Spectrum Conservation and Characteristics of Single-Sideband Phase Modulation

Phase modulation with an analytic signal, which is a Gaussian random process, is examined in order to determine the amount of spectrum conservation that may be achieved by using single-sideband phase modulation (SSB-PM) rather than conventional phase modulation (PM). The autocorrelation function is derived and found to be an analytic signal in terms of the autocorrelation function of the actual modulating signal and its Hilbert transform. When the modulating signal strength is very low, the sideband spectral distribution is the same as that of the actual modulating signal or single-sideband amplitude modulation. As the modulating signal mean-square value is increased, the sideband spectrum broadens and approaches a Gaussian shape. The average power output of an SSB-PM system increases exponentially with input modulating signal strength, while the carrier power remains constant. For the same modulating signal mean-square value, a greater fraction of power is in the one sideband of an SSB-PM system than in the two sidebands of conventional PM. Single-sideband phase or frequency modulation always effectuates spectrum conservation in the continuum when it is compared with conventional phase or frequency modulation on the basis of equal relative sideband power. A Fourier transform computer program is used to generate SSB-PM spectral distributions with varying modulating signal mean-square values, when the modulating signal spectrum is a low-pass rectangular spectrum, a narrowband pass spectrum, and the shape of an average voice spectrum. These examples illustrate the power series formulation of the output spectrum as well as the theoretical analysis of bandwidth.