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LP Standard Form

Now gather all of the constraints to form an LP problem:

\begin{displaymath}\begin{array}[t]{ll} \mathrm{minimize} & \left[\begin{array}{cccc} 0 & \cdots & 0 & 1\end{array} \right] \left[\begin{array}{c} h\\ \delta \end{array} \right]\\ [5pt] \mbox{subject to} & \begin{array}[t]{l} \left[\begin{array}{cc} d\left(0\right)^{T} & 0\end{array} \right]\left[\begin{array}{c} h\\ \delta \end{array} \right]=1\\ \left[\begin{array}{c} \left[\begin{array}{cc} -\mathbf{I} & \mathbf{0}\end{array} \right]\\ [5pt] \mathbf{A}_{sb}\end{array} \right]\left[\begin{array}{c} h\\ \delta \end{array} \right]\le \mathbf{0}\end{array} \end{array}\end{displaymath} (4.75)

where the optimization variables are $ [h, \delta]^T$ .

Solving this linear-programming problem should produce a window that is optimal in the Chebyshev sense over the chosen frequency samples, as shown in Fig.3.37. If the chosen frequency samples happen to include all of the extremal frequencies (frequencies of maximum error in the DTFT of the window), then the unique Chebyshev window for the specified main-lobe width must be obtained. Iterating to find the extremal frequencies is the heart of the Remez multiple exchange algorithm, discussed in the next section.

Figure 3.37: Normal Chebyshev Window

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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