Spectra of Digital Phase Modulation by Matrix Methods

We derive the spectral density of a sinusoidal carrier phase modulated by a random baseband pulse train in which the signaling pulse duration is finite and the signaling pulses may have different shapes. The spectral density is expressed as a compact Hermitian form in which the Hermitian matrix is a function of only the symbol probability distribution, and the associated column vector is a function of only the signal pulse shapes. If the baseband pulse duration is longer than one signaling interval, we assume that the symbols transmitted during different time slots are statistically independent. The applicability of the method to compute the spectral density is illustrated by examples for binary, quaternary, octonary, and 16-ary PSK systems with different pulse overlap. Similar methods yield the spectral density of the output of a nonlinear device whose input is a random baseband pulse train with overlapping pulses.