Window-Presum Parametric Discrete Fourier Transform

The article gives an analysis of the advantages and disadvantages of the original method for implementing a practical spectrum analyzer based on the discrete Fourier transform (DFT). There are two names used for this method: the weighted overlap-add structure, and the window-presum FFT method. It is shown that the main disadvantage of the weighted superimposition-addition method is the fixation of the central frequencies of the filters of the realized spectrum analyzer. The theoretical foundations of this method have been discovered and investigated. It is shown that the reason for the disadvantage of the method is the procedure used for preliminary data processing in the time domain. Based on the analysis of the DFT matrix, it is shown that the preprocessing procedure used in the time domain is only one of the possible procedures. A generalization of the weighted superposition-addition (FFT method with preliminary summation) is proposed on the basis of a parametric discrete Fourier transform.