Optimal sequential transmission over broadcast channel with nested feedback

We consider the optimal design of sequential transmission over broadcast channel with nested feedback. Nested feedback means that the channel output of the outer channel is also available at the decoder of the inner channel. We model the communication system as a decentralized team with three decisionmakers—the encoder and the two decoders. Structure of encoding and decoding strategies that minimize a total distortion measure over a finite horizon are determined. The results are applicable for real-time communication as well as for the information theoretic setup. I. Problem formulation and main result In this paper, we study real-time broadcast of correlated sources over physically degraded channel with nested noiseless feedback. The communication system is shown in Figure 1. It operates in discrete time for a horizon T . The source is a first-order time-homogeneous Markov chain. The source outputs (Ut, Vt) take values in U × V . The initial output of the source is distributed according to PU1V1 ; the transition matrix of the source is PUV . The source output is transmitted over a discrete memoryless broadcast channel that is physically degraded. Let Xt ∈ X denote the channel input at time t and (Yt, Zt) ∈ Y × Z denote the channel outputs at time t. Since the channel is memoryless, we have Pr(Yt = yt, Zt = zt |U t = u, V t = v, X = x, Y t−1 = yt−1, Zt−1 = zt−1) = Pr (Yt = yt, Zt = zt |Xt = xt) =:QY Z|X(yt, zt|xt). Moreover, the channel is physically degraded, so QY Z|X(y, z|x) = QY |X(y|x)QZ|X(z|x). Sometimes it is more convenient to describe the channel in a functional form as Yt = q1(Xt, N1,t), Zt = q2(Yt, N2,t). The channel noises {N1,t, t = 1, . . . , T} and {N2,t, t = 1, . . . , T} are i.i.d. sequences that are mutually independent and also independent of the source outputs. The channel functions q1 and q2 and the distribution of the noises are consistent with the conditional distributions QY |X and QZ|Y . The communication system consists of an encoder and two decoders, all of which operate causally and in real-time. The decoder that receives Yt is called the inner decoder while the decoder that receives Zt is called the outer decoder. The channel is used with feedback, i.e., Yt is available to the encoder after a unit delay and Zt is available to the encoder and the inner decoder after a unit delay. The encoder is described by an encoding strategy c := (c1, . . . , cT ) where ct : U × V ×X t−1 × Yt−1 ×Zt−1 7→ X . The encoded symbol at time t is generated according to the encoding rule ct as follows Xt = ct(U , V , Xt−1, Y t−1, Zt−1). (1) The inner decoder is described by a decoding strategy g 1 := (g1,1, . . . , g1,T ) where g1,t : Y ×Zt−1 7→ Û . Similarly, the outer decoder is described by a decoding strategy g 2 := (g2,1, . . . , g2,T ) where g2,t : Z 7→ V̂. Thus, the decoded symbols at time t are generated as follows Ût = g1,t(Y , Zt−1); (2) V̂t = g2,t(Z). (3) The fidelity of reconstruction at the two decoders is determined by distortion functions ρ1,t : U × Û 7→ [0, ρmax] and ρ2,t : V × V̂ 7→ [0, ρmax], where ρmax <∞. For any communication strategy (c , g 1 , g 2 ), the system incurs an expected distortion given by J(c , g 1 , g 2 ) := E(c T ,g 1 ,g T 2 ) { T ∑ t=1 [ ρ1,t(Ut, Ût) + ρ2,t(Vt, V̂t) ]} . (4) We are interested in the optimal design of the above communication system. Specifically, we are interested in the following optimization problem. Problem 1: Given the statistics of the source and the channel, the distortion functions ρ1,t and ρ2,t, and the time horizon T , choose a communication strategy (c∗T , g∗T 1 , g∗T 2 ), with encoders of the form (1) and decoders of the form (2) and (3), such that (c∗T , g∗T 1 , g∗T 2 ) minimizes the expected total distortion given by (4). Since the alphabets U , V , X , Y , and Z are finite, the number of communication strategies are finite. Therefore, in principle, we can evaluate the performance of all of them and choose the one with the best performance. Consequently, Problem 1 is well posed. The domain of the encoding and decoding functions of the form (1), (2), (3) increases exponentially with time. As a result, the number of communication strategies increase doubly exponentially with time. Furthermore, implementing a communication strategy for a large horizon becomes difficult. In this paper, we find structural properties of optimal communication strategies that will allow us to “compress” Source Encoder Inner Channel Outer Channel Inner Decoder Outer Decoder Ut