Parallel Relay Network with Orthogonal Components: Capacity and Power Allocation

Abstract

The authors consider a relay network in which the source-destination pair is augmented by a set of parallel relays. In this relay network, the source transmits to the destination and the relays over multiple orthogonal channel components. Practically, orthogonal channel components is motivated by the fact that a relay cannot receive and then transmit by using the same frequency band. For this relay network, lower bound on the capacity is shown to match the upper bound, and thus establish the capacity of the discrete memoryless physically degraded relay channel. Then, the capacity result is extended to the case of additive white Gaussian noise channel. Finally, rigorous simulation results are carried out to verify the analytical results.

Appendices

Appendix 1: Achievability Part

The source uses superposition block Markov encoding and binning scheme. Each relay can employ the decode-and-forward encoding scheme in order to get the signal and then forward it to the destination. In particular, regular encoding with backward decoding is employed by the relays and the destination. In this encoding technique, the source divides its signal into B equally sized blocks. Then, the transmission to the destination and the two relays happens over $B+1$ blocks. Further, for a given block index b, where $b\in [1,B]$, the source divides its signal w into tuple $\left( w_D,w_{R_1},w_{R_2}\right)$ $\in$ $[1,2^{nR_D}]\times [1,2^{nR_{R_1}}]\times [1,2^{nR_{R_2}}]$ such that $w_D$ is transmitted directly to the destination, and $w_{R_1}$ and $w_{R_2}$ are the transmitted signals to the destination via the relays $R_1$ and $R_2$, respectively. We note that the average rate triple $\left( {R_D} \frac{B}{B+1}, R_{R_1} \frac{B}{B+1}, R_{R_2} \frac{B}{B+1} \right)$ approaches $\left( R_D , R_{R_1} , R_{R_2} \right)$ as $B \longrightarrow \infty$. Indeed, the channels to both relays and the destination are assumed to be orthogonal.

Random Codebook Generation: First, fix a choice of $p(x_1)p(x_2)p(x_{R_1 } |x_1)p(x_{R_2} |x_2)p(x_D |x_1,x_2)$.

• Generate $2^{nR_{R_1}}$ independent identically distributed n-sequences $x_1^n$, each drawn according to $P(x_1^n )= \prod _{t=1}^{n} P(x_{1,t})$. Index them as $x_1^n(i)$, $i \in [1:2^{nR_{R_1}}]$.

• Generate $2^{nR_{R_2}}$ independent identically distributed n-sequences $x_2^n$, each drawn according to $P(x_2^n )= \prod _{t=1}^{n} P(x_{2,t})$. Index them as $x_2^n(j)$, $j \in [1:2^{nR_{R_2}}]$.

• For each $x_1^n (i)$, generate $2^{nR_{R_1}}$ conditional independent n-sequences $x_{R_1}^n$, drawn according to $P(x_{R_1}^n|x_1^n (i))= \prod _{t=1}^{n} P(x_{R_1,t}|x_{1,t} (i))$. Index them as $x_{R_1}^n (i,k)$, where $i,k\in [1:2^{nR_{R_1} }]$.

• For each $x_2^n (j)$, generate $2^{nR_{R_2} }$ conditional independent n-sequences $x_{R_2}^n$, drawn according to $P(x_{R_2}^n|x_2^n (j))= \prod _{t=1}^{n} P(x_{R_2,t}|x_{2,t} (j))$. Index them as $x_{R_2}^n (j,l)$, where $j,l\in [1:2^{nR_{R_1} }]$.

• For each $x_1^n (i)$ and $x_2^n (j)$, generate $2^{nR_D}$ conditional independent n-sequences $x_D^n$, drawn according to $P(x_D^n | x_1^n (i), x_2^n (j) )= \prod _{t=1}^{n} P(x_{D,t} | x_{1,t} (i),x_{2,t} (j))$. Index them as $x_D^n (u,v)$, $u,v \in [1:2^{nR_D}]$.

