An optimal discrete window for the calculation of power spectra

Let g be a function defined upon R, with values in C and G its Fourier transform. Let be the distribution upon R defined by = =0N-1 (kT/N) where and δ α is the Dirac function at abscissa α. is a discrete "time window" and its Fourier transform is a periodic function of the frequency (period N/T). Taking the Fourier transform of the product of by g, we obtain F₆ = F Fg = G (* means convolution product). F₆ is also a periodic function of the frequency (period N/T) and F₆ (jT) = =0N-1 g - 2i jkN where gk = g (k (T/N) ). F₆ (j/T) for j=0,. . . , N-1 is obtained very efficiently using the FFT algorithm of Cooley and Tukey. Cleverly choosing the weights, |F (j/T) |^2 for j = -N/2,. . . , N/2-1 is a good estimator of the power spectrum of g. The vector γ (with components, k=0,. . . , N-1) that maximize the ratio -1/T1/T| () |^{2 d}-N/2TN/2T| () |^{2 d} gives us an optimal discrete window. Then γ is the eigenvector corresponding to the greatest eigenvalue λ 0 of a matrix M defined by Mₐk = \ 2 ({q-k{N) } (q-k), k=0,. . . , N - 1}2{N if q=k, q = 0,. . . , N - 1} The method for calculating this eigenvector is shown for large values of N (N = 2048).