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Spectral Audio Signal Processing
    FIR Digital Filter Design
       Optimal FIR Digital Filter Design
          Chebyshev Optimal Linear Phase filter Design

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Chebyshev Optimal Linear Phase filter Design

Consider the $ L-infinity$-norm minimization problem:

$\displaystyle \min_x \left\Vert\,Ax-b\,\right\Vert _\infty $

We said earlier that the $ \infty$ norm of a vector is the maximum element of that vector, hence minimizing the $ \infty$ norm is minimizing the maximum element of a vector:

$\displaystyle \min_x \max_k \vert a_k^T x - b_k\vert $

where $ a_k^T$ denotes the $ k$th row of the matrix $ A$.

We can then write this as:

\begin{eqnarray*} \min t & \\ s.t.& \vert a_k^T x-b_k\vert<t \end{eqnarray*}

If we introduce a new variable

$\displaystyle \tilde{x} \mathrel{\stackrel{\Delta}{=}} \left[ \begin{array}{c} x \\ t \end{array} \right], $

then

$\displaystyle t = f^T \tilde{x} = [ \begin{array}{cccc} 0 & \dots & 0 & 1 \end{array} ] \tilde{x}, $

and our optimization problem becomes:

\begin{eqnarray*} \min_{\tilde{x}} & & f^T \tilde{x} \\ s.t. & & \left\vert [\b... ...} ] \cdot \tilde{x} -b_k \right\vert < \tilde{x}^T f^T\tilde{x} \end{eqnarray*}

Hence we are minimizing a linear objective, subject to a set of linear constraints. This is known as a linear programming problem (linprog in Matlab).


\begin{psfrags} % latex2html id marker 38451\psfrag{Linear}{\normalsize Linear... ...phics[width=3.5in]{eps/lp} \caption{} \end{figure} % was epsfbox \end{psfrags}

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Next: More General Real FIR Filters

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About the Author: 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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