7.5 OPTIMAL EQUIRIPPLE DESIGN TECHNIQUE
7.5.4 MATLAB IMPLEMENTATION The Parks-McClellan algorithm is available in MATLAB as a function
[h] = firpm(N,f,m,weights,ftype)
There are several versions of this syntax
• [h] = firpm(N,f,m)designs anNth-order (note that the length of the filter is M = N + 1) FIR digital filter whose frequency response is specified by the arrays fand m. The filter coefficients (or the impulse response) are returned in array h of length M. The array f contains band-edge frequencies in units of π, that is, 0.0≤f≤1.0. These fre- quencies must be in increasing order, starting with 0.0 and ending with 1.0. The arraym contains the desired magnitude response at frequen- cies specified inf. The lengths offandmarrays must be the same and must be an even number. The weighting function used in each band is equal to unity, which means that the tolerances (δi’s) in every band are the same.
• [h] = firpm(N,f,m,weights)is similar to the preceding case except that the array weightsspecifies the weighting function in each band.
• [h] = firpm(N,f,m,ftype) is similar to the first case except when ftype is the string ‘differentiator’ or ‘hilbert’, it designs digi- tal differentiators or digital Hilbert transformers, respectively. For the digital Hilbert transformer, the lowest frequency in the farray should not be 0, and the highest frequency should not be 1. For the digital
differentiator, the m vector does not specify the desired slope in each band but the desired magnitude.
• [h] = firpm(N,f,m,weights,ftype) is similar to the above case ex- cept that the array weights specifies the weighting function in each band.
To estimate the filter order N, the SP toolbox provides the function firpmord, which also estimates other parameters that can be used in the firpm function. The basic syntax is
[N,f0,m0,weights] = firpmord(f,m,delta);
The function computes the window order N, the normalized frequency band edges in f0, amplitude response in a0, and the band weights in weights. The vector f is a vector of normalized band edges and m is a vector specifying the desired amplitude on the bands defined by f. The length offis two less than twice the length ofm; i.e.,fdoes not contain 0 or 1. The vectordeltaspecifies tolerances in each band (not in decibels).
The estimated parameters can now be used in the firpmfunction.
As explained during the description of the Parks-McClellan algorithm, we have to first guess the order of the filter using (7.48) to use the function firpm. After we obtain the filter coefficients in arrayh, we have to check the minimum stopband attenuation and compare it with the given As
and then increase (or decrease) the filter order. We have to repeat this procedure until we obtain the desired As. We illustrate this procedure in the following several MATLAB examples. These examples also use the ripple conversion functiondb2delta, which is developed in Problem P7.1.
EXAMPLE 7.23 Let us design the lowpass filter described in Example 7.8 using the Parks- McClellan algorithm. The design parameters are
ωp= 0.2π , ωs= 0.3π ,
Rp= 0.25 dB As= 50 dB We provide a MATLAB script to design this filter.
>> wp = 0.2*pi; ws = 0.3*pi; Rp = 0.25; As = 50;
>> [delta1,delta2] = db2delta(Rp,As);
>> [N,f,m,weights] = firpmord([wp,ws]/pi,[1,0],[delta1,delta2]);
>> h = firpm(N,f,m,weights);
>> [db,mag,pha,grd,w] = freqz_m(h,[1]);
>> delta_w = 2*pi/1000; wsi=ws/delta_w+1; wpi = wp/delta_w;
>> Asd = -max(db(wsi:1:501)) Asd = 47.8404
>> N = N+1 N = 43
>> h = firpm(N,f,m,weights); [db,mag,pha,grd,w] = freqz_m(h,[1]);
>> Asd = -max(db(wsi:1:501)) Asd = 48.2131
>> N = N+1 N = 44
>> h = firpm(N,f,m,weights); [db,mag,pha,grd,w] = freqz_m(h,[1]);
>> Asd = -max(db(wsi:1:501)) Asd = 48.8689
>> N = N+1 N = 45
>> h = firpm(N,f,m,weights); [db,mag,pha,grd,w] = freqz_m(h,[1]);
>> Asd = -max(db(wsi:1:501)) Asd = 49.8241
>> N = N+1 N = 46
>> h = firpm(N,f,m,weights); [db,mag,pha,grd,w] = freqz_m(h,[1]);
>> Asd = -max(db(wsi:1:501)) Asd = 51.0857
>> M = N+1 M = 47
Note that we stopped this iterative procedure when the computed stopband attenuation exceeded the given stopband attenuationAs, and the optimal value of M was found to be 47. This value is considerably lower than the window design techniques (M = 61 for a Kaiser window) or the frequency sampling technique (M = 60). In Figure 7.35 we show the time- and the frequency- domain plots of the designed filter along with the error function in both the passband and the stopband to illustrate the equiripple behavior.
