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Window Design Method

Module by: Douglas L. Jones. E-mail the author

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The truncate-and-delay design procedure is the simplest and most obvious FIR design procedure.

Exercise 1

Is it any Good?

Solution

Yes; in fact it's optimal! (in a certain sense)

L2 optimization criterion

find ∀n,0≤n≤M−1:hn n 0 n M 1 h n , maximizing the energy difference between the desired response and the actual response: i.e., find minhn{∫-ππ| H d ω−Hω|2dω} h n ω H d ω H ω 2 by Parseval's relationship

minhn{∫-ππ| H d ω−Hω|2dω}=2π∑n=-∞∞| h d n−hn|2=2π∑n=-∞-1| h d n−hn|2+∑n=0M−1| h d n−hn|2+∑n=M∞| h d n−hn|2 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 (1)
Since ∀n,n<0n≥M:=hn n n 0 n M h n this becomes minhn{∫-ππ| H d ω−Hω|2dω}=∑h=-∞-1| h d n|2+∑n=0M−1|hn− h d n|2+∑n=M∞| h d n|2 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

Note:

hn h n has no influence on the first and last sums.

The best we can do is let hn= h d nif0≤n≤M−10ifelse h n h d n 0 n M 1 0 else Thus hn= h d nwn h n h d n w n , wn=1if0≤nM−10ifelse w n 1 0 n M 1 0 else is optimal in a least-total-sqaured-error ( L 2 L 2 , or energy) sense!

Exercise 2

Why, then, is this design often considered undersirable?

Solution: Gibbs Phenomenon

Figure 1
(a) Aω A ω , small M (b) Aω A ω , large M
Figure 1(a) (WindowDesignFig2.png) Figure 1(b) (WindowDesignFig1.png)

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

Window Design Method

Apply a more gradual truncation to reduce "ringing" (Gibb's Phenomenon) ∀n0≤n≤M−1hn= h d nwn n 0 n M 1 h n h d n w n

Note:

Hω= H d ω*Wω 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.

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