Spectra of Digital Phase Modulation by Matrix Methods

In recent years, digital phase-modulation techniques have been playing an increasingly important role in the transmission of information in radio and waveguide systems. Various methods have been developed recently for computing spectra of a sinusoidal carrier phase modulated by a random baseband pulse train. 1 - 8 In this paper, we derive the spectral density of a carrier phase modulated by a random baseband pulse stream 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 and the pulses from different time slots overlap, we assume that the symbols transmitted during different time slots are statistically independent. 899 The present method also yields the spectral density of the output of a nonlinear device whose input is a similar baseband pulse train. The work reported here generalizes and simplifies prior results. The form of the present results provides an appropriate division between analysis and machine computation that enhances physical understanding and simplifies numerical computations. We compute the spectra of binary, quaternary, octonary, and 16-aryPSIC systems, with overlapping baseband modulation pulses of several shapes.