An optimal discrete window for the calculation of power spectra
André Eberhard · IEEE Transactions on Audio and Electroacoustics · 1973
Let g be a function defined upon R, with values in C and G its Fourier transform. Let ablabe the distribution upon R defined by abla = \sum\min{k=0}\max{N-1} \gamma\kappa\delta(kT/N)where\gamma\kappa\inRand δαis the Dirac function at abscissa α. ablais a discrete "time window" and its Fourier transform is a periodic function\Gammaof the frequency (period N/T). Taking the Fourier transform of the product of ablaby g, we obtainF[ abla_{g}] = F [ abla] \ast F[g] = \Gamma \ast G(* means convolution product).F[ abla_{g}]is also a periodic function of the frequency (period N/T) andF[ abla_{g}] (\frac{j}{T}) = \sum\min{k=0}\max{N-1} \gamma\kappa \cdot g\kappa \cdot \exp - 2i\pi \frac{jk}{N}wheregk = g(k(T/N)). F[ abla_{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[ abla](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 byM_{q}k = \{\min\frac{\sin 2\pi(\frac{q-k}{N})}{\pi(q-k)}, k=0,..., N - 1}\max{\frac{2}{N} if q=k, q = 0,..., N - 1}The method for calculating this eigenvector is shown for large values of N (N = 2048).