مرکز منطقه ای اطلاع رساني علوم و فناوري - An optimal discrete window for the calculation of power spectra

Abstract :

Let g be a function defined upon R, with values in C and G its Fourier transform. Let $\\nabla$ be the distribution upon R defined by $\\nabla = \\sum \\min{k=0}\\max {N-1} \\gamma \\kappa \\delta (kT/N)$ where $\\gamma \\kappa \\in R$ and δαis the Dirac function at abscissa α. $\\nabla$ is a discrete "time window" and its Fourier transform is a periodic function $\\Gamma$ of the frequency (period N/T). Taking the Fourier transform of the product of $\\nabla$ by g, we obtain $F[\\nabla _{g}] = F [\\nabla ] \\ast F[g] = \\Gamma \\ast G$ (* means convolution product). $F[\\nabla _{g}]$ is also a periodic function of the frequency (period N/T) and $F[\\nabla _{g}] (frac{j}{T}) = \\sum \\min{k=0}\\max {N-1} \\gamma \\kappa \\cdot g\\kappa \\cdot \\exp - 2i\\pi frac{jk}{N}$ where $gk = g(k(T/N)). F[\\nabla _{g}](j/T)$ for j=0,..., N-1 is obtained very efficiently using the FFT algorithm of Cooley and Tukey. Cleverly choosing the weights $\\gamma \\kappa , |F[\\nabla ](j/T)|^{2}$ for j = -N/2, ..., N/2-1 is a good estimator of the power spectrum of g. The vector γ (with components $\\gamma \\kappa , k=0, ..., N-1$ ) that maximize the ratio $frac{\\int\\min{-1/T}\\max {1/T}|\\Gamma (\\lambda )|^{2} d\\lambda }{\\int\\min{-N/2T}\\max {N/2T}|\\Gamma (\\lambda )|^{2} d\\lambda }$ gives us an optimal discrete window. Then γ is the eigenvector corresponding to the greatest eigenvalue λ0of a matrix M defined by $M_{q}k = \\Bigg\\{ {{2 \\over N} \\hbox{ if } q = k, \\; q = 0, \\cdots , N - 1 \\atop {\\hbox{sin } 2\\pi ( {q - k \\over N} ) \\over \\pi(q - k)}, \\; k = 0, \\cdots , N - 1}$ The method for calculating this eigenvector is shown for large values of N (N = 2048).