Window Design Method 1.1 2005/01/27 14:06:21 US/Central 2005/05/12 15:50:45 GMT-5 Douglas L. Jones dl-jones@uiuc.edu Douglas L. Jones dl-jones@uiuc.edu Kyle Clarkson kclarks@rice.edu DSP filters FIR The truncate-and-delay design procedure is the simplest and most obvious FIR design procedure. Is it any Good? Yes; in fact it's optimal! (in a certain sense)

L2 optimization criterion find n 0 n M 1 h n , maximizing the energy difference between the desired response and the actual response: i.e., find h n ω H d ω H ω 2 by Parseval's relationship h n ω H d ω H ω 2 2 n h d n h n 2 2 n 1 h d n h n 2 n M 1 0 h d n h n 2 n M h d n h n 2 Since n n 0 n M h n this becomes h n ω H d ω H ω 2 h 1 h d n 2 n M 1 0 h n h d n 2 n M h d n 2 h n has no influence on the first and last sums. The best we can do is let h n h d n 0 n M 1 0 else Thus h n h d n w n , w n 1 0 n M 1 0 else is optimal in a least-total-sqaured-error ( L 2 , or energy) sense! Why, then, is this design often considered undersirable? Gibbs Phenomenon

A ω , small M A ω , large M

For desired spectra with discontinuities, the least-square designs are poor in a minimax (worst-case, or L ∞ ) error sense.

Window Design Method Apply a more gradual truncation to reduce "ringing" (Gibb's Phenomenon) n 0 n M 1 h n h d n w n H ω H d ω W ω The window design procedure (except for the boxcar window) is ad-hoc and not optimal in any usual sense. However, it is very simple, so it is sometimes used for "quick-and-dirty" designs of if the error criterion is itself heurisitic.