L-Infinity Norm of Derivative Objective

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

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

-norm only cares about the maximum derivative.
Large $\eta$ means we put more weight on the smoothness than the sidelobe level.

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

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

In matrix form,

$\displaystyle \left[\begin{array}{c} -\mathbf{D}\\ \mathbf{D}\end{array}\right]h-\sigma \mathbf1$

$\displaystyle \le$

$\displaystyle 0.$

Objective function becomes

$\displaystyle \mathrm{minimize}\quad \delta +\eta \sigma .$

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

**Figure 3.32:**
$\includegraphics[width=\twidth,height=6.5in]{eps/print_linf_chebwin_1}$

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

**Figure 3.33:**
$\includegraphics[width=\twidth,height=6.5in]{eps/print_linf_chebwin_2}$

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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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