On the accuracy of the high SNR approximation of the differential entropy of signals in additive Gaussian noise

One approach for analyzing the high signal-to-noise ratio (SNR) capacity of non-coherent wireless communication systems is to ignore the noise component of the received signal in the computation of its differential entropy. In this paper we consider the error incurred by this approximation when the transmitter and the receiver have one antenna each, and the noise has a Gaussian distribution. For a general instance of this case, we show that the approximation error decays as 1/SNR. In addition, we consider the special instance in which the received signal corresponds to a signal transmitted over a channel with additive Gaussian noise and a Gaussian fading coefficient. For that case, we provide an explicit expression for the second order term of the Taylor series expansion of the differential entropy. To circumvent the difficulty that arises in the direct computation of that term, we invoke Schwartz´s inequality to obtain an efficiently computable bound on it, and we provide examples that illustrate the utility of this bound.