Optimal (but poor if unweighted)

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 denote the impulse response of a length causal FIR filter that we wish to design. The sum of squared errors is given by

(5.4) |

where does not depend on . Note that . 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:

The same solution works also for any norm (§4.10.1). That is, the error

(5.6) |

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

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

(5.7) |

Therefore, is an optimal least-squares approximation to when is given by (4.5). In other words, the frequency response of the filter is optimal in the (least-squares) sense.

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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University