The digital computation of discrete spectra using the fast Fourier transform

This paper is devoted to a discussion of discrete spectrum analysis which is important in applicational areas such as sonar and replica correlation. The discrete Fourier transform is shown to arise naturally as a consequence of finite impulsive sampling and the fast Fourier transform is introduced as the most efficient means of computing the discrete Fourier transform. These are described in terms of parameters pertinent to digital sonar signal processing, including resolution, dynamic range, and processing gain. Computational accuracy is investigated as a function of word lengths associated with the data, kernels, and intermediate transforms for both conditional and automatic array scaling. In real-time equipment, it is frequently necessary to employ some sort of automatic gain control and such a device is investigated here. Results are presented which enable specification of word length and automatic gain control requirements as a function of desired dynamic range, input signal-to-noise ratio, and mean-square error at the quantizer output.