A duality theorem for joint source-channel coding

We consider joint source-channel coding for a memoryless Gaussian source and an additive white Gaussian noise (AWGN) channel. For a given code defined by an encoder-decoder pair (α,β), its dual code is obtained by interchanging the encoder and decoder: (β,α). It is shown that if a code (α,β) is optimal at rate ρ channel uses per source sample and if it satisfies a certain uniform continuity condition, then its dual code (β,α) is optimal for rate 1/ρ channel uses per source sample. It is demonstrated that there is a code which is optimal but its dual code is not optimal. Finally, using random coding, we show that there is an optimal code which has an optimal dual