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Window Design Method

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

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 h⁢n  ,   0≤n≤M−1    n 0 n M 1 h n , maximizing the energy difference between the desired response and the actual response: i.e., find min h⁢n h⁢n ∫−ππ| H d ⁢ω−H⁢ω|2d ω h n ω H d ω H ω 2 by Parseval's relationship

min h⁢n h⁢n ∫−ππ| H d ⁢ω−H⁢ω|2d ω =2⁢π⁢∑ n =−∞∞| h d ⁢n−h⁢n|2=2⁢π⁢(∑ n =−∞−1| h d ⁢n−h⁢n|2+∑ n =0M−1| h d ⁢n−h⁢n|2+∑ n =M∞| h d ⁢n−h⁢n|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 h⁢n  ,   n<0 n≥M    n n 0 n M h n this becomes min h⁢n h⁢n ∫−ππ| H d ⁢ω−H⁢ω|2d ω =∑ h =−∞−1| h d ⁢n|2+∑ n =0M−1|h⁢n− 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:

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

The best we can do is let h⁢n={ h d ⁢n  if  0≤n≤M−10  if  else h n h d n 0 n M 1 0 else Thus h⁢n= h d ⁢n⁢w⁢n h n h d n w n , w⁢n={1  if  0≤n⁢M−10  if  else 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) A⁢ω A ω , small M (b) A⁢ω 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)    n 0 n M 1 h n h d n w n

Note:

H⁢ω= H d ⁢ω*W⁢ω 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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