Truncate-and-Forward: A New Coding Strategy for the General Relay Channel

In this work we focus on the general relay channel. We present a new coding strategy that yields a rate increase over the point-to-point rate forany channel conditions without the use of auxiliary random vari ables. Specifically, even when Cover & El-Gamal’s decode-and-forward or estimate-an d forward strategies do not exceed the point-to-point rate, our method results in a rate to the destination that is g reater than that rate. We then consider the cooperative broadcast scenario with a single common message and show how the new relaying strategy can be used to derive a multi-step conference that achieves the full cooperation r ate bound for conference capacities which are less than the Slepian-Wolf condition. This simple method can be extended to other network scenarios to improve results for the case where the relay is very noisy. I. I NTRODUCTION The relay channel was introduced by van der Meulen in 1971 [1] . In this setup, a single transmitter with channel outputX 1 communicates with a single receiver with channel input Y , where the superscript n denotes the length of a vector. In addition, an external transceiver, called a r el y, listens to the channel and is able to output signals to the channel. We denote the relay input with Y n 1 and its output withX n 2 . This setup is depicted in figure 1. Relay Channel Encoder Decoder Relay p(y,y 1 |x 1 ,x 2 ) W W ^ Y 1 n X 2 n Y n X 1 n Fig. 1. The relay channel. The encoder sends a message W to the decoder. The authors are with the School of Electrical and Computer En gineering, Cornell University, Ithaca, NY. URL: http://cn.ece.cornell.edu/. This work was presented in part at the International Sympos ium on Information Theory, Seattle, WA, 2006. Work supported by the National Science Foundation , under awards CCR-0238271 (CAREER), CCR-0330059, and ANR0325556. February 14, 2008 DRAFT 2 A. Current Relay Strategies In [2] Cover & El-Gamal introduced two relaying strategies c ommonly referred to as decode-and-forward (DAF) and estimate-and-forward (EAF). In DAF the relay decodes th message sent from the transmitter and then, at the next time interval, transmits a codeword based on the decode d message. The rate achievable with DAF is given in [2, theorem 1]: Theorem 1: (achievability of [2, theorem 1]) For the general elay channel any rateR satisfying R ≤ min {I(X1, X2;Y ), I(X1;Y1|X2)} (1) for some joint distributionp(x1, x2, y, y1) = p(x1, x2)p(y, y1|x1, x2), is achievable. We note that for DAF to be effective, the rate to the relay has t o be greater than the point-to-point rate i.e. I(X1;Y1|X2) > I(X1;Y |X2), (2) otherwise higher rates could be obtained without using the r elay at all. For situations where DAF is not useful, [2] presented the EAF strategy. In this strategy, the relay send s a estimate of its channel input to the destination. The achievable rate with EAF is given in [2, theorem 6]: Theorem 2: ([2, theorem 6]) For the general relay channel any rateR satisfying R ≤ I(X1;Y, Ŷ1|X2), (3) subject to I(X2;Y ) ≥ I(Y1; Ŷ1|X2, Y ), (4) for some joint distributionp(x1, x2, y, y1, ŷ1) = p(x1)p(x2)p(y, y1|x1, x2)p(ŷ1|y1, x2), where ||Ŷ1|| < ∞, is achievable. Recently, we improved the EAF decoding scheme of [2, theorem 6] and obtained an increased feasible region for the EAF strategy. This region is given in [3]: Theorem 3: [3, theorem 2] For the general relay channel any ra teR satisfying R ≤ I(X1;Y |X2) + min ( I(X2;Y ) − I(Y1; Ŷ1|X1, X2, Y ), I(X1; Ŷ1|X2, Y ) ) , (5) subject to I(X2;Y ) ≥ I(Y1; Ŷ1|X2, Y ) − I(Ŷ1;X1|X2, Y ) = I(Y1; Ŷ1|X1, X2, Y ), (6) for some joint distributionp(x1, x2, y, y1, ŷ1) = p(x1)p(x2)p(y, y1|x1, x2)p(ŷ1|y1, x2), where ||Ŷ1|| < ∞, is achievable. As can be seen from the expressions above, even the improved E AF has several basic limitations: 1) finding the highest achievable rate requires a search over all possible mappingsp(ŷ1|y1, x2). This is not a convex optimization problem, therefore, finding the highest achievable rate wit h this scheme requires an exhaustive search. 2) Any mapping has to satisfy the feasibility condition of (6). The refore, EAF cannot be applied in all relay scenarios. In fact, the more information̂Y1 contains onY1, the less rate increase EAF provides, and at the maximal rate between February 14, 2008 DRAFT 3 Ŷ1 andY1, when (6) holds with equality, we get the point-to-point rat e of I(X1;Y |X2). This is a clear inefficiency: although we transmit information from the relay at rate I(X2;Y ), we are not able to exceed the point-to-point rate. Of course, one can combine the DAF and EAF schemes and obtain h igher rates, following [2, theorem 7]. B. Related Work In recent years, the research in relaying has mainly focused on extending the basic DAF and EAF strategies of [2] to multiple level relaying and to the MIMO relay channel. In a ddition, variations of the DAF method were introduced to the multiple-relay case. In [4] the DAF method of [2] was ex tended to the