The Capacity of the Relay Channel



Consider the following seemingly simple discrete memoryless relay channel:
Here Y 1, Y 2 are conditionally independent and conditionally identically distributed given X, that is, \(p(y_1,\, y_2\, |\, x) = p(y_1\, |\, x) p(y_2\, |\, x)\). Also, the channel from Y 1 to Y 2 does not interfere with Y 2. A (2nR, n) code for this channel is a map \(x : 2^{nR} \rightarrow X^n\), a relay function \(r :Y_1^{n}\rightarrow 2^{nC_{0}}\), and a decoding function \(g : 2^{nC_{0}} \times Y_2^{n} \rightarrow 2^{nR}\). The probability of error is given by
$$ P_e^{(n)} = P \{\,g(r(y_1),y_2) \ne W\}$$
, where W is uniformly distributed over 2nR and
$$ p(w, y_1, y_2) = 2^{-nR}\,\, \underset {i=1}{\overset {n}{\Pi}} p(y_{1i}\, |\, x_i(w)) \,\,\underset {i=1}{\overset {n}{\Pi}}\,\, p(y_{2i}\, |\, x_i(w))$$


  1. [1]
    T. Cover and A. El Gamal, “Capacity Theorems for the Relay Channel,” IEEE Trans. Inf. Theory, IT-25, No. 5, pp. 572–584 (Sept. 1979).CrossRefGoogle Scholar

Copyright information

© Springer-Verlag New York Inc. 1987

Authors and Affiliations

  1. 1.Departments of Electrical Engineering and StatisticsStanford UniversityStanfordUSA

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