The Use of Fast Fourier Transform for the Estimation of Power Spectra: A Method Based on Time Aver. aging Over Short, Modified Periodograms

The use of the fast Fourier transform in power spectrum analysis is described. Principal advantages of this method are a reduction in the number of computations and in required core storage, and convenient application in nonstationarity tests. The method involves sectioning the record and averaging modified periodograms of the sections. T INTRODLCTION HIS PAPER outlines a method for the application of the fast Fourier transform algorithm to the estimation of power spectra, which involves sectioning the record, taking modified periodograms of these sections, and averaging these modified periodograms. In many instances this method involves fewer computations than other methods. Moreover, it involves the transformation of sequences which are shorter than the whole record which is an advantage when computations are to be performed on a machine with limited core storage. Finally, it directly yields a potential resolution in the time dimension which is useful for testing and measuring nonstationarity. As will be pointed out, it is closely related to the method of complex demodulation described by Bingham, Godfrey, and Tukey.l THE METHOD Let X ( j ) , j = 0, N 1 be a sample from a stationary, second-order stochastic sequence. Assume for simplicity that E ( X ) 0. Let X ( j ) have spectral density Pcf), I f \ 5%. We take segments, possibly overlapping, of length L with the starting points of these segments D units apart. Let X,(j) , j=O, L 1 be the first such segment. Then X d j ) X($ j = O ; . , L l . Similarly, X d j ) X ( j j = o , . , L L and finally X&) X ( j + ( K 1 ) D ) j 0, , L 1 . We suppose we have K such segments; X l ( j ) , X,($, and that they cover the entire record, Le., that ( K 1 ) D f L N. This segmenting is illustrated in Fig. 1. The method of estimation is as follows. For each segment of length L we calculate a modified periodogram. That is, we select a data window W ( j ) , j = 0, L-1, and form the sequences Xl(j)W(j), X,(j) W(j ) . We then take the finite Fourier transforms A1(n), A K ( ~ ) of these sequences. Here ~ k ( n ) xk(j) w(j )e-z~c i jn lL 1 L-1 L j 0 and i = Finally, we obtain the K modified periodograms L U Ik(fn) I A h ( % ) k 1, 2, K ,