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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)
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!
Why, then, is this design often considered undersirable?
For desired spectra with discontinuities, the least-square designs are poor in a minimax(worst-case, or ${L}_{}$ ) error sense.
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.
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