Equiripple Filters A filter which has the Smallest

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

Fact: filters with the smallest maximum deviation from ideal characteristic are equiripple. . same ripple 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.

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

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.

Equiripple (Remez Algorithm) Hamming window

Frequency Response of the Non Ideal LPF ripple attenuation stop pass transition region LPF specified by: • passband frequency • passband ripple or • stopband frequency • stopband attenuation or stop

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. ripple attenuation Linear Interpolation

Example: Low Pass Filter Passband Stopband with attenuation 40 d. B Choose order Almost 40 d. B!!!

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