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Capacity theorems for the relay channel

01 Sep 1979-Vol. 80, pp 572-584
TL;DR: An achievable lower bound to the capacity of the general relay channel is established and superposition block Markov encoding is used to show achievability of C, and converses are established.
About: The article was published on 1979-09-01 and is currently open access. It has received 3918 citations till now. The article focuses on the topics: Relay channel & Channel capacity.
Citations
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Journal ArticleDOI
TL;DR: Using distributed antennas, this work develops and analyzes low-complexity cooperative diversity protocols that combat fading induced by multipath propagation in wireless networks and develops performance characterizations in terms of outage events and associated outage probabilities, which measure robustness of the transmissions to fading.
Abstract: We develop and analyze low-complexity cooperative diversity protocols that combat fading induced by multipath propagation in wireless networks. The underlying techniques exploit space diversity available through cooperating terminals' relaying signals for one another. We outline several strategies employed by the cooperating radios, including fixed relaying schemes such as amplify-and-forward and decode-and-forward, selection relaying schemes that adapt based upon channel measurements between the cooperating terminals, and incremental relaying schemes that adapt based upon limited feedback from the destination terminal. We develop performance characterizations in terms of outage events and associated outage probabilities, which measure robustness of the transmissions to fading, focusing on the high signal-to-noise ratio (SNR) regime. Except for fixed decode-and-forward, all of our cooperative diversity protocols are efficient in the sense that they achieve full diversity (i.e., second-order diversity in the case of two terminals), and, moreover, are close to optimum (within 1.5 dB) in certain regimes. Thus, using distributed antennas, we can provide the powerful benefits of space diversity without need for physical arrays, though at a loss of spectral efficiency due to half-duplex operation and possibly at the cost of additional receive hardware. Applicable to any wireless setting, including cellular or ad hoc networks-wherever space constraints preclude the use of physical arrays-the performance characterizations reveal that large power or energy savings result from the use of these protocols.

12,761 citations


Cites background from "Capacity theorems for the relay cha..."

  • ...This example might correspond to a snapshot of a wireless network in which a higher level network protocol has allocated bandwidth to two terminals for transmission to their intended destinations or next hops....

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  • ...…i.e., achievable rates, via three structurally different random coding schemes: • facilitation [5, Theorem 2], in which the relay does not actively help the source, but rather, facilitates the source transmission by inducing as little interference as possible; • cooperation [5, Theorem 1],…...

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  • ...Such an approach might provide diversity in a wireless setting because, even if the fading is severe between and , the information might be successfully transmitted through ....

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Journal ArticleDOI
TL;DR: Results show that, even though the interuser channel is noisy, cooperation leads not only to an increase in capacity for both users but also to a more robust system, where users' achievable rates are less susceptible to channel variations.
Abstract: Mobile users' data rate and quality of service are limited by the fact that, within the duration of any given call, they experience severe variations in signal attenuation, thereby necessitating the use of some type of diversity. In this two-part paper, we propose a new form of spatial diversity, in which diversity gains are achieved via the cooperation of mobile users. Part I describes the user cooperation strategy, while Part II (see ibid., p.1939-48) focuses on implementation issues and performance analysis. Results show that, even though the interuser channel is noisy, cooperation leads not only to an increase in capacity for both users but also to a more robust system, where users' achievable rates are less susceptible to channel variations.

6,621 citations


Cites background from "Capacity theorems for the relay cha..."

  • ...4, we consider a situation where users face statistically similar channels toward the BS, and we plot the probability of outage versus the service sustainability ratefor the equal rate point ( ), both for the cooperation and the no-cooperation schemes....

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  • ...It is also complicated by the fact that both partners have information of their own to send; this is not a simple relay [11], [12] problem....

