Sidelobe Specification

Free Books Spectral Audio Signal Processing

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

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

(4.71)

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

(4.72)

where

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

(4.73)

We need $K\gg L$ to obtain many frequency samples per side lobe. Stacking inequalities for all $\omega_{i}$ ,

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