LP Standard Form

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Spectral Audio Signal Processing

Optimal Window Design by linprog

Sidelobe Specification

Remez Exchange Algorithm

The main lobe in the Fourier transform of a window function, such as the Chebyshev window, is the central local maximum in the transform magnitude (response about dc). Outside of the main lobe are usually side lobes (ripples in the magnitude spectrum). — Click for https://ccrma.stanford.edu/~jos/sasp/Rectangular_Window_Side_Lobes.html

Click for https://ccrma.stanford.edu/~jos/sasp/Dolph_Chebyshev_Window.html#10532

The Discrete Time Fourier Transform (DTFT) is the appropriate Fourier transform for discrete-time signals of arbitrary length. It can be obtained as the limit of a Discrete Fourier Transform (DFT) as its length goes to infinity. — Click for https://ccrma.stanford.edu/~jos/mdft/Discrete_Time_Fourier_Transform.html

Linear programming refers to a class of computational optimization procedures. — Click for http://www-unix.mcs.anl.gov/otc/Guide/faq/linear-programming-faq.html

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Spectral Audio Signal Processing

Optimal Window Design by linprog

Sidelobe Specification

Remez Exchange Algorithm

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
$\includegraphics[width=\twidth,height=6.5in]{eps/print_normal_chebwin}$

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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

LP Standard Form

``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

``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