Filter Design Toolbox 4.6

Design of Peaking and Notching Filters

Filters that peak or notch at a certain frequency are useful to retain or eliminate a particular frequency component of a signal. The design parameters for the filter are the frequency at which the peak or notch is desired, and either the 3-dB bandwidth or the filter's Q-factor. Moreover, given these specifications, by increasing the filter order, it is possible to obtain designs that more closely approximate an ideal filter.

Contents

Notch Filters

Suppose you need to eliminate a 60 Hz interference in a signal sampled at 3000 Hz. A notch filter can be used for such purpose.

F0 = 60;   % interference is at 60 Hz
Fs = 3000; % sampling frequency is 3000 Hz
f = fdesign.notch('N,F0,Q',2,60,10,3000);
h = design(f);
hfvt= fvtool(h,'Color','white');

The quality factor or Q-factor of the filter is a measure of how well the desired frequency is isolated from other frequencies. For a fixed filter order, a higher Q-factor is accomplished by pushing the poles closer to the zeros.

f.Q = 100;
h1 = design(f);
set(hfvt,'Filters',[h h1]);
legend(hfvt,'Q = 10','Q = 100');

An equivalent way of specifying the quality factor it to specify the 3-dB bandwidth, BW. They are related by Q = F0/BW. Specifying the bandwidth may be a more convenient way of achieving exactly the desired shape for the filter that is designed.

f = fdesign.notch('N,F0,BW',2,60,5,3000);
h2 = design(f);
set(hfvt,'Filters',[h h1 h2]);
legend(hfvt,'Q = 10','Q = 100','BW = 5 Hz');

Since it is only possible to push the poles so far and remain stable, in order to improve the brickwall approximation of the filter, it is necessary to increase the filter order.

f = fdesign.notch('N,F0,Q',2,.4,100);
h = design(f);
f.FilterOrder = 6;
h1 = design(f);
set(hfvt,'Filters',[h h1]);
legend(hfvt,'2nd-Order Filter','6th-Order Filter');

For a given order, we can obtain sharper transitions by allowing for passband and/or stopband ripples.

N = 8; F0 = 0.4; BW = 0.1;
f = fdesign.notch('N,F0,BW',N,F0,BW);
h = design(f);
f1 = fdesign.notch('N,F0,BW,Ap,Ast',N,F0,BW,0.5,60);
h1 = design(f1);
set(hfvt,'Filters',[h h1]);
legend(hfvt,'Maximally Flat 8th-Order Filter',...
    '8th-Order Filter With Passband/Stopband Ripples','Location','NorthWest'
);
axis([0 1 -90 0.5]);

Peak Filters

Peaking filters are used if we want to retain only a single frequency component (or a small band of frequencies) from a signal. All specifications and tradeoffs mentioned so far apply equally to peaking filters. Here's an example:

N = 6; F0 = 0.7; BW = 0.001;
f = fdesign.peak('N,F0,BW',N,F0,BW);
h = design(f);
f1 = fdesign.peak('N,F0,BW,Ast',N,F0,BW,80);
h1 = design(f1);
set(hfvt,'Filters',[h h1]);
legend(hfvt,'Maximally Flat 6th-Order Filter',...
    '6th-Order Filter With 80 dB Stopband Attenuation','Location','North');