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 σω for a given temporal RMS width σt. The window family yields as a byproduct the minimum time-bandwidth product σωσt for given σt and N. The new windows interpolate for decreasing σt between the popular Cosine-window and a nearly Gaussian window. The new “confined Gaussian” window function gk(cG) (with k=0,…,N−1) is extremely well approximated by gk(acG)∝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[−δt2(x−(N−1)/2)2/(4s2)], the temporal width s≈σt, and time step δt.