Optimal (but poor) Least-Squares Impulse Response Design

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Optimal (but poor) Least-Squares Impulse Response Design

Perhaps the most commonly employed error criterion in signal processing is the least-squares error criterion.

Let denote some ideal filter impulse response, possibly infinitely long, and let ${\hat h}(n)$ denote the impulse response of a length causal FIR filter we wish to design. The sum of squared errors is given by

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

where does not depend on ${\hat h}$ . Note that $J({\hat h})\geq c_2$ . We can minimize the error by simply matching the first terms in the desired impulse response. That is, the optimal least-squares FIR filter has the following ``tap'' coefficients:

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

(B.2)

The same solution works also for any norm. That is, the error

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

is also miminized by matching the leading terms of the desired impulse response.

In the case, we have, by the Fourier energy theorem (§2.3.8),

$\displaystyle J_2({\hat h}) \isdef \sum_{n=-\infty}^\infty\left\vert h(n)-{\hat... ...i}\int_{-\pi}^{\pi}\left\vert H(\omega)-{\hat H}(\omega)\right\vert^2 d\omega.$

Therefore, ${\hat H}(\omega)=\hbox{\sc DTFT}({\hat h})$ is an optimal least-squares approximation to $H(\omega)$ when ${\hat h}$ is given by (B.2). In other words, the frequency response of the filter ${\hat H}$ is optimal in the sense.

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written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.

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Spectral Audio Signal Processing
FIR Digital Filter Design
Optimal (but poor) Least-Squares Impulse Response Design