Numerical calculation of information rates and capacity of quadrature Gaussian mixture channels

This paper presents novel methods to accurately calculate the information rates and capacity of quadrature Gaussian mixture (GM) noise channels without the need of time-consuming Monte Carlo simulations or numerical integrations. The focus is on three important input signals: i) a Gaussian input; ii) a complex input with discrete amplitude and independent uniform phase, which is a capacity-achieving input; and iii) finite-alphabet signaling schemes, such as practical quadrature amplitude modulation (QAM). To this end, a novel piecewise-linear curve fitting (PWLCF) method is first proposed to estimate the entropy of a complex GM random variable to achieve any desired level of accuracy. The result can then be used to calculate the information rate when a Gaussian input is used. For a complex input with discrete amplitude and independent uniform phase, the output entropy is estimated in a similar manner but using polar coordinates and the Kernel function. When a finite-alphabet input is used, we exploit the Laguerre-Gauss quadrature formula for an effective calculation of the output entropy. Combining with the noise entropy, we show that in all cases, the information rates can be computed accurately.