Encoding: We now describe the encoding at the source and the two relays. The encoding is performed in $B+1$ blocks and can be explained as follows:

• Source terminal: In the block b, where $b\in [1,B]$, the message is split into B equally sized blocks $w_{R_1,b} ,w_{R_2,b}$, and $w_{D,b}$ for $b=1, 2, \ldots , B$. In block $b = 1, 2,\ldots , B + 1$, the sender transmits $x_{R_1,b}^n (w_{R_1,b-1},w_{R_1,b} )$, $x_{R_2,b}^n (w_{R_2,b-1}, w_{R_2,b} )$ and $x_{D,b}^n (w_{D,b-1},w_{D,b} )$ where $w_{R_1,0}=w_{R_1,B+1}=w_{R_2,0}=w_{R_2,B+1}=w_{D,0}=w_{D,B+1}=1$.

• Relay $R_1$ : After the transmission of block b is completed, relay $R_1$ has received $y_{1,b}^n$. The relay tries to find $(w_{R_1,b})$ such that

  $$\begin{aligned} \left( x_{R_1,b}^n (\hat{w}_{R_1,b-1},\hat{w}_{R_1,b}), x_{1,b}^n (\hat{w}_{R_1,b-1}), y_{1,b}^n \right) \in A_{\epsilon }^{n} \left( X_{R_1},X_1,Y_1\right) \end{aligned}$$

  (15)

  where $A_\epsilon ^n$ denotes the strongly jointly $\epsilon -$typical set of length n. Moreover, $\hat{w}_{R_1,b-1}$ are the relay terminals estimate of $w_{R_1,b-1}$.

• Relay $R_2$: This relay performs the same encoding and decoding as does the relay $R_1$. Then, the relay $R_2$ transmits $x_{2,b}^n (\hat{w}_{R_2,b-1})$.

Decoding:

• The Relay $R_1$ can reliably decode $w_{R_1,b}$ for sufficiently large n and if the past estimate $( \hat{w}_{R_1,b-1})$ is a correct estimate of $(w_{R_1,b-1})$ and

  $$\begin{aligned} R_{R_1} \le I(X_{R_1};Y_1|X_1 ). \end{aligned}$$

  (16)

• The Relay $R_2$ can reliably decode $w_{R_2,b}$ for sufficiently large n and if the past estimate $( \hat{w}_{R_2,b-1})$ is a correct estimate of $(w_{R_2,b-1})$ and

  $$\begin{aligned} R_{R_2} \le I(X_{R_2};Y_2|X_2 ). \end{aligned}$$

  (17)

• Using backward decoding, the destination knows $w_{D,b-2}$, $w_{R_1,b-2}$ and $w_{R_2,b-2}$ and searches for a unique $\left( w_{D,b-1}, w_{R_2,b-1}, w_{D,b-1} \right)$ such that

  $$\begin{aligned} \left( x_{1}^n \left( \hat{w}_{R_1,b-2},\hat{w}_{R_1,b-1}\right) ,x_{2}^n \left( \hat{w}_{R_2,b-2}, \hat{w}_{R_2,b-1}\right) , x_{D}^n \left( \hat{w}_{D,b-1}\right) , y_{D,b-1}^n\right) \in A_{\epsilon }^{n} \left( X_1,X_2,X_D,Y_D\right) . \end{aligned}$$

  Then, the destination can decode reliably as long as n is large and

  $$\begin{aligned} R \le \left( I(X_D,X_1,X_2;Y_D\right) . \end{aligned}$$

  (18)

  We remind that the transmission from the relays and the source to the destination forms a MAC channel in which the sum rate has been considered.

Appendix 2: Converse Part

The best known upper bound on the capacity of the relay channel is the max-flow min-cut upper bound (also referred to as the cut-set bound). This upper bound is tight in all the cases the capacity of the relay channel is known. Although this bound is not believed to be the capacity of the relay channel in general, there is not a single example proving suboptimality of this upper bound. In this appendix, we present an upper bound on the capacity of the discrete memoryless two-parallel relay channel with orthogonal components using the cut-set bound. The transmission have two steps, which are: i) the broadcast term from the source to the relay nodes and the destination, and ii) the MAC term from the relay nodes and the source to the destination.

