L-Infinity Norm of Derivative Objective

We can add a smoothness objective by adding $\ensuremath{L_\infty}$ -norm of the derivative to the objective function.

$$\mathrm{minimize}\quad \delta +\eta \left\Vert \Delta h\right\Vert _{\infty }.$$ (4.79)

The $\ensuremath{L_\infty}$ -norm only cares about the maximum derivative. Large $\eta$ means we put more weight on the smoothness than the side-lobe level.

This can be formulated as an LP by adding one optimization parameter $\sigma$ which bounds all derivatives.

$$-\sigma \leq \Delta h_{i}\leq \sigma \qquad i=1,\ldots ,L-1.$$ (4.80)

In matrix form,

$$\left[\begin{array}{r} -\mathbf{D}\\ \mathbf{D}\end{array}\right]h-\sigma \mathbf1 \le 0.$$

Objective function becomes

$$\mathrm{minimize}\quad \delta +\eta \sigma .$$ (4.81)

The result of adding the Chebyshev norm of diff(h) to the objective function to be minimized ($\eta =1$ ) is shown in Fig.3.39. The result of increasing $\eta$ to 20 is shown in Fig.3.40.

Figure: Chebyshev norm of diff(h) added to the objective function to be minimized ($\eta =1$ )

Figure: Twenty times the norm of diff(h) added to the objective function to be minimized ($\eta =20$ )