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44 ADVANCES IN ELECTRONICS AND TELECOMMUNICATIONS, VOL. 1, NO. 1, APRIL 2010 IV. MAXIMUM ENTROPY CHANNEL ESTIMATION The essential derivations of this section will consist in establishing, at time t, the posterior probability p(ht|yt, yt−1, . . . , y1, I) (14) For readability we only treat the case t = 2, the general case being a trivial extension. Assume only pilots are transmitted or, at least, that the information about I carried by the nonpilot signals are rather uninformative. Discarding the terms I for readability, we have in that case p(h2|y2, y1, I) = p(h2|h ′ 2 , h′ 1 , I) (15) = p(h2)p(h ′ 2 |h2)p(h ′ 1 |h2) p(h ′ 1h′ 2 ) (16) = p(h2)p(h ′ 2 |h2) � h1 p(h′ 1 |h1)p(h1|h2)dh1 p(h ′ 1 h′ 2 ) (17) When the exact time evolution model for ht is known, p(h1|h2) has an explicit expression which allows then to perform the above calculus. In practice however, it is rarely the case that such a time-evolution model be perfectly known. One then needs here an automatic method to provide a consistent expression for p(h1|h2, I) when little is known about the interaction between h1 and h2. The method we shall use here is referred to as the maximum entropy principle [8]. Consider a given parameter x, whose knowledge is limited to the information contained in I. The maximum entropy principle allows one to assign a unique probability distribution p(x|I) as follows, 1) among the set of all probability distributions, consider those distributions that satisfy the constraints about x given in I. This is, we exclude all distributions that do not satisfy the prior information I. The remaining set of probability distributions is denoted Q. 2) in this remaining set of acceptable distributions Q, p(x|I) is assign the distribution which has maximum entropy, i.e. p(x|I) = arg max q∈Q − � q(y) log q(y)dy (18) Choosing the distribution that has maximum entropy allows one not to make undesired assumptions on the unknown system variables, and as such allows one to remain neutral with regard to unaccessible information; see [11] for further details on the maximum entropy principle. In the model described in Section III, since only the total power is known about both the channel delay profile and the additive noise, both are assigned Gaussian distributions with zero mean and variance consistent with prior knowledge, as required by the maximum entropy principle. We then have nt ∼ CN(0, σ2IN ) and νt ∼ CN(0, 1 LIL), which in the frequency domain, applying Fourier transform, translates into ht ∼ CN(0, Q), with Q defined as Qnm = E � L−1 � L−1 � νkν k=0 l=0 ∗ l kn−lm e−2πi N � = 1 L−1 � L k=0 n−m −2πik e N (19) Note that Q is singular, since L < N as OFDM requires. If the coherence time function λ(τ) is perfectly known, the maximum entropy principle then assign to p(ht|ht+τ ) a Gaussian distribution of mean λ(τ)ht+τ and variance E[hth H t+τ ] = (1 − λ(τ)2 )Q (20) In the problem with t = 2, we denote λ = λ(1). We then find that p(h1|h2, I) is Gaussian and satisfies p(h1|h2) = lim ˜Φ→Φ 1 π N det( ˜ Φ) e−(h1−λh2)H ˜ Φ −1 (h1−λh2) (21) for { ˜ Φ} any sequence converging to Φ. This allows then to compute the full expression of p(h2|y2, y1, I) and a closedform expression of some conventional estimators. In particular, the MMSE estimator ˆ h (MMSE) 2 for h2, defined as in this case expresses as [7] ˆh (MMSE) 2 =M −1 2 with M2 satisfying ˆh (MMSE) 2 = E [h2|y] (22) � P2h ′ 2 σ2 +(IN + 1−λ2 −1 λ P1Q) σ2 σ2 P1h ′ � 1 (23) QM2 = IN λ2 − 1 − λ2 1 − λ2 (IN 1 − λ2 + σ2 QP1) −1 + QP2 σ2 (24) This expression generalizes to t ≥ 1, for which we have the MMSE estimator ˆ h (MMSE) t given in Equation (25). However it often occurs that the assumption that L and λ(1), λ(2), . . . are known a priori at the receiver is not realistic. In truth, rather limited information is known a priori on these parameters. We may then reconsider (17) to include the uncertainty on L and/or λ(1), λ(2), . . .. In the previous t = 2 setup, we have in particular ˆh (MMSE) 2 = E[h2|y2, y1, I] (26) � = p(λ|I)p(L|I)E[h2|y2, y1, λ, L, I]dλdL (27) λ which leads to yet other expressions derived thoroughly in [7]. Through the OFDM example, we therefore developed a rather automatic method to derive consistent estimators under any prior knowledge at the receiver which we claim optimal on an information theoretic viewpoint, i.e. those estimators are taking into consideration all prior information I and are made such that no ad-hoc assumption is taken regarding imperfectly known parameters, while being compliant with the Bayesian principles. However, an underlying assumption of the previous approach is that infinite storage is available at the receiver. Indeed, for large t, we still need to consider all events from time instants 1 to t; if we decide to discard the oldest data, we then depart from the optimality of the proposed scheme. We then need to reconsider the whole framework to include an additional feature: the receiver is oblivious to part of the past events. The natural way to handle this modification of the current maximum entropy framework is to consider updated distribution assignments for posterior probabilities instead of absolute distribution assignments. This is the topic of the next and our main section.
