Aperiodic Discrete Fourier Transform in Problems of Measuring the Energy Spectrum of Random Signals

Abstract

The paper studies two inter-related but not replacing each other ways of measuring the energy spectrum of random discrete signals which are stationary according to A. Ya. Khinchin: the frequency approach and correlation approach. Advantages and drawbacks of each of these approaches are considered. The paper describes the following division of estimation of characteristics for random discrete signals which are stationary in the broad sense: sampling (estimate obtained in a particular experiment, “estimates”), evaluation considered as a random variable (“estimators”), true (theoretical) values of characteristics of stationary random signals ("the theoretical value"). Two versions are considered for interpreting  implementation of the random discrete signal for the finite interval , : as the value  (here  is the symbol of transposing operation) of the-dimension random value , and as the segment ,  of the random discrete signal for the finite interval. It is proved that the estimation of the energy spectrum of the random discrete signal for the finite interval obtained by discrete Fourier transform is not the transform of the estimation of the aperiodic correlation function; it is the transform of some new estimation of the correlation function – the cyclic correlation function. The paper proposes the solution of problems appearing in this case by introduction of a new transform – aperiodic discrete Fourier transform.