First, the broadcast term may be obtained as

$$\begin{aligned} \begin{aligned} nR&= H(W), \\&=I(W;Y^n )+H(W|Y^n ), \\&\quad \le I(W;Y^n )+n \epsilon _3 , \\&= I(W_{R_1},W_{R_2},W_{D}; Y^n)+n \epsilon _3, \\&= {}^{a}I(W_{R_1};Y^n )+I(W_{R_2};Y^n| W_{R_1} ),\\&\quad +I(W_D;Y^n|W_{R_1},W_{R_2} ) +n \epsilon _3,\\&= {}^{b}I(W_{R_1};Y^n )+I(W_{R_2};Y^n )+I(W_D|Y^n)+n \epsilon _3,\\&= I(W_{R_1};Y_1^n )+I(W_{R_2};Y_2^n )+I(W_D;Y^n )+n \epsilon _3,\\&= R_{R_1}+R_{R_2}+R_D, \end{aligned} \end{aligned}$$

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where a: follows from chain rule. b: follows from the fact that $W_{R_1},W_{R_2}$,and $W_D$ are independent. In addition, $R_{R_1}$, $R_{R_2}$, and $R_D$ are to be derived.

Then, the signal $X_{R_1}$ can be estimated at the first relay, $R_1$, with rate bounded as

$$\begin{aligned} \begin{aligned} nR_{R_1}&= H(W_{R_1}), \\&= I\left( W_{R_1};Y_1^n \right) +H\left( W_{R_1 }| Y_1^n \right) ,\\&\le I\left( W_{R_1};Y_1^n \right) +n \epsilon _1, \\&=\sum _{i=1}^{n} I\left( W_{R_1};Y_{1,i}|Y_{1,{i-1}}\right) +n \epsilon _1, \\&= {}^{a}\sum _{i=1}^{n} I\left( W_{R_1};Y_{1,i}| Y_{1,{i-1}},X_{1,i}\right) +n \epsilon _1,\\&= \sum _{i=1}^{n} H\left( Y_{1,i}|Y_{1,{i-1}},X_{1,i}\right) -H\left( Y_{1,i}| Y_{1,{i-1}},X_{1,i},W_{R_1} \right) +n \epsilon _1, \\&\le {}^{b}\sum _{i=1}^{n} H\left( Y_{1,i}|X_{1,i} \right) -H\left( Y_{1,i} |W_{R_1},Y_{1,{i-1}},X_{R_1,i},X_{1,i}\right) +n \epsilon _1,\\&= {}^{c}\sum _{i=1}^{n} H\left( Y_{1,i}|X_{1,i} \right) -H\left( Y_{1,i} |X_{R_1,i},X_{1,i}\right) +n \epsilon _1,\\&= \sum _{i=1}^{n} I\left( X_{R_1,i};Y_{1,i}|X_{1,i}\right) +n \epsilon _1, \end{aligned} \end{aligned}$$

(20)

where a: follows from the fact that $X_{1,i}$ is function of $f_i (Y_{1,{i-1}})$. b: follows from the fact that conditioning reduces entropy. c: follows from a Markov chain $(W_{R_1},Y_{1,{i-1}} )\longrightarrow (X_{R_1,i},X_{1,i} ) \longrightarrow Y_{1,i}$.