COUILLET et al.: BAYESIAN FOUNDATIONS OF CHANNEL ESTIMATION FOR SMART RADIOS 45 ˆh (MMSE) t = tX λ(k) 1 + k=1 2 1 − λ(k) 2 ! tX λ(k) IN − k=1 2 1 − λ(k) 2 „ 1 − λ(k)2 IN + σ2 QPk «−1 !−1 Q V. MINIMAL CHANNEL ESTIMATION UPDATE A. Introduction to Bayesian minimal update When it comes to update probability assignments, Caticha proposes an extension of the maximum entropy principle, namely the minimum cross entropy principle (ME) [9]. When p(ht|I1) has been assigned for some side information I1, and new cogent information I2 is later available, then the ME principle consists in assigning to p(ht|I2) the distribution where p(ht|I2) = arg min q S[q, p(ht|I1)] (28) � � � p(x) S[q, p] = q(x) log dx (29) q(x) The functional S[q, p] is referred to as the cross-entropy between the probability distributions q and p. This method is based on a minimal update requirement, which in essence assigns to p(ht|I2) the unique distribution which minimizes the changes brought to p(ht|I1) while satisfying the new constraints given by I2. In the following, additional side information on ht (which possibly varies over time) will come from new available pilots at later time positions. B. Perfect system parameters knowledge We assume here that channel estimation is performed at different time instants t = 1, 2, . . .. Denote Ik the knowledge at time k. Since memory restrictions impose to discard past received data, we decide here only to consider at time k the last received pilot data symbols, the last assigned probability p(hk|Ik−1), and the supposedly known time correlation λ = λ(1) between the current channel hk and the past channel hk−1. Assume prior assigned distribution p(hk−1|Ik−1) at time index k − 1. We have in general p(hk|yk, Ik) (30) = p(yk|hk, Ik) · p(hk|Ik) p(yk|Ik) (31) = p(yk|hk, Ik) · � p(hk|hk−1, Ik) · p(hk−1|Ik)dhk−1 p(yk|Ik) (32) We use Caticha’s ME principle [9] and set p(hk−1|Ik) to the previous p(hk−1|yk−1, Ik−1). The reason lies in the minimal update principle: if no additional information is given in Ik, compared to Ik−1, then p(hk−1|yk−1, Ik−1) is the distribution q that minimizes the cross-entropy S[q, p(hk−1|yk−1, Ik−1)] 1 . 1 if new statistical information comes in, the minimum cross-entropy distribution with p(hk−1|Ik) satisfying those new constraints should be computed and used instead. tX k=1 „ λ(k) IN + 1 − λ(k)2 σ2 «−1 1 PkQ σ2 Pkh ′ ! k (25) Let us now perform a recursive reasoning over the channel estimates at time indexes k ∈ N. Assume that p(hk−1|yk−1, Ik−1) is Gaussian CN(kk−1, Mk−1). We will show that this implies p(hk|yk, Ik) is still Gaussian. This will therefore be denoted CN(kk, M −1 k ). We have p(hk|yk, Ik) � (33) = α1p(yk|hk, Ik) · p(hk|hk−1, Ik) · p(hk−1, Ik)dhk−1 ′ (hk−h = lim e ˜Q→Q k )H Pk ′ σ2 (hk−h k ) × α2e (hk−1−kk−1) H M −1 k−1 (hk−1−kk−1) dhk−1 � (34) e (hk−λhk−1) H ˜ Q −1 1−λ2 (hk−λhk−1) (35) where the ˜ Q’s are taken from a set of invertible matrices in the neighborhood of Q, and the αi’s are constants. First we need to write the exponents of the Gaussian products in the integrand in a single Gaussian exponent form of the vector hk−1 times a constant independent of hk−1. By expansion and simplification, this is (hk − λhk−1) H ˜ Q−1 1 − λ2 (hk − λhk−1) + (hk−1 − kk−1) H M −1 k−1 (hk−1 − kk−1) = (hk−1 − l) H N(hk−1 − l) + C(hk) (36) with ⎧ N = ⎪⎨ ⎪⎩ λ2Q˜ −1 1−λ2 + M −1 k−1 l = N−1 ( λ 1−λ2 ˜ Q−1hk + M −1 k−1kk−1) C = hH ˜Q k −1 1−λ2 hk + kH k−1M−1 k−1kk−1− (hk + 1−λ2 ˜QM λ −1 H λ2 k−1kk−1) � 2 λ Q˜ −1 1−λ2 + M −1 �−1 ˜Q −1 k−1 (hk + 1−λ2 λ (1−λ 2 ) 2 ˜ Q −1 ˜QM −1 k−1 kk−1) The integral (35) is then a constant times e C , which depends on hk. The term C must then be written again into a quadratic expression of hk. This is C = (hk − j) H R(hk − j) + B (37) with ⎧ ⎪⎨ R ⎪⎩ � = λ2Mk−1 + (1 − λ2 ) ˜ j B �−1 Q = λkk−1 = 0 (38) Together with the term outside the integral (35), this is p(hk|yk, Ik) = α · e (hk−kk) H M −1 k (hk−kk) (39) with α = ( � p(hk|yk, Ik)dhk) −1 . Finally, after some arithmetic derivation, in the limit ˜ Q → Q, ⎧ ⎪⎨ Mk = λ ⎪⎩ 2 � Mk−1 + 1−λ2 λ2 � Q � 2 λ × σ2 Pk(Mk−1 + 1−λ2 λ2 �−1 Q) + (40) IN kk = λkk−1 + 1 σ 2 MkPk(h ′ k − λkk−1)
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