Precoding for the AWGN Channel With Discrete Interference

For a state-dependent discrete memoryless channel with input alphabet ${\cal X}$, state alphabet ${\cal S}$, and output alphabet ${\cal Y}$ where the independent and identically distributed (i.i.d.) state sequence is known causally at the transmitter, it is shown that by using at most $\min \{\vert{\cal X}\vert \vert{\cal S}\vert -\vert{\cal S}\vert +1,\vert{\cal Y}\vert \}$ out of $\vert{\cal X}\vert^{\vert{\cal S}\vert}$ inputs of the Shannon's derived channel, the capacity is achievable. As an example of state-dependent channels with side information at the transmitter, $M$-ary signal transmission for the additive white Gaussian noise (AWGN) channel with additive $Q$-ary interference where the sequence of i.i.d. interference symbols is known causally at the transmitter is considered. The optimal precoding scheme is derived under the constraint that the channel input given any current interference symbol is uniformly distributed over the channel input alphabet. It is shown that at low signal-to-noise ratio (SNR) not doing precoding is optimal. For the special case where the Gaussian noise power is zero, it is shown that the rate $\log_2 M$ is achievable by a one-shot coding scheme if ${\cal X}$ is an arithmetic progression.