On the Capacity Regions of Degraded Relay Broadcast Channels with and without Feedback

In the classical three-node relay channel [1,2], a relay node helps the communication between the transmitter and receiver. If the relay node also wants to decode a private message sent by the transmitter, then this channel turns out to be a partially cooperative relay broadcast channel (RBC) [3,4]. The capacity region of partially cooperative RBC is established for the case of degraded message sets [4], where the transmitter has a common message for both destinations and a private message for the relay. The fully cooperative RBC [3] is a more general model where both destinations can serve as the relay and receiver. A two-user degraded broadcast channel with conferencing and degraded message sets was studied in [5]. In [6], a degraded RBC was studied and an achievable rate region and distortion performance based on the decode-forward relay strategies proposed. In [7], the physically degraded and state-dependent RBC with cooperating decoders was investigated. A compress-forward inner bound was established for RBC with confidential messages in [8]. Partially and fully cooperative state-dependent relay broadcast channels with perfect causal channel state information (CSI) were considered in [9]. In [10], rate regions for two types of partially cooperative RBC with non-causal side information were established. The fully and partially cooperative RBCs with rate-limited feedback were studied in [11].

In many practical scenarios, receivers are unwilling to serve as relay nodes for other users due to the concern for privacy leakage or energy overuse. It is thus more practical to introduce exclusive relay nodes to assist the communication, leading to the so-called dedicated RBCs [12,13]. This model is particularly common in modern communication systems such as 4G or 5G cellular network, where relay nodes are incorporated into networks to improve the communication rate and enlarge the service range as well. The dedicated RBCs have been widely studied in the literature. In [14], the capacity region of a single-relay dedicated Gaussian RBC was established when one receiver’s observed signal was a degraded form of the other receiver’s, and the stronger receiver’s observed signal was a degraded form of the relay’s. In [15], an RBC with three parallel unmatched subchannels was studied, which was shown to enlarge the family of RBCs and relay channels for which the capacity region was known. For r-relay ($r\le 2$) dedicated RBCs, the general outer bounds and capacity results for several classes of broadcast relay channels were investigated in [16], in which the capacity region of the so-called semi-degraded BC was established. The achievable rate region for K-receiver RBCs with one dedicated relay node was proposed in [17]. Multiple-input multiple-output (MIMO) RBC with direct links based on a weighted sum-rate criterion was studied in [18]. A new achievable rate region generalizing Marton’s coding to two-relay two-user RBCs was proposed in [19]. For multiple-relay RBC, single relay selection schemes using either selective digital relaying or selective analog relaying were proposed, which were shown to improve significantly the bit error probability performance [20]. Inner bounds were proposed for a single-relay dedicated RBC with fast fading and state feedback in [21].

In spite of the enormous amount of prior work on dedicated RBCs, the capacity region is still unknown. Moreover, it has been shown that feedback cannot enlarge the capacity region for some three-node degraded communication channels, such as degraded BCs and the degraded relay channel. Whether this result still holds for the four-node degraded RBCs is also an interesting problem. In this paper, we revisit the four-node dedicated RBC $p_{Y_1{Y}_2{Y}_3|X{X}_3}(y_1,y_2,y_3|x,x_3)$, as depicted in Figure 1, where X is a symbol sent by the transmitter, $Y_3$ and $Y_k$, for $k\in \{1,2\}$, are symbols observed by the relay and receiver k, respectively, and $X_3$ is a symbol created by observing $Y_3$ and to be sent by the relay. Even for the degraded case, the capacity region is still unknown, except for some special cases. The main challenges to establish its capacity region mainly come from the following three parts:

Now, we summarize the main contributions of this paper as follows:

The rest of the paper is organized as follows. The system model is introduced in Section 2. Various definitions of degraded RBCs are given in Section 3. In Section 4, we state our main capacity results for DM and Gaussian PDRBC. The result on RBC with feedback is give in Section 5, and its proof is given in Section 6. The proof of the outer bounds on the capacity regions of DM-PDRBC with feedback are stated in Section 7. The proof of capacity regions for Gaussian PDRBC is presented in Section 8. Section 9 concludes this paper.

Notation: We use capital letters to denote random variables and small letters for their realizations, e.g., X and x. Define a function $C\left(x\right):=\frac{1}{2}{log}_2(1+x)$. For nonnegative integers $k,j$, let $X_k^j:=(X_{k,1},\dots ,X_{k,j})$. Given a random variable X, we write its probability mass function (pmf) $p_X\left(x\right)$ as $p\left(x\right)$ for short. For an integer N, we use $[1:N]$ to denote {1,…,N}. Given a real value $\alpha$ with $0\le \alpha \le 1$, let $\overline{\alpha }:=1-\alpha$. Considering K sequences $X_1^n,\dots ,X_K^n$ each of n-length, the joint typical set ${\mathcal{T}}_{ϵ}^{\left(n\right)}\left(p\left(x_1,x_2,\dots ,x_K\right)\right)$ denotes the set of all sequences ($X_1^n,\dots ,X_K^n$) when they are join typical, where the parameter $ϵ$ is a positive value involved in the definition of joint typicality and tends to zero as the length n goes to infinity. More details and properties about ${\mathcal{T}}_{ϵ}^{\left(n\right)}\left(p(x_1,x_2,\dots x_K)\right)$ can be found in [22] (p. 29).

