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
\nabla
be the distribution upon R defined by
\nabla = \sum\min{k=0}\max{N-1} \gamma\kappa\delta(kT/N)
where
\gamma\kappa\inR
and δ
α
is 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 λ
0
of a matrix M defined by
M_{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).