We first review some of the existing results in the literature for the source coding version of the problem under consideration, in which the fading channel is substituted by an error-free channel of finite capacity. We then focus on the scenario in which the channel is noisy but static, i.e., the channel gain is constant and known both at the encoder and the decoder. We show that separate source and channel coding is optimal in the case of a static channel.
3.2.1
Background: Lossy Source Coding with Fading Side In-
formation
The source-coding version of this problem in which the fading channel is substituted by an error-free channel of rate R and a time-varying side information sequence Tn is available at the destination is considered in [43]. Here we briefly review the results of [43] which will be used later in the chapter.
Let the distribution pΓ(γ) be discrete with M states γ1≤ · · · ≤ γM with probabilities
Pr[Γ = γi] = pi. We define the side information sequence available at the decoder when
the realization of the side information fading gain is γsi as Ti,1n ,
√
γiSn+ Zn 1. Note
that the side information has a degraded structure, characterized by the Markov chain T1,j− · · · − TM −1,j− TM,j− Sj, j = 1, ..., n. (3.5)
This is equivalent to the Heegard-Berger source coding problem with degraded side information [5], in which an encoder is connected by an error-free channel of rate R to M receivers, and receiver i has access to side information Tn
i,1. The minimum expected
distortion is given by the solution to the following problem,
ED∗(R) = min
D:RHB(D)≤R
pTD, (3.6)
where p , [p1, ..., pM], D = [D1, ..., DM] with Didefined as the achievable distortion at
receiver i and RHB(D) is the Heegard-Berger rate-distortion function given by
RHB(D) = min WM 1 ∈P(D) M X i=1 I(S; Wi|W1i−1, Ti), (3.7)
where W1i denotes the auxiliary random variables W1, ..., Wi, and P(D) is the set of
random variables WM
1 satisfying the Markov chain condition
WM− · · · − W1− S − TM − TM −1− · · · − T1,
for which there exist source reconstructions ˆSi(Ti, W1i) satisfying E[di(S, ˆSi)] ≤ Di,
i = 1, ..., M .
When the source Xnis Gaussian, it can be shown that the optimal auxiliary random
variables W1M minimizing (3.6) are jointly Gaussian. Then, the minimum expected
distortion for a Gaussian source with finite number of side information states can be
1To avoid confusion in the indexing, we use Tn
i,1 , [Ti,1, ..., Ti,n] to denote all the elements Ti,j,
found by solving the following convex optimization problem [43, Eq. (59)-(62)]: ED∗F(R) = min D1,...,DM∈R++ M X i=1 piDi s.t. −1 2 M −1 X i=0 log(1 + (γi+1− γi)Di) − 1 2log DM ≤ R, Di≤ (Di−1−1 + γi− γi−1)−1, i = 1, ..., M, (3.8)
where D0 , σ2x = 1 and γ0 , 0. The Heegard-Berger rate distortion function also
extends to the set of infinitely many degraded fading states, γ1 ≤ γ2 ≤ · · · with
P∞
i=1pi = 1 [43]. For a countable number of states, the expected distortion is given in
[43, Eq. (75)-(78)] as the solution to
ED∗C(R) = min D1,D2...∈R++ ∞ X i=1 piDi s.t. −1 2 ∞ X i=0
(log(D−1i−1+ γi− γ,i−1) + log Di) ≤ R,
Di≤ (D−1i−1+ γi− γ,i−1)−1, i = 1, 2, ... (3.9)
When the side information distribution pΓ(γ) is continuous and quasiconcave2, the op-
timal expected distortion is achieved by single-layer rate allocation such that all the available rate R is targeted to a single side information state ¯γ [43]. Then, the optimal expected distortion is given by
EDQ∗(R) = Z ¯γ 0 pΓ(γ) 1 + γdγ + Z ∞ ¯ γ pΓ(γ) (¯γ + 1)22R+ γ − ¯γdγ, (3.10)
where ¯γ minimizing (3.10) is determined as follows: Let a super-level set be defined as [γl(α), γr(α)] , {γ|pΓ(γ) ≥ α}. Then, ¯γ is defined as the left endpoint of the super-level
set induced by α∗, i.e., ¯γ = γl(α∗), where α∗ ∈ [0, max pΓ(γ)] is found by solving the
equation Z ∞ γl(α∗) pΓ(γ) − α∗ ((1 + γl(α∗))22R+ γ − γl(α∗))2 dγ = 0. (3.11)
When the side information state is Rayleigh distributed, the side information gain Γ is exponentially distributed. Then it can be seen that ¯γ = 0 and the optimal expected distortion becomes ED∗Ray(R) = 1 E[Γ]e 22R E[Γ]E 1 22R E[Γ] , (3.12)
where E1(x) ,R ∞ x t
−1e−tdt is the exponential integral [43].
In the following sections we use ED∗
F(R), EDQ∗(R) and EDRay∗ (R) to generate lower
bounds on the expected distortion. To unify these results, we define the function ED∗s(R) as the minimum expected distortion in the source coding problem for these three setups. Therefore, while the achievability results are valid for any distribution, the optimality results in this chapter are valid for discrete, i.e., finite or countable num- ber of states, as well as continuous quasiconcave distributions of the side information.
3.2.2
Static Channel and Fading Side Information
In this section we consider a static channel and prove the optimality of separate source and channel coding in this setting. We consider a channel from Xn to Yn, not neces-
sarily the fading Gaussian channel characterized in (3.1), of fixed capacity C. The side information is still block-fading as in (3.2) with the side information gain following a distribution pΓ(γ). Note that it is a JSCC generalization of the source coding problem
reviewed in Section 3.2.1. We denote the minimum expected distortion in the case of a static channel by ED∗sta.
Optimality of separate source and channel coding can be proven when Γ, the side information gain, has finite or countable number of states, or when it has a continuous quasiconcave distribution. This reduces the problem to the source coding problem of Section 3.2.1 with R = C.
Theorem 2. Assume that the channel is static with capacity C. When the side infor- mation gain Γ has a discrete number of states, or a continuous quasiconcave pdf pΓ(γ),
the minimum expected distortion, ED∗sta, is achieved by separate source and channel coding, and is given by
ED∗sta= EDs∗(C). (3.13)
Proof. The theorem is first proven when Γ has a discrete distribution. Then, to show the optimality of separation when pΓ(γ) is continuous and quasiconcave we construct a
lower bound on the expected distortion ED∗
staby discretizing the continuum of analog
side information states, and show that this bound is achievable in the limit of finer discretizations. See Appendix C for details.