Consider a four-node discrete memoryless relay broadcast channel (DM-RBC) in which there is one transmitter, two receivers, and one relay that helps the transmitter to communicate with the receivers, as depicted in Figure 1. This channel consists of five finite alphabets ($\mathcal{X},{\mathcal{X}}_3,{\mathcal{Y}}_1,{\mathcal{Y}}_2,{\mathcal{Y}}_3$) and a collection of pmf $p(y_1,y_2,y_3|x,x_3)$ on ${\mathcal{Y}}_1\times {\mathcal{Y}}_2\times {\mathcal{Y}}_3$, one for each $\mathcal{X}\times {\mathcal{X}}_3$. Here, $x\in \mathcal{X}$ is a symbol sent by the transmitter, $y_3\in {\mathcal{Y}}_3$ and $y_k\in {\mathcal{Y}}_k$, for $k\in \{1,2\}$, are symbols observed by the relay and receiver k, respectively, and $x_3\in {\mathcal{X}}_3$ is a symbol created by observing $y_3$ and will be sent by the relay.

The transmitter wishes to communicate a message $M_k\in [1:2^{n{R}_k}]$ to receiver k, for $k\in \{1,2\}$, with the assistance of a relay node, where n denotes the block length of transmission. In the following subsections, we introduce the DM-RBC without and with feedback, respectively.

In this paper, we mainly focus on the PDRBC. Let ${\mathcal{C}}_{\mathrm{PD}}$ denote the capacity region of PDRBC. Without loss of generality, assume $Y_2$ is a random degradation of $Y_1$, which implies that output $Y_1$ is “stronger” than $Y_2$. According to the degradation order among $Y_1,Y_2$ and $Y_3$, we consider three types of PDRBC.

Type-I PDRBC and Type-II PDRBC are not new, which were introduced by [16]. In Type-I DM-PDRBC, the relay observes a “stronger” symbol than both receivers, and its capacity region is still unknown, except for the Gaussian case [14]. In Type-II PDRBC, the relay observes a “stronger” signal than one receiver, but a “weaker” signal than another receiver. For the DM case, the capacity region of Type-II PDRBC was established in [16] (Theorem 7), which is the set of rate pairs $(R_1,R_2)$ such that: for some pmf $p(u,x_3)p\left(x\right|u)$. Type-III PDRBC is newly defined, in which the relay’s observed signal is “weaker” than both receivers’. Type-II and Type-III PDRBCs could characterize some practical communication scenarios when some receivers are closer to the transmitter and thus have a better received signal than relay nodes.

More generally, stochastically degraded RBC is defined as below.

Our scheme can strictly enlarge Behboodi and Piantanida’s rate regions in [16]. This can be seen by the following example.

In [14], it was shown that for Type-I PDRBC with feedback from the users to the relay and transmitter, feedback cannot enlarge the capacity region. We obtain similar results for Type-II and Type-III PDRBCs as follows:

We present a scheme based on block-Markov coding [22] that consists of $B+1$ blocks, where messages $M_{1,b}\in [1:2^{n{R}_1}]$ and $M_{2,b}\in [1:2^{n{R}_2}]$, for $b\in [1:B]$, are sent to the receivers over $B+1$ blocks. Split message $M_{1,b}$ into $(M_{1,c,b},M_{1,p1,b},M_{1,p2,b})$, where $M_{1,c,b}\in [1:2^{n{R}_{1,c}}]$ and $M_{1,pi,b}\in [1:2^{n{R}_{1,pi}}]$, for $i=1,2$, are independent of each other. Split message $M_{2,b}$ into $(M_{2,c,b},M_{2,p,b})$, where $M_{2,c,b}\in [1:2^{n{R}_{2,c}}]$ and $M_{2,p,b}\in [1:2^{n{R}_{2,p}}]$, are independent of each other. Define: Let messages $M_{1,B+1}=M_{2,B+1}=1$, and assume $(M_{1,B+1},M_{2,B+1})$ are known by the relay and receivers before communication. Note that the transmission takes place over $B+1$ blocks, and the messages $(M_{1,B+1},M_{2,B+1})$ are deterministic; thus, the transmission rates of messages $M_1$ and $M_2$ are in fact equal to $R_1\frac{B}{B+1}$ and $R_2\frac{B}{B+1}$, respectively. In the limit $n\to \infty$ and $B\to \infty$, $R_1$ and $R_2$ approach these transmission rates. Therefore, we neglect this technicality in the following.