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Concept# Relay channel

Summary

In information theory, a relay channel is a probability model of the communication between a sender and a receiver aided by one or more intermediate relay nodes.
A discrete memoryless single-relay channel can be modelled as four finite sets, and , and a conditional probability distribution on these sets. The probability distribution of the choice of symbols selected by the encoder and the relay encoder is represented by .
o------------------o
| Relay Encoder |
o------------------o
Λ |
| y1 x2 |
| V
o---------o x1 o------------------o y o---------o
| Encoder |--->| p(y,y1|x1,x2) |--->| Decoder |
o---------o o------------------o o---------o
There exist three main relaying schemes: Decode-and-Forward, Compress-and-Forward and Amplify-and-Forward. The first two schemes were first proposed in the pioneer article by Cover and El-Gamal.
Decode-and-Forward (DF): In this relaying scheme, the relay decodes the source message in one block and transmits the re-encoded message in the following block. The achievable rate of DF is known as .
Compress-and-Forward (CF): In this relaying scheme, the relay quantizes the received signal in one block and transmits the encoded version of the quantized received signal in the following block. The achievable rate of CF is known as subject to .
Amplify-and-Forward (AF): In this relaying scheme, the relay sends an amplified version of the received signal in the last time-slot. Comparing with DF and CF, AF requires much less delay as the relay node operates time-slot by time-slot. Also, AF requires much less computing power as no decoding or quantizing operation is performed at the relay side.
The first upper bound on the capacity of the relay channel is derived in the pioneer article by Cover and El-Gamal and is known as the Cut-set upper bound. This bound says where C is the capacity of the relay channel. The first term and second term in the minimization above are called broadcast bound and multi-access bound, respectively.
A relay channel is said to be degraded if y depends on only through and , i.

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Related publications (3)

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Network information theory explores the fundamental data transport limits over communication networks. Broadcasting and relaying are two natural models arising in communication contexts where multiple users share a common communication medium. Number of relay and broadcasting models are considered in this thesis. We first collect the existing methodologies of communication over a discrete memoryless relay channel. A new communication strategy over this channel is also proposed, which achieves the same data-rates as the best known schemes in literature, incurring a lower delay at the same time. We then consider the Gaussian broadcast channel (GBC), and show that its capacity region is enlarged by the availability of feedback, even if the feedback is from only one of the receivers. We further show a low complexity, explicit capacity achieving strategy over a GBC with degraded receivers. A combination of relay and broadcast models, called the relay-broadcast channel (RBC) is analyzed next. Achievable regions are proposed for different RBC scenarios. For the Gaussian version of an RBC, we characterize the complete capacity region when the receivers are degraded with respect to the relay. The final part of this thesis investigates communication over fading channels. We propose upper bounds for non-coherent, as well as lower bounds for coherent communication over a MIMO fading channel.

While both fundamental limits and system implementations are well understood for the point-to-point communication system, much less is developed for general communication networks. This thesis contributes towards the design and analysis of advanced coding schemes for multi-user communication networks with structured codes. The first part of the thesis investigates the usefulness of lattice codes in Gaussian networks with a generalized compute-and-forward scheme. As an application, we introduce a novel multiple access technique --- Compute-Forward Multiple Access (CFMA), and show that it achieves the capacity region of the Gaussian multiple access channel (MAC) with low receiver complexities. Similar coding schemes are also devised for other multi-user networks, including the Gaussian MAC with states, the two-way relay channel, the many-to-one interference channel, etc., demonstrating improvements of system performance because of the good interference mitigation property of lattice codes. As a common theme in the thesis, computing the sum of codewords over a Gaussian MAC is of particular theoretical importance. We study this problem with nested linear codes, and improve upon the currently best known results obtained by nested lattice codes. Inspired by the advantages of linear and lattice codes in Gaussian networks, we make a further step towards understanding intrinsic properties of the sum of linear codes. The final part of the thesis introduces the notion of typical sumset and presents asymptotic results on the typical sumset size of linear codes. The results offer new insight to coding schemes with structured codes.

In this paper, we investigate the diversity-multiplexing tradeoff (DMT) of the multiple-antenna (MIMO) static half-duplex relay channel. A general expression is derived for the DMT upper bound, which can be achieved by a compress-and-forward protocol at the relay, under certain assumptions. The DMT expression is given as the solution of a minimization problem in general, and an explicit expression is found when the relay channel is symmetric in terms of number of antennas, i.e., the source and the destination have n antennas each, and the relay has m antennas. It is observed that the static half-duplex DMT matches the full-duplex DMT when the relay has a single antenna, and is strictly below the full-duplex DMT when the relay has multiple antennas. Besides, the derivation of the upper bound involves a new asymptotic study of spherical integrals (that is, integrals with respect to the Haar measure on the unitary group u(n)), which is a topic of mathematical interest in itself.