S.I. Ziatdinov – Dr.Sc.(Eng.), Professor, 

Department of Information and Net Technology, Saint-Petersburg State University of Aerospace Instrumentation E-mail: kaf53@guap.ru

We consider the transmission of information over a communication channel using modulation of a high-frequency carrier oscillation. Modulation refers to a change in one or more parameters of the carrier oscillation by a transmitted message. In practice, either amplitude or angular modulation is used. Currently, the most common is phase modulation, which as a kind of angular modulation is one of the main methods of transmission of messages over the information channel. In the existing literature, spectral-correlation characteristics of phase-modulated signals are well studied for special cases when the modulating signal is either a harmonic oscillation or a random process with zero average frequency, which limits the use of theoretical results in solving a variety of practical problems. The spectral-correlation properties of the carrier oscillation in the case of phase modulation by a normal narrowband random process with a nonzero average frequency are considered in the article from a General standpoint. Analytical relations for the correlation function and the energy spectrum of a phase-modulated signal by a random process with a nonzero average frequency are obtained. As a carrier signal, a harmonic oscillation with a given amplitude, frequency and initial phase is taken. For phase modulation, a normal narrowband stationary random process with random amplitude and initial phase at a constant average frequency is used. An expression for the correlation function of the phase-modulated carrier is obtained for the received signal models. By applying the WienerHinchen inverse probing to the phase-modulated carrier correlation function, an expression for the energy spectrum of the phasemodulated carrier is obtained. To concretize the results, the most frequently used Gaussian model of the envelope of the correlation coefficient of the fluctuating phase of a modulating random process was taken. Based on the received signal models, the energy spectrum of the phase-modulated carrier is calculated and constructed. The obtained results show that the energy spectrum of the phase-modulated harmonic oscillation is infinitely wide and contains a discrete spectral line and a partial region at the frequency of the modulated oscillation, as well as partial regions corresponding to the harmonics of the average frequency of the modulating narrowband random signal relative to the frequency of the modulated oscillation. The width of the spectral regions depends on the dispersion of the modulating signal and the number of the corresponding harmonic of the average frequency of the modulating signal. In General, the shape and width of the spectral density of the phase-modulated signal are determined by the dispersion of the fluctuating phase. When the RMS value of the fluctuating phase is much less than one radian, in the spectrum of the phase-modulated signal there are practically only the lower and upper side partial regions determined by the average frequency of the modulating random signal. With increasing dispersion of the fluctuating phase in the spectrum of the phase-modulated signal, additional lateral partial regions appear at frequencies multiple of the average frequency of the modulating signal relative to the frequency of the modulated signal. The analytical relations for the correlation function and the energy spectrum of the phase-modulated signal presented in General form are easily reduced to special cases of use for phase modulation of a harmonic signal or a random process with zero average frequency. The theoretical results obtained in the analysis of spectral-correlation properties of phase-modulated signals can be used in the research and development of information transmission systems with angular modulation of the carrier oscillation.

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