1. Equiripple Filters A filter which has the Smallest Maximum Approximation Error among all filters over the frequencies of interest: Define: where

2. same ripple Fact: filters with the smallest maximum deviation from ideal characteristic are equiripple.. They are computed as follows: B=firpm(N,F,M) F=[F(1),F(2),…], M=[M(1), M(2), …]; F=1 corresponding to N = filter order.

3. Fact: the error is miminal in minmax sense, provided there exist L+2 frequencies such that

4. Example: see the same example we saw for the FIR filter with window. Recall the specs: 1. Pass band 2. Stopband 3. Sampling Frequency Apply Remez Algorithm. You have to determine the filter order a priori, and let’s choose the same order N=81: h=firpm(80,[0,0.4,0.5,1],[1,1,0,0]); The impulse response obtained is shown. The frequency response is shown in the next slide, compared with the one we obtained by using the hamming window. Notice that the attenuation in the stopband is higher in the equiripple.

5. Equiripple (Remez Algorithm) Hamming window

6. Frequency Response of the Non Ideal LPF passstop transition region attenuation ripple LPF specified by: passband frequency passband ripple or stopband frequency stopband attenuation or

7. Best Design tool for FIR Filters: the Equiripple algorithm (or Remez). It minimizes the maximum error between the frequency responses of the ideal and actual filter. attenuation ripple Linear Interpolation

8. Example: Low Pass Filter Passband Stopband with attenuation 40dB Choose order Almost 40dB!!!

9. Example: Low Pass Filter Choose order N=40 > 37 OK!!!