On the approach of a filtered pulse train to a stationary Gaussian process

A narrow-band process is conveniently characterized in terms of a complex envelope whose magnitude is the envelope, and whose angle is the phase variation of the actual narrow-band process. When the narrow-band process is normally distributed, the complex envelope has the properties of a complex normally distributed process. This paper investigates the approach to the complex normally distributed form of the complex envelope of the output of a narrow-band filter when the input is wide-band non-Gaussian noise of a certain class, and the bandwidth of the narrow-band filter approaches zero. The non-Gaussian input consists of a train of pulses having identical waveshapes, but random amplitudes and phases. While the derivations assume statistical independence between pulses, it is shown that the results are valid for a certain interesting class of dependent pulses. The Central Limit Theorem is proved in the multidimensional case for the output process.