Arbitrarily Tight Bounds on Differential Entropy of Gaussian Mixtures

A sequence of lower and upper bounds is derived on the differential entropy of a Gaussian mixture where the Gaussian components only differ in mean values. As the sequence index increases, the computational complexity of the bounds increases; however, the gap between the lower and upper bounds becomes vanishingly small. We address the applications of these bounds in several communication scenarios where the transmitters utilize Pulse Amplitude Modulation (PAM) constellations to transmit data.