Approximation of Achievable Rates in Additive Gaussian Mixture Noise Channels

In this paper, we detail effective methods to approximate the achievable rates of channels with additive Gaussian mixture (GM) noise for both real and complex channels to achieve any desired level of accuracy. Attention is paid to a Gaussian input, a discrete real input, and a complex input with discrete amplitude and independent uniform phase. Such discrete inputs represent a wide range of input distributions and they include the capacity-achieving inputs as special cases. At first, we propose a simple technique to accurately calculate the noise entropy. Specifically, when the noise level is high, a lower bound on the integrand of the entropy is established and the noise entropy can be estimated using a closed-form solution. In the low noise region, the piecewise-linear curve fitting (PWLCF) method is applied. We then extend this result to calculate the achievable rate when the input is Gaussian distributed, which is shown to be asymptotically optimal. Next, we propose a simple PWLCF-based method to approximate the output entropy for a real GM channel when the input is discrete, and for a complex GM channel when the input is discrete in amplitude with independent uniform phase. In particular, for the real channel, the output entropy is evaluated by examining the output in high and low regions of amplitude using a lower bound on the integrand of the output entropy and PWLCF, respectively. For the complex channel, the output entropy is approximated a similar manner but using polar coordinates and the Kernel function. It is demonstrated that the output entropy, and consequently, the achievable rates, can be computed to achieve any given accuracy level.