Example Zero-Phase Filter Design

Figure 10.1 shows the impulse response and frequency response of a length 11 zero-phase FIR lowpass filter designed using the Remez exchange algorithm.^11.2 The matlab code for designing this filter is as follows:

N = 11;                % filter length - must be odd
b = [0 0.1 0.2 0.5]*2; % band edges
M = [1  1   0   0 ];   % desired band values
h = remez(N-1,b,M);    % Remez multiple exchange design

The impulse response h is returned in linear-phase form, so it must be left-shifted $(N-1)/2=5$ samples to make it zero phase.

Figure 10.1: Impulse response and frequency response of a length 11 zero-phase FIR lowpass filter. Note that the frequency response is real because the filter is zero phase. Also plotted (in dashed lines) are the desired passband and stopband gains.

Figure 10.2 shows the amplitude and phase responses of the FIR filter designed by remez. The phase response is zero throughout the passband and transition band. However, each zero-crossing in the stopband results in a phase jump of $\pi$ radians, so that the phase alternates between zero and $\pi$ in the stopband. This is typical of practical zero-phase filters.

Figure 10.2: Amplitude response and phase response of the length 11 zero-phase FIR lowpass filter in Fig.10.1.