Therefore, $R_{R_1} \le I(X_{R_1};Y_{1}|X_{1} )$ is obtained. In a similar manner, we may easily show that the achievable rate at the second relay, $R_2$ is bounded as $R_{R_2} \le I(X_{R_2};Y_{2}|X_{2} )$. Next, the signal $X_D$ may be estimated at the destination with rate bounded as

$$\begin{aligned} \begin{aligned} nR_{D}&= H(W_{R_D}), \\&= I(W_{D};Y_D^n )+H(W_{D}| Y_D^n ),\\&\le I(W_{D};Y_D^n )+n \epsilon _2 , \\&= \sum _{i=1}^{n} I(W_{D};Y_{D,i}|Y_{D,{i-1}})+n \epsilon _2, \\&= \sum _{i=1}^{n} I(W_D;Y_{D,i} |Y_{D,{i-1}},X_{1,i},X_{2,i},X_{D,i})+n \epsilon _2, \\&= \sum _{i=1}^{n} H(Y_{D,i}|Y_{D,{i-1}},X_{1,i},X_{2,i},X_{D,i} )\\&\quad -H(Y_{D,i} |Y_{D,{i-1}},X_{1,i},X_{2,i},X_{D,i},W_D)+n \epsilon _2,\\&= \sum _{i=1}^{n} H(Y_{D,i}|X_{1,i},X_{2,i})-H(Y_{D,i} |X_{D,i}) +n \epsilon _2, \\&= \sum _{i=1}^{n} I(X_{D,i};Y_{D,i}|X_{1,i},X_{2,i})+n \epsilon _2. \end{aligned} \end{aligned}$$

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Thus, $R_{D}\le I(X_{D};Y_D|X_{1},X_{2})$ is obtained. Therefore, the broadcast term can be bounded as

$$\begin{aligned} \begin{aligned} R&\le R_{R_1}+R_{R_2}+R_{D} , \\&\le I\left( X_{R_1};Y_{1}|X_{1} \right) +I\left( X_{R_2};Y_{2}|X_{2} \right) , + I\left( X_{D};Y_D|X_{1},X_{2}\right) . \end{aligned} \end{aligned}$$

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Now, the multiple access term can be bounded as

$$\begin{aligned} \begin{aligned} nR&=H(W), \\&=I\left( W;Y_D^n \right) +H\left( W|Y_D^n \right) , \\&\le ^{a} I\left( W;Y_D^n\right) +n \epsilon _4, \\&=\sum _{i=1}^{n} I\left( W;Y_{D,i}|Y_{D,i-1}\right) +n \epsilon _4,\\&=\sum _{i=1}^{n} H\left( Y_{D,i}| Y_{D,i-1}\right) -H\left( Y_{D,i}|W,Y_{D,i-1}\right) +n \epsilon _4,\\&\le ^{b} \sum _{i=1}^{n} H(Y_{D,i})-H\left( Y_{D,i} |W,Y_{D,i-1}\right) +n \epsilon _4, \\&\le ^{c} \sum _{i=1}^{n} H(Y_{D,i})-H\left( Y_{D,i} |X_{1,i},X_{2,i},X_{D,i},W,Y_{D,i-1}\right) +n \epsilon _4, \\&=^{d}\sum _{i=1}^{n}\left( H(Y_{D,i} )-H\left( Y_{D,i} | X_{D,i},X_{1,i},X_{2,i}\right) \right) +n \epsilon _4, \\&= I(X_{D,i},X_{1,i},X_{2,i};Y_i)+n \epsilon _4, \end{aligned} \end{aligned}$$

(23)

where a: follows from Fano’s inequality. b: follows from the fact that conditioning reduces entropy. c: follows from $(X_{1,i},X_{2,i},X_{D,i} )$ are function of $W_i$. d: follows from the fact that $(W,Y_{D,i-1} ) \longrightarrow (X_{1,i},X_{2,i},X_{D,i} )\longrightarrow Y_{D,i}$ form a Markov chain.

Finally, the total upper bound at the destination is attained by taking the minimum of the broadcast term and the multiple access term. Therefore, the following upper bound is achieved

$$\begin{aligned} R \le min\left\{ I\left( X_D, X_1, X_2;Y_D\right) , I\left( X_{R_1};Y_1|X_1 \right) +I\left( X_{R_2};Y_2|X_2 \right) +I\left( X_D;Y_D|X_1,X_2 \right) \right\} \end{aligned}$$