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.