multiple-relay case. Later work [5], [6] and [7] applied the so-called regular encoding/sliding -window decoding and the regular encoding/backward decoding techniques to the multiple-relay scenario. For th e single relay case, however, all these strategies converge to the achievable rate of theorem 1 (see [8]). The DAF strateg y of theorem 1 was also extended to the MIMO relay channel in [9]. As for the EAF strategy, again the focus was on the multiple-relay scenario. In [10] the EAF was considered in the multiple-relay setup and in [11] the au thors considered communication over two parallel relay channels to a destination, without a direct link betwe n the source and the destination. Another approach applied recently to the relay channel is that of iterative de coding. In [12] the three-node network in the half-duplex regime was considered. In the relay case [12] uses a feedback scheme where the receiver first uses EAF to send information to the relay and then the relay decodes and uses D AF at the next time interval to help the receiver decode its message. Combinations of EAF and DAF were also con sidered in [13], where conferencing schemes over orthogonal relay-receiver channels were analyzed and compared. Both of these papers are restricted to the Gaussian case. In [3] we applied simultaneous decoding to th e EAF method which resulted in an increased feasible region for this strategy compared to [2, theorem 6]. Another wo k that should be noted in that context is [14] where simultaneous decoding is used to improve upon Cover and El-G amal’s combined DAF/EAF result of [2, theorem 7]. However, when specialized to the EAF setup, the result of [14] converges to [2, theorem 6]. In contrast, in [3], we used simultaneous decoding at the receiver to improve the EAF region itself. An extension of the relay scenario to a hybrid broadcast/rel ay system was introduced in [15] in which the authors applied a combination of EAF and DAF strategies to the indepe ndent broadcast channel setup, and then extended this strategy to the multi-step conference. In [16] we used both a single-step and a two-step conference with orthogonal conferencing channels in the discrete memoryless framewor k. The work in [16] used an improved EAF method, therefore the results in [16] exceed previous results based on the EAF procedure of [2, theorem 6]. A thorough investigation of the broadcast-relay channel was done in [1 7]. In this work the authors applied the DAF strategy to the case where only one user is helping the other user, and als o pre ented an upper bound for this case. Then, the fully cooperative scenario was analyzed. The authors appli ed both the DAF and the EAF methods of [2] to that case. However, none of this work (except [3] and [14]) introduced a ny modification to the basic DAF and EAF strategies and their fundamental limitations, as describe d above, still constitute the basic limitations inherent to all recent results. These basic limitations motivated the sear ch for a different relaying strategy which will produce a February 14, 2008 DRAFT 4 rate increase over the point-to-point rate for any relay channel . Such a technique would be especially useful when the relay channel is noisy so that both DAF and EAF are not appl icable. C. Main Contributions In this paper we present a new relaying strategy for the gener al r lay channel that does not require auxiliary random variables. In this strategy, instead of compressing the relay input signal in the Wyner-Ziv sense [18], which eventually gives rise to the bound on the feasible region, we simply send to the receiver a fraction of the symbols received at the relay, without compression. The receiver th en uses that fraction to improve the decoding of the source message, compared to the point-to-point case. This strateg y results in a rate increase over the point-to-point rate for any channel conditions. Furthermore, whenever theorem 3 ac hieves the full cooperation bound of I(X1;Y, Y1|X2), so does our result. However, our result is definitely not opti mal and we expect it to intersect the rate of theorem 3 at some value of I(X2;Y ). Combining our result with theorems 1 and 3, yields the best a chievable rate for the general relay channel, which is always greater than the poin t-t -point rate. In the second part of this paper we demonstrate our new strate gy in the cooperative broadcast channel with a single common message scenario. For this setup we present an xplicit three-step cooperation scheme that does not require auxiliary random variables. This new cooperation s cheme yields a rate increase over the non-cooperative rate for any given cooperation capacity. In addition, this s cheme achieves the full cooperation bound when the conference capacities are less than those given by the Slepi an-Wolf theorem [19, theorem 14.4.1]. The rest of this paper is organized as follows: in section II w e define the mathematical framework. In section III we present the new relay strategy followed by discussion in s ection IV.