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Optimal (but Poor) Least-Squares Impulse Response Design


Sum of squared errors:

$\displaystyle J_2({\hat h})
\isdef \sum_{n=-\infty}^\infty\left\vert h(n)-{\hat h}(n)\right\vert^2
= \sum_{n=0}^{L-1}\left\vert h(n)-{\hat h}(n)\right\vert^2 + c_2

where $ c_2\isdef \sum_{n=-\infty}^{-1}\left\vert h(n)\right\vert^2 + \sum_{n=L}^\infty\left\vert h(n)\right\vert^2$ does not depend on $ {\hat h}$ . Note that $ J_2({\hat h})\geq c_2$ .

Result: The error is minimized (in the least-squares sense) by simply matching the first $ L$ terms in the desired impulse response.

Optimal least-squares FIR filter:

$\displaystyle {\hat h}(n) \isdef \left\{\begin{array}{ll}
h(n), & 0\leq n \leq L-1 \\ [5pt]
0, & \hbox{otherwise} \\
\end{array} \right.

Also optimal under any $ \ensuremath{L_p}$ norm with any error weighting:

$\displaystyle J_p({\hat h}) = \sum_{n=0}^{L-1}w(n)\left\vert h(n)-{\hat h}(n)\right\vert^p + c_p \geq c_p

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``The Window Method for FIR Digital Filter Design}'', by Julius O. Smith III, (From Lecture Overheads, Music 421).
Copyright © 2020-06-27 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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