 1 − 1+|y 2 ³ −jθk ´ 2 σn log p(yk |θk ) = e cosh < y e . (23) k πσn2 σn2 Taking the second derivative of (23), one easily obtains that ³ 2 ´ ∂ 2 log p(yk |θk ) 2 = − 2
