Capacity-achieving signals of non-coherent rayleigh fading channels with additive gaussian mixture noise

This paper investigates the detailed characterization of the capacity-achieving input signals for a non-coherent Rayleigh fading channel with Gaussian mixture noise where neither the transmitter nor the receiver has the knowledge of fading coefficients. The considered model is suited for wireless networks having multi-tier heterogeneous architectures in which the channel conditions change rapidly. By first establishing an integrable upper bound on the absolute function of the integrand in the output entropy equation, we demonstrate that there exists a unique input distribution that achieves the channel capacity. By formulating the Kuhn-Tucker condition (KTC), we then examine in detail the number of mass points in the optimal input distribution. Specifically, by establishing a diverging lower bound on the KTC, we show that it is not possible for the optimal input distribution to have an infinite number of mass points. As a result, the capacity-achieving input distribution is discrete having a finite number of mass points. Finally, we develop a simple numerical method to evaluate the optimal input and compute the capacity.