### Lp norms

The norm of an -dimensional vector (signal) is defined as

 (5.27)

#### Special Cases

• norm

 (5.28)

• Sum of the absolute values of the elements
• City block'' distance
• norm

 (5.29)

• Euclidean'' distance
• Minimized by Least Squares'' techniques

• norm

 (5.30)

In the limit as , the norm of is dominated by the maximum element of . Optimal Chebyshev filters minimize this norm of the frequency-response error.

#### Filter Design using Lp Norms

Formulated as an norm minimization, the FIR filter design problem can be stated as follows:

 (5.31)

where
• FIR filter coefficients
• suitable discrete set of frequencies
• desired (complex) frequency response
• obtained frequency response (typically fft(h))
• (optional) error weighting function
An especially valuable property of FIR filter design under norms is that the error norm is typically a convex function of the filter coefficients, rendering it amenable to a wide variety of convex-optimization algorithms [22]. The following sections look at some specific cases.

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Optimal Chebyshev FIR Filters
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