An alternative proof of de Bruijn's identity for additive Gaussian noise channels with independent component

Additive noise channels are the most commonly used channels in signal processing. In these channels the received signal, random variable Y, is composed of a transmitted signal, random variable X, and an additive noise, random variable Z. One of the important problems studied on the received signal is the entropy of random variable Y. When additive noise Z is an independent Gaussian random variable with zero mean and unit variance, the elegant algebraic connection between differential entropy of output signal Y and Fisher information is stated through a relation known as the De Bruijn’s identity. In this paper, we first obtain a general relation for differentials of conditional distribution of output signal and use it to prove the relationship between the first derivative of differential entropy of output signal and its Fisher information. This method can be used for extending De Bruijn’s identity when the additive noise is distributed as another useful statistical distributions.