An Improvement of Cover/El Gamals Compress-and-Forward Relay Scheme

1 An Impro v ement of Co ve r/El Gamal’ s Compress-and-F orward Rela y Scheme Liang-Liang Xie Departmen t of E lectrical an d Com puter En gineerin g University of W aterloo, W aterloo, ON, Canada N2L 3G1 Email: llxie@ece. uwaterloo.ca Abstract The compress-an d-for ward relay scheme developed by (Cover and El Gamal, 1979 ) is improved with a modificatio n o n th e decoding process. The impr ovement follows as a result of realizing that it is not necessary f or the destination to d ecode the compr essed ob servation of the relay ; a nd ev en if the compressed observation is to be decoded, it can be mo re easily don e by joint decodin g with the o riginal message, r ather than in a successive way . An extension to multiple relays is also discussed. I . I N T RO D U C T I O N The relay channel, orig inally proposed i n [1], models a communication scenario where there is a relay node that can help the information transm ission b etween the source and the destinatio n, as shown in Fig. 1. T wo fundamentally differe nt relay strategies were developed in [2], wh ich, depending on whether the relay decodes the informati on or not, are generally known as d ecode- and-forwar d and compr ess-and-forwar d respectively . The compress-and-forward relay strategy is used when the relay cannot decode th e mess age sent by the so urce, but still can help by compressing and forwarding its ob serva tion to the destination. r s d Fig. 1. The relay channel. 2 In the compress-and-forward coding scheme developed in [2 ], t he relay first compresses i ts observation Y 1 into ˆ Y 1 , and then forwards this compressed version to the destinatio n via X 1 . This compression is generally necessary since the destinati on m ay not be able t o completely recover Y 1 . Instead, th e compressed version ˆ Y 1 can be recovered, as long as the following cons traint is satisfied: I ( X 1 ; Y ) > I ( Y 1 ; ˆ Y 1 | X 1 , Y ) . (1) Then, based on ˆ Y 1 and Y , the destination can decode the original message X if the rate R < I ( X ; ˆ Y 1 , Y | X 1 ) . (2) In this paper , we propose a modification of this compress-and-forward cod ing scheme by realizing that it is not necessary to recover ˆ Y 1 since the orig inal problem is to d ecode X only; and e ven if ˆ Y 1 is to be decoded, it can be done by join tly decoding ˆ Y 1 and X , i nstead o f successiv ely decoding ˆ Y 1 and then X . W e wi ll sh ow that witho ut decodin g ˆ Y 1 , the const raint (1) is not n eeded, and the achie vable rate is m ore generally given by R < I ( X ; ˆ Y 1 , Y | X 1 ) − max { 0 , I ( Y 1 ; ˆ Y 1 | X 1 , Y ) − I ( X 1 ; Y ) } . (3) Obviously , an y rate satisfyi ng (1)-(2) also satisfies (3). Ho wev er , it remain s a question whether there are interesting channel models where (3) is strictly larger than (1)-(2). This problem will not be addressed here. Instead, we point out an immediate advantage o f (3) over (1)-(2). For (1)-(2), the relay needs to kn ow the value o f I ( Y 1 ; ˆ Y 1 | X 1 , Y ) in order to decide on t he appropriate compressed version ˆ Y 1 to choose. Thi s requires the k nowledge of the channel dynami cs from X to Y , which m ay be dif ficult to obtain for the relay , e.g. , in wireless communications. Howe ver , this is not necessary for (3), where the relay can choo se any version ˆ Y 1 that is sufficiently clo se to Y 1 , since ˆ Y 1 is n ot t o be decoded. What if we also want to decode ˆ Y 1 ? It turns out that by j ointly decoding ˆ Y 1 and X , the constraint (1) is not necessary; ins tead, we need a less strict inequality as the following: I ( X 1 ; Y ) > I ( Y 1 ; ˆ Y 1 | X 1 , Y , X ) (4) where, obviously , the difference from (1) is the additional i nformation provided by X . 3 I I . T H E S I N G L E R E L AY C A S E Formally , the single-relay channel d epicted in Fig. 1 can be denoted by ( X × X 1 , p ( y , y 1 | x, x 1 ) , Y × Y 1 ) where, X and X 1 are the t ransmitter alphabets of the sou rce and the relay respecti vely , Y and Y 1 are the receiver alphabets o f the destin ation and the relay respectiv ely , and a collection of probability di stributions p ( · , ·| x, x 1 ) on Y × Y 1 , one for each ( x, x 1 ) ∈ X × X 1 . The int erpretation is t hat x i s t he inpu t to t he channel from the source, y is the output of the channel to the destination, and y 1 is the output received by the relay . The relay sends an input x 1 based on wh at it h as recei ved: x 1 ( t ) = f t ( y 1 ( t − 1) , y 1 ( t − 2) , . . . ) , for e very time t, (5) where f t ( · ) can be any causal function. No te that a one-step t ime delay is assumed in (5) to account for the signal processing tim e at the relay . Theor em 2.1: F or the single-relay channel depicted in Fig. 1, by the modified comp