A Linear Filtering Approach to the Computation of Discrete Fourier Transform

It is shown in this paper that the discrete equivalent of a chirp filter is needed to implement the computation of the discrete Fourier transform (DFT) as a linear filtering process. We show further that the chirp filter should not be realized as a transversal filter in a wide range of cases; use instead of the conventional FFT permits the computation of the DFT in a time proportional to N logz N for any N , N being the numberyof points in the array that is transformed. Another proposed implementation of the chirp filter requires N to be a perfect square. The number of operations required for this algorithm is proportional to N312. Manuscript received October 15,1968; revised August 20,1970. Introduction There is currently a good deal of interest [l], [2] in a technique known as the fast Fourier transform (FFT), which is a method for rapidly computing the discrete Fourier transform of a time series (discrete data samples). This transform is the array of N numbers, A,, n=O, 1, . . , N 1 which are defined by the relation A, = X k exp (27rjnk/N) where Xk is the kth sample of a time series consisting of N (possibly complex) samples and j=.\/?. Quite obviously, computing the A , in a brute force manner requires N2 operations,la numberwhich is too large to make the use of the discrete transform attractive. The fast Fourier transform discovered by Cooley and Tukey [3], on the other hand, requires about N log2 N operations (if N is chosen to be a power of 2) a saving which is so great when N is large that it makes the use of the discrete transform attractive in such fields as digital filtering, computation of power spectra and autocorrelation functions and the like. If N is not chosen optimally, that is, if N is a composite number of the form 1\71