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Lp norms

The $ Lp$ norm of an $ N$ -dimensional vector (signal) $ x$ is defined as

$\displaystyle \left\Vert\,x\,\right\Vert _p \isdefs \left( \sum_{i=0}^{N-1} \vert x_i\vert^p \right)^ \frac{1}{p}.$ (5.27)


Special Cases

  • $ L1$ norm

    $\displaystyle \left\Vert\,x\,\right\Vert _1 \isdefs \sum_{i=0}^{N-1} \vert x_i\vert$ (5.28)

    • Sum of the absolute values of the elements
    • ``City block'' distance
  • $ L2$ norm

    $\displaystyle \left\Vert\,x\,\right\Vert _2 \isdefs \sqrt{ \sum_{i=0}^{N-1} \vert x_i\vert^2 }$ (5.29)

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

  • $ L-infinity$ norm

    $\displaystyle \left\Vert\,x\,\right\Vert _\infty \isdefs \lim_{p\to\infty} \left( \sum_{i=0}^{N-1} \vert x_i\vert^p \right)^\frac{1}{p} \protect$ (5.30)

    In the limit as $ p \rightarrow \infty$ , the $ Lp$ norm of $ x$ is dominated by the maximum element of $ x$ . Optimal Chebyshev filters minimize this norm of the frequency-response error.

Filter Design using Lp Norms

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

$\displaystyle \min_h \left\Vert W(\omega_k)\left[H(\omega_k) - D(\omega_k)\right]\right\Vert _p$ (5.31)

where
  • $ h = $ FIR filter coefficients
  • $ \omega_k = $ suitable discrete set of frequencies
  • $ D(\omega_k) = $ desired (complex) frequency response
  • $ H(\omega_k) = $ obtained frequency response (typically fft(h))
  • $ W(\omega_k) = $ (optional) error weighting function
An especially valuable property of FIR filter design under $ Lp$ 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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Conclusions