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A channel with output Y = X + S + Z is examined, The state S \sim N(0, QI) and the noise Z \sim N(0, NI) are multivariate Gaussian random variables ( I is the identity matrix.). The input X \in R^{n} satisfies the power constraint (l/n) \sum _{i=1}^{n}X_{i}^{2} \leq P . If S is unknown to both transmitter and receiver then the capacity is frac{1}{2} \ln (1 + P/( N + Q)) nats per channel use. However, if the state S is known to the encoder, the capacity is shown to be C^{\ast } =frac{1}{2} \ln (1 + P/N) , independent of Q . This is also the capacity of a standard Gaussian channel with signal-to-noise power ratio P/N . Therefore, the state S does not affect the capacity of the channel, even though S is unknown to the receiver. It is shown that the optimal transmitter adapts its signal to the state S rather than attempting to cancel it.

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IEEE Transactions on Information Theory  (Volume:29 ,  Issue: 3 )