On the Accuracy of the High-SNR Approximation of the Differential Entropy of Signals in Additive Gaussian Noise: Real and Complex Cases

One approach to the analysis of the high signal-to-noise ratio (SNR) capacity of noncoherent 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 when the noise has a Gaussian distribution. We consider the complex and real cases, and we show that when the probability density function (pdf) of the signal component of the received signal is piecewise differentiable, the approximation error decays as 1/SNR, which tightens the available result that the error decays as o(1). In addition, we consider the special instance in which the signal component of the received signal corresponds to a signal transmitted over a channel with a Gaussian fading coefficient. For that case, we provide explicit expressions for the first nonconstant term of the Taylor expansion of the differential entropy, and we invoke Schwartz´s inequality to obtain an efficiently computable bound on it. Our results are supported by numerical examples.