In Oppenheim & Schafer's Discrete-Time Signal Processing, only type I lowpass filters are explained in detail in the chapter "Optimum Approximations of FIR filters".

I want to know what happens not only with the other three types of generalized-phase filters, but also understand how the equiripple approximation works with bandpass and highpass filters.

In the previously mentioned book, it says:

Specifically, we will show that for type I lowpass filters:

  • The maximum possible number of alternations of the error is $L+3$.
  • Alternations will always occur at $\omega_p$ and $\omega_s$.
  • [...] the filter will be equiripple, except possibly at $\omega=0$ and $\omega =\pi$.

I have a bunch of questions, for example:

  • Is there a maximum possible number of alternations for every type (II, III, IV) and kind (lowpass, highpass, bandpass) of filter?
  • Do the other two properties (alternations at edges and equiripple-ness except possibly at $0$ and $\pi$) apply to other types and kinds of filters?
  • Do other types and kinds of filters have other special properties (i.e., properties that differ from the ones in the quote I wrote)?

It would be great if you could give me some insight in the subject, or if you could recommend a book to which I could fall back on.


The only place where I found the answer to this question was in Theory and Application of Digital Signal Processing by Rabiner & Gold.

In its chapter about theory and approximation of FIR filters, there is a section where all of my questions are explicitely answered:

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