Power Spectra of Multilevel Digital Phase-Modulated Signals

Power spectra of digital angle-modulated signals can be calculated in many ways. The direct way of defining the power spectrum is to find the Fourier transform of a sample of the signal on a finite time interval T,, . The magnitude square of this Fourier transform is then divided by T,, and averaged over all possible values of the signal. The power spectrum is finally obtained by taking the limit of the previous result as T,, tends to infinity. Power spectra of binary frequency shiftkeyed signals have been calculated by this method by W. R. Bennett and S. 0. Rice. 1 R. R. Anderson and J. Saltz 2 have extended the analysis to multilevel digital frequency-modulated signals by using the same technique. Power spectra of digital phase-modulated signals can also be obtained 2857 2858 T H E BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1971 from the Fourier transform of the signal autocorrelation function. This second method has been used by L. Lundquist 3 to calculate the power spectra of signals phase-modulated by pulse stream akg{t -- kT). The results obtained in his analysis apply to the case of overlapping pulses providing that the random discrete variables a k are independent and have identical probability distributions. In this paper, a general expression is derived for the power spectrum of multilevel digital phase-modulated signals by using the Fourier transform technique. The only restriction in these calculations is that the signal is modulated by independent non-overlapping pulses.