Some thoughts on least squared error optimal windows

Windowing methods give simple designs of FIR filters. Window designs are generally not considered optimal in any meaningful sense. A typical desired FIR lowpass filter is specified by giving the passband, stopband and transitionband responses of the filter. A particular class of transition responses viz., spline transition functions, give rise to windows that are optimal in a least squared sense. Do all transition functions have associated windows that are least squared optimal? More importantly, given a window does there exist a (meaningful) transition function with respect to which the window is least squared optimal? In trying to answer the second question this note also characterizes all possible lowpass extensions of a given sequence and exhibits the unique minimum norm extension. This is used to show that a given finitely supported window w_b(n), and a desired response with prescribed transition band edges (but no transition function) there exist infinitely many transition functions with respect to which the windowed FIR filter (obtained by windowing an ideal frequency response) is least squared optimal. The unique minimum norm lowpass extension of w _b(n) is used to exhibit a particular transition function with an extremal property. Since it is desirable to have a monotone transition function we ask the question: can a given window be least squared optimal with respect to a monotone desired transition response? This question leads to a few open problems on non-negative definite lowpass extensions of sequences