Minimum-Phase Filter Design
Above, we used the Hilbert transform to find the imaginary part of an analytic signal from its real part. A closely related application of the Hilbert transform is constructing a minimum phase [263] frequency response from an amplitude response.
Let
denote a desired complex, minimum-phase frequency response
in the digital domain (
plane):
![]() |
(5.23) |
and suppose we have only the amplitude response
![]() |
(5.24) |
Then the phase response
![$ \Theta(\omega)$](http://www.dsprelated.com/josimages_new/sasp2/img809.png)
![$ \ln G(\omega)$](http://www.dsprelated.com/josimages_new/sasp2/img810.png)
![]() |
(5.25) |
If
![$ \Theta$](http://www.dsprelated.com/josimages_new/sasp2/img812.png)
![$ G$](http://www.dsprelated.com/josimages_new/sasp2/img813.png)
![$ \ln
H(e^{j\omega})$](http://www.dsprelated.com/josimages_new/sasp2/img814.png)
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Minimum-Phase and Causal Cepstra
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Generalized Window Method