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.

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Filter Design using Lp Norms
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Windowing a Desired Impulse Response Computed by the Frequency Sampling Method