Sidelobe Specification

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

$\displaystyle -\delta \leq d\left(\omega _{i}\right)^{T}h\leq \delta$ (4.71)

or

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

I.e.,

$\displaystyle \zbox {\mathbf{A}_{sb}\left[\begin{array}{c} h\\ \delta \end{array} \right] \le \mathbf{0}.}$ (4.74)


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