ress-and- forward coding scheme, a rate R is achie v able if it satisfies R < I ( X ; ˆ Y 1 , Y | X 1 ) − max { 0 , I ( Y 1 ; ˆ Y 1 | X 1 , Y ) − I ( X 1 ; Y ) } (6) for some p ( x ) p ( x 1 ) p ( ˆ y 1 | y 1 , x 1 ) . In addi tion, th e compressed version ˆ Y 1 can be decoded if I ( X 1 ; Y ) > I ( Y 1 ; ˆ Y 1 | X 1 , Y , X ) . (7) In the modified scheme, the cod ebook generation and encoding process is exactly th e same as that in the proof of Theorem 6 of [2]. The mo dification is only on t he decoding process at the destination: i ) The destination finds th e unique X sequence that is jointly typical with th e Y sequence recei ved, and also with a ˆ Y 1 sequence from the specific bin sent by the relay via X 1 ; ii) If t he ˆ Y 1 sequence is to be decoded, t he desti nation finds the unique pair of X sequence and ˆ Y 1 sequence from th e specific bin that are jointly typical with the Y sequence receiv ed. I I I . E X T E N S I O N T O M U LT I P L E R E L A Y S An extension of Cov er/El Gamal’ s compress-and-forward codi ng scheme to multi ple relays was presented in [3]. W e can also e xtend the mod ified scheme to multiple relays. A multiple-relay channel is depicted in Fig. 2, whi ch can be denoted by ( X × X 1 × · · · × X n , p ( y , y 1 , . . . , y n | x, x 1 , . . . , x n ) , Y × Y 1 × · · · × Y n ) 4 where, X , X 1 , . . . , X n are t he transmitt er alphabets of th e source and the relays respectively , Y , Y 1 , . . . , Y n are the receiv er alphabets of the desti nation and the relays respectively , and a collection of probabil ity distributions p ( · , · , . . . , ·| x, x 1 , . . . , x n ) on Y × Y 1 × · · · × Y n , one for each ( x, x 1 , . . . , x n ) ∈ X × X 1 × · · · × X n . The interpretation is that x is the input to the channel from the source, y is the out put of the channel to the destination, and y i is the output receive d by the i -th relay . The i -th relay sends an input x i based on w hat it has recei ved: x i ( t ) = f i,t ( y i ( t − 1) , y i ( t − 2) , . . . ) , for e very time t, (8) where f i,t ( · ) can be any causal function. r 1 s d r n … Fig. 2. A multiple-relay channel. Before presenting the achie vability result, we introduce some simplified notations. Denot e the set N = { 1 , 2 , . . . , n } , and for any s ubset S ⊆ N , let X S = { X i , i ∈ S } , and use similar notations for other variables. W e ha ve the following achie vability result. Theor em 3.1: F or the mul tiple-relay channel depi cted in Fig. 2, b y the modified compress- and-forward coding scheme, a rate R is achie v able if for some p ( x ) p ( x 1 ) · · · p ( x n ) p ( ˆ y 1 | y 1 , x 1 ) · · · p ( ˆ y n | y n , x n ) , there e xists a rate vector { R i , i = 1 , . . . , n } satisfying X i ∈S 1 R i < I ( X S 1 ; Y | X S c 1 ) (9) for an y subset S 1 ⊆ N , such that for any sub set S ⊆ N , R < I ( X ; ˆ Y N , Y | X N ) − H ( ˆ Y S | ˆ Y S c , Y , X N ) + X i ∈S H ( ˆ Y i | Y i , X i ) + X i ∈S R i . (10) In addition, a subset of the compressed version ˆ Y D for som e D ⊆ N can be decoded, if for any S ⊆ N wit h S ∩ D 6 = ∅ , H ( ˆ Y S | ˆ Y S c , Y , X, X N ) − X i ∈S H ( ˆ Y i | Y i , X i ) < X i ∈S R i . (11) 5 It is easy t o check that Theorem 3.1 implies Th eorem 2.1, by noti ng the Markov Chain ( X , Y ) → ( X 1 , Y 1 ) → ˆ Y 1 . I V . F U RT H E R I M P R OV E M E N T Furthermore, we can even consider j oint decoding wi th X N . Then the const raint (9) is not necessary for the decoding of X N , with the help of X and ˆ Y N from the previous block. For this, we ha ve the following achie vability result. Theor em 4.1: For the multi ple-relay channel depicted in Fig. 2, a rate R is achie vable i f for some p ( x ) p ( x 1 ) · · · p ( x n ) p ( ˆ y 1 | y 1 , x 1 ) · · · p ( ˆ y n | y n , x n ) , there e xists a rate vector { R i , i = 1 , . . . , n } such that for any S 1 ⊆ S ⊆ N , R < I ( X ; ˆ Y N , Y | X N ) − H ( ˆ Y S | ˆ Y S c , Y , X N ) + X i ∈S H ( ˆ Y i | Y i , X i ) + X i ∈S \S 1 R i + I ( X S 1 ; Y | X S c 1 ) (12) and H ( ˆ Y S | ˆ Y S c , Y , X, X N ) − X i ∈S H ( ˆ Y i | Y i , X i ) − X i ∈S \S 1 R i − I ( X S 1 ; Y | X S c 1 ) < 0 . (13) In addition, a subset of the compressed version ˆ Y D for som e D ⊆ N can be decoded, if for any S ⊆ N wit h S ∩ D 6 = ∅ , H ( ˆ Y S | ˆ Y S c , Y , X, X N ) − X i ∈S H ( ˆ Y i | Y i , X i ) < X i ∈S R i . (14) R E F E R E N C E S [1] E . C. va n der Meulen, “Three-terminal communication channels, ” Adv . A ppl. Prob . , vol. 3, pp. 120–154, 1971. [2] T . Cov er and A. El Gamal, “Capacity theorems for the relay channel, ” IEEE T rans. Inform. Theory , vol. 25, pp. 572–584, 1979. [3] G. Kramer , M. Gastpar , and P . Gupta, “Cooperativ e strategies and capacity theorems for relay networks, ” IEEE T rans. Inform. Theory , vol. 51, pp. 3037–3 063, September 2005.