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Journal ArticleDOI
TL;DR: This work develops and analyzes space-time coded cooperative diversity protocols for combating multipath fading across multiple protocol layers in a wireless network and demonstrates that these protocols achieve full spatial diversity in the number of cooperating terminals, not just theNumber of decoding relays, and can be used effectively for higher spectral efficiencies than repetition-based schemes.
Abstract: We develop and analyze space-time coded cooperative diversity protocols for combating multipath fading across multiple protocol layers in a wireless network. The protocols exploit spatial diversity available among a collection of distributed terminals that relay messages for one another in such a manner that the destination terminal can average the fading, even though it is unknown a priori which terminals will be involved. In particular, a source initiates transmission to its destination, and many relays potentially receive the transmission. Those terminals that can fully decode the transmission utilize a space-time code to cooperatively relay to the destination. We demonstrate that these protocols achieve full spatial diversity in the number of cooperating terminals, not just the number of decoding relays, and can be used effectively for higher spectral efficiencies than repetition-based schemes. We discuss issues related to space-time code design for these protocols, emphasizing codes that readily allow for appealing distributed versions.

4,385 citations

Journal ArticleDOI
TL;DR: An overview of the developments in cooperative communication, a new class of methods called cooperative communication has been proposed that enables single-antenna mobiles in a multi-user environment to share their antennas and generate a virtual multiple-antenn transmitter that allows them to achieve transmit diversity.
Abstract: Transmit diversity generally requires more than one antenna at the transmitter. However, many wireless devices are limited by size or hardware complexity to one antenna. Recently, a new class of methods called cooperative communication has been proposed that enables single-antenna mobiles in a multi-user environment to share their antennas and generate a virtual multiple-antenna transmitter that allows them to achieve transmit diversity. This article presents an overview of the developments in this burgeoning field.

3,130 citations


Cites methods from "Capacity theorems for the relay cha..."

  • ...The partners may be assigned mutually by the base station, or via some other technique....

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Journal ArticleDOI
TL;DR: The capacity results generalize broadly, including to multiantenna transmission with Rayleigh fading, single-bounce fading, certain quasi-static fading problems, cases where partial channel knowledge is available at the transmitters, and cases where local user cooperation is permitted.
Abstract: Coding strategies that exploit node cooperation are developed for relay networks. Two basic schemes are studied: the relays decode-and-forward the source message to the destination, or they compress-and-forward their channel outputs to the destination. The decode-and-forward scheme is a variant of multihopping, but in addition to having the relays successively decode the message, the transmitters cooperate and each receiver uses several or all of its past channel output blocks to decode. For the compress-and-forward scheme, the relays take advantage of the statistical dependence between their channel outputs and the destination's channel output. The strategies are applied to wireless channels, and it is shown that decode-and-forward achieves the ergodic capacity with phase fading if phase information is available only locally, and if the relays are near the source node. The ergodic capacity coincides with the rate of a distributed antenna array with full cooperation even though the transmitting antennas are not colocated. The capacity results generalize broadly, including to multiantenna transmission with Rayleigh fading, single-bounce fading, certain quasi-static fading problems, cases where partial channel knowledge is available at the transmitters, and cases where local user cooperation is permitted. The results further extend to multisource and multidestination networks such as multiaccess and broadcast relay channels.

2,842 citations


Cites background or result from "Capacity theorems for the relay cha..."

  • ...We summarize the history of information theory for such channels, as well as some recent developments concerning coding strategies....

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  • ...Many other recent results can be found in [26]–[36], [38]–[76], and references therein (see especially [72]–[74] for new capacity theorems)....

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References
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Journal ArticleDOI
David Slepian1, Jack K. Wolf
TL;DR: The minimum number of bits per character R_X and R_Y needed to encode these sequences so that they can be faithfully reproduced under a variety of assumptions regarding the encoders and decoders is determined.
Abstract: Correlated information sequences \cdots ,X_{-1},X_0,X_1, \cdots and \cdots,Y_{-1},Y_0,Y_1, \cdots are generated by repeated independent drawings of a pair of discrete random variables X, Y from a given bivariate distribution P_{XY} (x,y) . We determine the minimum number of bits per character R_X and R_Y needed to encode these sequences so that they can be faithfully reproduced under a variety of assumptions regarding the encoders and decoders. The results, some of which are not at all obvious, are presented as an admissible rate region \mathcal{R} in the R_X - R_Y plane. They generalize a similar and well-known result for a single information sequence, namely R_X \geq H (X) for faithful reproduction.

