Sidelobe Specification

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

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

or

$$\left\{ \begin{array}{ccc} -d\left(\omega _{i}\right)^{T}h-\delta & \leq & 0\\ \;d\left(\omega _{i}\right)^{T}h-\delta & \leq & 0\end{array} \right.$$ (4.72)

where

$$\omega _{sb}\leq \omega _1,\omega _{2}\ldots ,\omega _{K}\leq \pi .$$ (4.73)

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

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

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

I.e.,

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