Discrete-time windows with minimal RMS bandwidth for given RMS temporal width

We derive a family of discrete window functions for the N-point Fourier transform for application in spectral analysis that optimize the root mean square (RMS) frequency width sigma(omega) for a given temporal RMS width sigma(t). The window family yields as a byproduct the minimum time-bandwidth product sigma(omega)sigma(t) for given sigma(t) and N. The new windows interpolate for decreasing sigma(t) between the popular Cosine-window and a nearly Gaussian window. The new confined Gaussian window function g(k)((cG)) (with k=0,...,N-1) is extremely well approximated by g(k)((acG)) proportional to G(k)-G(-1/2)[G(k+N)+G(k-N)]/[G(-1/2+N)+G(-1/2-N)] with the Gaussian G(x) = exp[delta t(2)(x-(N-1)/2)(2)/(4s(2))], the temporal width s approximate to sigma(t), and time step delta t. (C) 2014 Elsevier B.V. All rights reserved.