A cauchy input achieves the capacity of a Cauchy channel under a logarithmic constraint

Abstract

In this work, we consider a discrete-time memoryless communication channel where the input is subjected to an independent additive Cauchy noise. We find the input constraint under which a Cauchy input is capacity achieving. The constraint is logarithmic and depends on a scalar parameter k which we interpret as a power measure. We draw a parallelism between this setup and that of the Gaussian channel under the second moment constraint. In fact, a Cauchy input yields a Cauchy output over this channel and achieves a capacity value of “log(1 + SNR)”.