Estimation of achievable rates in additive Gaussian mixture noise channels

This paper details novel methods to accurately estimate the achievable rates of channels with additive Gaussian mixture (GM) noise. Attention is paid to a Gaussian input and discrete inputs. Such discrete inputs represent a wide range of signaling strategies and include the capacity-achieving input as a special case. At first, we propose a simple technique to calculate the GM noise entropy. Specifically, when the noise level is high, a lower bound on the integrand of the noise entropy is established and the noise entropy can be estimated in closed-form. In the low noise region, the piecewise-linear curve fitting (PWLCF) method is applied to calculate the noise entropy. It is then demonstrated this can be estimated in both regions with a predetermined accuracy. We then extend this result to calculate the output entropy and 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 estimate the output entropy for a given discrete input. In particular, 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. It is demonstrated that the output entropy, and consequently, the achievable rates, can be computed to achieve any desired accuracy level.