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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)

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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 H d H 2 by Parseval's relationship

h n H d H 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 H d H 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 , small M
A , large M
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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 H d W

The window design procedure (except for the boxcar window) is ad-hoc and not optimal in any usual sense. However, it isvery simple, so it is sometimes used for "quick-and-dirty" designs of if the error criterion is itself heurisitic.

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Source:  OpenStax, Fundamentals of signal processing. OpenStax CNX. Nov 26, 2012 Download for free at http://cnx.org/content/col10360/1.4
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