Capacity-Achieving Signal and Capacity of Gaussian Mixture Channels with 1-bit Output Quantization

This paper addresses the optimal signaling scheme and capacity of an additive Gaussian mixture (GM) noise channel using 1-bit output quantization. The considered GM distribution is a weighted sum Gaussian component densities with arbitrary means, and it can be used to represent any non-Gaussian channel of engineering interest. By first establishing a necessary and sufficient Kuhn-Tucker condition (KTC) for an input signal to be optimal, we demonstrate that the maximum number of mass points in the capacity-achieving signal is four. Our proof relies on novel bounds on the product of Q functions and Dubin’s theorem. By considering a special case of GM with zero mean Gaussian components, which is a realistic accurate model for co-channel interference in heterogeneous wireless networks and impulsive interference, it is shown that the optimal input is $\pi$/2 circularly symmetric. As a result, in this case, the capacity-achieving signal has exactly four mass points forming a square centered at the origin. By further checking the first and second derivatives of the modified KTC, it is then shown that the phase of the optimal mass point located in the first quadrant is $\pi$/4. Thus, with zero-mean GM, the capacity-achieving input signal is QPSK, and the channel capacity can be established in closed-form.