3.8 Window design method

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

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

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!

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 ) $$\forall n0\le n\le M-1h(n)=h_d(n)w(n)$$

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.