S.B. Makarov - Dr. Sc. (Eng.), Professor, Head of Institute of Physics, Nanotechnology and Telecommunications, St. Petersburg State Polytechnical University. E-mail: makarov@cee.spbstu.ru
S.V. Zavyalov - Post-graduate Student, St. Petersburg State Polytechnical University. E-mail: sz5@mail.ru

The study of the energy spectra of random sequences of multi-frequency PSK 4 signals with non-orthogonal location of subcarriers for different form of envelope has been conducted. Application of envelope a(t) = cos2(πt/T) can reduce bandwidth at level of −50 dB in one and a half times in comparison with a(t) = cos(πt/T). Bandwidth at level of −30 dB is three times less in comparison with a(t) = 1. Occupied bandwidth for the level of −30 dB decreases with decreasing of value Δf, but decreasing rate of level of out-of-band emissions of energy spectrum remains practically constant for fixed type of envelope. Optimization problem of the synthesis of the complex envelope of signals with duration T in accordance with the criterion of minimization of out-of-band emissions formulated and solved. The expansion coefficients of envelope and the corresponding energy spectra for fixed values of decreasing rate of level of out-of-band emissions of energy spectrum are found. The main lobe of the energy spectrum expands with increasing of decreasing rate of level of out-of-band emissions (energy is concentrated near the basic frequency with increasing of requirements for decreasing rate of level of out-of-band emissions of the energy spectrum). Transition to the optimization problem in the presence of conditions of limitation the cross-correlation coefficient of signals with intersymbol interference is performed. The expansion coefficients of envelope a(t) and the corresponding energy spectra for fixed values of decreasing rate of level of out-of-band emissions of energy spectrum and fixed value of cross-correlation coefficient are found. The duration of main lobe of the energy spectrum reduce with application of limitation on cross-correlation coefficient. Zero values of energy spectrum can be obtained at 1/T, 2/T, 3/T,? by choosing duration of signals with envelope a(t) equal to 2.3T. Level of interchannel interference has minimum value for the case of location of subcarriers on these points. Obtained signals has higher decreasing rate of level of out-of-band emissions in comparasion with OFDM signals for equal data rate. Proposed multi-frequency signals have the same form of envelope on each subcarrier frequency. This fact allows us to use same digital structures, which are used for OFDM signals. Peak-to-average power ratio of random sequence of signals for obtained envelopes are studied. Mathematical expectation of peak-to-average power ratio can be reduced on 2,5 dB by application envelope of signals obtained as solution of the optimization problem with limitation on cross-correlation coefficient. BER performance for multi-frequency non-orthogonal signals for the case of application nonlinear bit-by-bit coherent detection algorithm on the decision diagram has been studied. For application of obtained optimal PSK 4 signals with duration 2T (with limitation on cross-correlation coefficient) on each subcarrier energy losses are equal to 0.2 dB in comparison with OFDM signals, energy gain is equal to 4 dB in area of BER performance p = 10−5 in comparison with application of obtained optimal PSK 4 signals with duration 2T (without limitation on cross-correlation coefficient). Thus application of obtained optimal PSK 4 signals (with limitation on cross-correlation coefficient) is rational.