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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Special Cases