1.1 Window design method

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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)

L2 optimization criterion

find $\forall n, 0\le n\le M-1\colon h(n)$ , maximizing the energy difference between the desired response and the actual response: i.e., find $\min\{h(n) , \int_{-\pi }^{\pi } \left|{H}_{d}()-H()\right|^{2}\,d \}$ by Parseval's relationship

$\min\{h(n) , \int_{-\pi }^{\pi } \left|{H}_{d}()-H()\right|^{2}\,d \}=2\pi \sum$ 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 $\forall n, n< 0n\ge M\colon h(n)$ this becomes $\min\{h(n) , \int_{-\pi }^{\pi } \left|{H}_{d}()-H()\right|^{2}\,d \}=\sum_{h=()}^{-1}$ 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 $h(n)=\begin{cases}{h}_{d}(n) & \text{if 0\le n\le M-1}\\ 0 & \text{if \text{else}}\end{cases}$ Thus $h(n)={h}_{d}(n)w(n)$ , $w(n)=\begin{cases}1 & \text{if 0\le n(M-1)}\\ 0 & \text{if \text{else}}\end{cases}$ is optimal in a least-total-sqaured-error ( ${L}_{2}$ , or energy) sense!

Why, then, is this design often considered undersirable?

Gibbs phenomenon

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 ) $\forall n0\le n\le M-1h(n)={h}_{d}(n)w(n)$

$H()=({H}_{d}(), W())$

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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