# Filters using equiripple approximations (different from type I lowpass)

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.