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
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
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Optimal Chebyshev FIR Filters
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Conclusions