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 sω for a given temporal RMS width st. The window family yields as a byproduct the minimum time-bandwidth product sωst for given st and N. The new windows interpolate for decreasing st between the popular Cosine-window and a nearly Gaussian window. The new “confined Gaussian” window function gðcGÞ k (with k1⁄4 0;...;N 1) is extremely well approximated by gðacGÞ k pGðkÞ Gð 1=2Þ1⁄2GðkþNÞþGðk NÞ =1⁄2Gð 1=2þNÞþGð 1=2 NÞ with the Gaussian GðxÞ 1⁄4 exp1⁄2 δt2ðx ðN 1Þ=2Þ=ð4s2Þ , the temporal width s st , and time step δt. & 2014 Elsevier B.V. All rights reserved.