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See Also

Embedded SystemsFPGAElectronics

Spectral Audio Signal Processing
    Spectrum Analysis Windows
       Window Design by Linear Programming
          Sidelobe Specification

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Sidelobe Specification

Likewise, side-lobe specification can be enforced at frequencies $ \omega_{i}$ in the stop-band.

$\displaystyle -\delta \leq d\left(\omega _{i}\right)^{T}h\leq \delta $

or

$\displaystyle \left\{ \begin{array}{ccc} -d\left(\omega _{i}\right)^{T}h-\delta & \leq & 0\\ d\left(\omega _{i}\right)^{T}h-\delta & \leq & 0\end{array}\right.$

where,

$\displaystyle \omega _{sb}\leq \omega _1,\omega _{2}\ldots ,\omega _{K}\leq \pi .$

Stacking inequalities for all $ \omega_{i}$,

$\displaystyle \left[\begin{array}{c} -d\left(\omega _1\right)^{T}\\ \vdots \\ ... ...ta \\ \vdots \\ -\delta \\ -\delta \\ \vdots \\ -\delta \end{array}\right]$ $\displaystyle \le$ $\displaystyle \mathbf{0}$  
$\displaystyle \left[\begin{array}{cc} -d\left(\omega _1\right)^{T} & -1\\ \vdo... ...T} & -1 \end{array}\right]\left[\begin{array}{c} h\\ \delta \end{array}\right]$ $\displaystyle \le$ $\displaystyle \mathbf{0}.$  

I.e.,

$\displaystyle \zbox {\mathbf{A}_{sb}\left[\begin{array}{c} h\\ \delta \end{array}\right] \le \mathbf{0}.} $

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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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