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 <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∇</tex> be the distribution upon R defined by <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∇ = ∑min{k=0}max{N-1} γκδ(kT/N)</tex> where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">γκ</tex> and δ <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">α</inf> is the Dirac function at abscissa α. <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∇</tex> is a discrete "time window" and its Fourier transform is a periodic function <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Γ</tex> of the frequency (period N/T). Taking the Fourier transform of the product of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∇</tex> by g, we obtain <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">F[∇g] = F [∇] F[g] = Γ G</tex> (* means convolution product). <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">F[∇g]</tex> is also a periodic function of the frequency (period N/T) and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">F[∇g] (j/T) = ∑min{k=0}max{N-1} γκ · gκ · exp - 2iπ jk/N</tex> where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">gk = g(k(T/N)). F[∇g](j/T)</tex> for j=0,..., N-1 is obtained very efficiently using the FFT algorithm of Cooley and Tukey. Cleverly choosing the weights <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">γκ, |F[∇](j/T)|²</tex> for j = -N/2, ..., N/2-1 is a good estimator of the power spectrum of g. The vector γ (with components <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">γκ, k=0, ..., N-1</tex> ) that maximize the ratio <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">{∫min{-1/T}max{1/T}|Γ(λ)|² dλ}{∫min{-N/2T}max{N/2T}|Γ(λ)|² dλ}</tex> gives us an optimal discrete window. Then γ is the eigenvector corresponding to the greatest eigenvalue λ <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</inf> of a matrix M defined by <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Mqk = \{min{sin 2π(q-k/N)}{π(q-k)}, k=0,..., N - 1}max{2/N if q=k, q = 0,..., N - 1}</tex> The method for calculating this eigenvector is shown for large values of N (N = 2048).