Measuring of Amplitudes, Frequencies and Phases of Simple Periodic Components of the Complex Signal at Spectral Analysis by Fast Fourier Transform

Abstract

The method of estimation of amplitudes, frequencies and phases of final number of simple periodic components of a difficult signal on the discrete spectrum received by fast Fourier transformation (FFT) is offered. In order to apply the offered algorithms it is necessary to know only aprioristic data (the minimum volume of realization and the step of digitization) and the calculated data - the valid and imaginary parts of readout of the spectrum laying about the greatest maxima.

Modeling of the offered algorithm operation is carried out. The total signal consisting of seven sinusoids with individual amplitudes, different frequencies and phases was modeled. Channel noise was modeled by the addition of normally distributed random variable to the exact readout of the total signal, and the noise of 50 Hz – by the addition of realization of a sinusoid with the random amplitude and phase.

Definition of parameters of composed sinusoids was made by two iterative methods. At the first method the total spectrum of a signal was considered as the sum of spectra of seven independent signals. At the second method the spectrum of a total signal was considered as the sum of spectra of independent pairs of signals. Errors of phase definition did not surpass 3 %; and errors of measurement of amplitudes and frequencies did not surpass 1 %.

The study can serve as a theoretical foundation for the algorithm for measuring the amplitudes, frequencies, and phases of a finite number of simple periodic components of a complex signal using the fast Fourier transformation in the absence of a priori information about the signal parameters using only real and imaginary components of the discrete spectrum of the signal.