Recursive computation of the Fourier transform

Efficient recursive methods and circuits for computing a continuously updated discrete Fourier transform (DFT) of an input-digital signal are considered. The Fourier transform (FT) is recomputed at each sample input time, with only O(N) operations being required to compute the transform, where N is the number of frequency bins. Various window functions are considered for windowing the input-wave form, namely a rectangular window, a triangular window, and an exponential window. The last type of window has not been widely considered in the past, partly due to its asymmetrical shape, and hence nonlinear phase response. Nevertheless, it is shown to have certain advantages in ease of computation and in flexibility. For the exponential window, a circuit that conveniently allows zooming in to particularly interesting parts of the frequency spectrum is shown. By appropriately loading a multiplier storage RAM, arbitrarily fine resolution may be achieved in any part of the spectrum, thus permitting closely adjacent peaks to be distinguished. The general approach is based on the interpretation of the FT as a set of simultaneous bandpass filters. Though these filters are generally finite-impulse-response filters, computational advantages are derived from formulating them as recursive filters.<>