Theory of Analytic Modulation Systems

In 1962, Bedrosian proposed a modulation system called single sideband phase (or frequency) modulation.* (See Ref. 5.) The modulated signal is of the form Re jei(o,ct+x(t)+iyU)) = e-y(t) cos (Wct + x(t)) where x(t) is the "baseband" signal, y(t) is the Hilbert transform of x(t), and coc is the carrier frequency. The special relation between the amplitude modulation e~y{t) and the phase modulation x(t) results in the modulated signal having no spectrum in the interval (--coc,a>c); i.e., the amplitude modulation removes the lower sideband. However, the spectrum is still infinite in extent. We adopt the terminology "single sideband exponential modulation" (SSBEM) for this system. Bedrosian pointed out that SSBEM was compatible with conventional FM receivers, and suggested that the single-sideband system might offer * K. H. Powers received a U. S. Patent (No. 3,054,073) on such a system shortly before the appearance of Bedrosian's paper. See Voelcker.21 491 some savings in transmission bandwidth over the conventional system. However, since filtering operations could radically alter the zero crossings of the modulated signal, it was not clear to what degree one could maintain compatibility and at the same time realize some saving in bandwidth. · Others 2,10,17 have compared the spectral distribution of single sideband and conventional frequency modulated signals for the cases of sinusoidal and Gaussian noise modulation. They have shown that the "effective" bandwidths, as measured by central second moments, of the single-sideband signals may be greater or less than that of the conventional FM signal depending on the nature of the modulation.