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Least-Squares Linear-Phase FIR Filter Design

Another versatile, effective, and often-used case is the weighted least squares method, which is implemented in the matlab function firls and others. A good general reference in this area is [204].

Let the FIR filter length be $ L+1$ samples, with $ L$ even, and suppose we'll initially design it to be centered about the time origin (``zero phase''). Then the frequency response is given on our frequency grid $ \omega_k$ by

$\displaystyle H(\omega_k) \eqsp \sum_{n=-L/2}^{L/2} h_n e^{-j\omega_kn}, \; k=0,1,2,\ldots,N-1, \; N\gg L.$ (5.33)

Enforcing even symmetry in the impulse response, i.e., $ h_n =
h_{-n}$ , gives a zero-phase FIR filter that we can later right-shift $ L/2$ samples to make a causal, linear phase filter. In this case, the frequency response reduces to a sum of cosines:

$\displaystyle H( \omega_k ) \eqsp h_0 + 2\sum_{n=1}^{L/2} h_n \cos (\omega_k n), \quad k=0,1,2,\ldots, N-1,$ (5.34)

or, in matrix form:

$\displaystyle \underbrace{\left[ \begin{array}{c} H(\omega_0) \\ H(\omega_1) \\ \vdots \\ H(\omega_{N-1}) \end{array} \right]}_{{\underline{d}}} = \underbrace{\left[ \begin{array}{ccccc} 1 & 2\cos(\omega_0) & \dots & 2\cos[\omega_0(L/2)] \\ 1 & 2\cos(\omega_1) & \dots & 2\cos[\omega_1(L/2)] \\ \vdots & \vdots & & \vdots \\ 1 & 2\cos(\omega_{N-1}) & \dots & 2\cos[\omega_{N-1}(L/2)] \end{array} \right]}_\mathbf{A} \underbrace{\left[ \begin{array}{c} h_0 \\ h_1 \\ \vdots \\ h_{L/2} \end{array} \right]}_{{\underline{h}}} \protect$ (5.35)

Recall from §3.13.8, that the Remez multiple exchange algorithm is based on this formulation internally. In that case, the left-hand-side includes the alternating error, and the frequency grid $ \omega_k$ iteratively seeks the frequencies of maximum error--the so-called extremal frequencies.

In matrix notation, our filter-design problem can be stated as (cf. §3.13.8)

$\displaystyle \min_{{\underline{h}}} \left\Vert \mathbf{A}{\underline{h}}-{\underline{d}}\right\Vert _2^2$ (5.36)

where these quantities are defined in (4.35). We can denote the optimal least-squares solution by

$\displaystyle {\underline{\hat{h}}}\isdefs \arg \min_{\underline{h}}\left\Vert\,\mathbf{A}{\underline{h}}-{\underline{d}}\,\right\Vert _2 \eqsp \arg \min_{\underline{h}}\left\Vert\,\mathbf{A}{\underline{h}}-{\underline{d}}\,\right\Vert _2^2$ (5.37)

To find $ {\underline{\hat{h}}}$ , we need to minimize
$\displaystyle \left\Vert\,\mathbf{A}{\underline{h}}-{\underline{d}}\,\right\Vert _2^2$ $\displaystyle =$ $\displaystyle (\mathbf{A}{\underline{h}}-{\underline{d}})^T(\mathbf{A}{\underline{h}}-{\underline{d}})$  
  $\displaystyle =$ $\displaystyle ({\underline{h}}^T\mathbf{A}^T-{\underline{d}}^T)(\mathbf{A}{\underline{h}}-{\underline{d}})$  
  $\displaystyle =$ $\displaystyle {\underline{h}}^T\mathbf{A}^T\mathbf{A}{\underline{h}}
\protect$ (5.38)

This is a quadratic form in $ {\underline{h}}$ . Therefore, it has a global minimum which we can find by setting the gradient to zero, and solving for $ {\underline{h}}$ .5.14Assuming all quantities are real, equating the gradient to zero yields the so-called normal equations

$\displaystyle \mathbf{A}^T\mathbf{A}{\underline{h}}\eqsp \mathbf{A}^T{\underline{d}}$ (5.39)

with solution

$\displaystyle \zbox {{\underline{\hat{h}}}\eqsp \left[(\mathbf{A}^T\mathbf{A})^{-1}\mathbf{A}^T\right]{\underline{d}}.}$ (5.40)

The matrix

$\displaystyle \mathbf{A}^\dagger \isdefs (\mathbf{A}^T\mathbf{A})^{-1}\mathbf{A}^T$ (5.41)

is known as the (Moore-Penrose) pseudo-inverse of the matrix $ \mathbf {A}$ . It can be interpreted as an orthogonal projection matrix, projecting $ {\underline{d}}$ onto the column-space of $ \mathbf {A}$ [264], as we illustrate further in the next section.

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``Spectral Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2011, ISBN 978-0-9745607-3-1.
Copyright © 2022-02-28 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University