4,165 citations

Journal ArticleDOI
TL;DR: A coding theorem and weak converse are proved and a necessary and sufficient condition for a positive capacity is derived and upper and lower bounds on the capacity are obtained, which coincide for channels with symmetric structure.
Abstract: Summary The problem of transmitting information in a specified direction over a communication channel with three terminals is considered. Examples are given of the various ways of sending information. Basic inequalities for average mutual information rates are obtained. A coding theorem and weak converse are proved and a necessary and sufficient condition for a positive capacity is derived. Upper and lower bounds on the capacity are obtained, which coincide for channels with symmetric structure.

1,727 citations

Journal ArticleDOI
TL;DR: In Section H of the paper, a characterization of the capacity region for degraded broadcast channels (DBC's) is given, which was conjectured by Bergmans and is somewhat sharper than the one obtained by Gallager.
Abstract: Let \{(X_i, Y_i,)\}_{i=1}^{\infty} be a memoryless correlated source with finite alphabets, and let us imagine that one person, encoder 1, observes only X^n = X_1,\cdots,X_n and another person, encoder 2, observes only Y^n = Y_1,\cdots,Y_n . The encoders can produce encoding functions f_n(X^n) and g_n(Y^n) respectively, which are made available to the decoder. We determine the rate region in case the decoder is interested only in knowing Y^n = Y_1,\cdots,Y_n (with small error probability). In Section H of the paper we give a characterization of the capacity region for degraded broadcast channels (DBC's), which was conjectured by Bergmans [11] and is somewhat sharper than the one obtained by Gallager [12].

498 citations

Journal ArticleDOI
TL;DR: It is established that the Slepian-Wolf theorem is true without change for arbitrary ergodic processes \{(X_i,Y_i)\}_{i=1}^{\infty} and countably infinite alphabets.
Abstract: If \{(X_i, Y_i)\}_{i=1}^{\infty} is a sequence of independent identically distributed discrete random pairs with (X_i, Y_i) ~ p(x,y) , Slepian and Wolf have shown that the X process and the Y process can be separately described to a common receiver at rates R_X and R_Y hits per symbol if R_X + R_Y > H(X,Y), R_X > H(X\midY), R_Y > H(Y\midX) . A simpler proof of this result will be given. As a consequence it is established that the Slepian-Wolf theorem is true without change for arbitrary ergodic processes \{(X_i,Y_i)\}_{i=1}^{\infty} and countably infinite alphabets. The extension to an arbitrary number of processes is immediate.

429 citations

Journal ArticleDOI
A.D. Wyner1
TL;DR: The characterization of the family of rate triples (R_0,R_1, R_2) for which this system can deliver essentially perfect reproductions of X and Y and the principal result is a characterization of this family via an information-theoretic minimization.
Abstract: Let \{(X_k, Y_k, V_k)\}_{k=1}^{\infty} be a sequence of independent copies of the triple (X,Y,V) of discrete random variables. We consider the following source coding problem with a side information network. This network has three encoders numbered 0, 1, and 2, the inputs of which are the sequences \{ V_k\}, \{X_k\} , and \{Y_k\} , respectively. The output of encoder i is a binary sequence of rate R_i, i = 0,1,2 . There are two decoders, numbered 1 and 2, whose task is to deliver essentially perfect reproductions of the sequences \{X_k\} and \{Y_k\} , respectively, to two distinct destinations. Decoder 1 observes the output of encoders 0 and 1, and decoder 2 observes the output of encoders 0 and 2. The sequence \{V_k\} and its binary encoding (by encoder 0) play the role of side information, which is available to the decoders only. We study the characterization of the family of rate triples (R_0,R_1,R_2) for which this system can deliver essentially perfect reproductions (in the usual Shannon sense) of \{X_k\} and \{Y_k\} . The principal result is a characterization of this family via an information-theoretic minimization. Two special cases are of interest. In the first, V = (X, Y) so that the encoding of \{V_k \} contains common information. In the second, Y \equiv 0 so that our problem becomes a generalization of the source coding problem with side information studied by Slepian and Wo1f [3].

423 citations