EXAMPLE 7.24 Let us design the bandpass filter described in Example 7.10 using the Parks- McClellan algorithm. The design parameters are:
ω1s= 0.2π ω1p= 0.35π ω2p= 0.65π ω2s= 0.8π
; Rp= 1 dB
; As= 60 db
Solution The following MATLAB script shows how to design this filter.
>> ws1 = 0.2*pi; wp1 = 0.35*pi; wp2 = 0.65*pi; ws2 = 0.8*pi;
>> Rp = 1.0; As = 60;
>> [delta1,delta2] = db2delta(Rp,As);
>> f = [ws1,wp1,wp2,ws2]/pi; m = [0,1,0]; delta = [delta2,delta1,delta2];
>> [N,f,m,weights] = firpmord(f,m,delta); N N = 26
>> h = firpm(N,f,m,weights);
>> [db,mag,pha,grd,w] = freqz_m(h,[1]);
>> delta_w=2*pi/1000;
>> ws1i=floor(ws1/delta_w)+1; wp1i = floor(wp1/delta_w)+1;
>> ws2i=floor(ws2/delta_w)+1; wp2i = floor(wp2/delta_w)+1;
>> Asd = -max(db(1:1:ws1i)) Asd = 54.7756
>> N = N+1;
>> h = firpm(N,f,m,weights);
>> [db,mag,pha,grd,w] = freqz_m(h,[1]);
>> Asd = -max(db(1:1:ws1i)) Asd = 56.5910
>> N = N+1;
>> h = firpm(N,f,m,weights);
>> [db,mag,pha,grd,w] = freqz_m(h,[1]);
Asd = -max(db(1:1:ws1i))
>> Asd = 61.2843
>> M = N+1 M = 29
0 46
−0.1 0 0.1 0.2 0.3
n
h(n)
Actual Impulse Response
0 0.2 0.3 1
50 0
Magnitude Response in dB
frequency in π units
Decibels
0 0.2 0.3 1
0 1
Amplitude Response
frequency in π units
Hr(w)
0 0.2 0.3 1
−0.0144
−0.0032 0 0.0032 0.0144
Error Response
frequency in π units
Hr(w)
FIGURE 7.35 Plots for equiripple lowpass FIR filter in Example 7.23
0 30
−0.4
−0.2 0 0.2 0.4
n
h(n)
Actual Impulse Response
0 0.2 0.35 0.65 0.8 1
60 0
Magnitude Response in dB
frequency in π units
Decibels
0 0.2 0.35 0.65 0.8 1
0 1
Amplitude Response
frequency in π units
Hr(w)
0 0.2 0.35 0.65 0.8 1
−1.0575 0
1.0575× 10−3 Weighted Error
frequency in π units
Hr(w)
FIGURE 7.36 Plots for equiripple bandpass FIR filter in Example 7.24
The optimal value ofMwas found to be 29. The time- and the frequency-domain plots of the designed filter are shown in Figure 7.36.
EXAMPLE 7.25 Design a highpass filter that has the following specifications:
ωs= 0.6π, ωp= 0.75π,
As= 50 dB Rp= 0.5 dB
Solution Since this is a highpass filter, we must ensure that the length M is an odd number. This is shown in the following MATLAB script.
>> ws = 0.6*pi; wp = 0.75*pi; Rp = 0.5; As = 50;
>> [delta1,delta2] = db2delta(Rp,As);
>> [N,f,m,weights] = firpmord([ws,wp]/pi,[0,1],[delta2,delta1]); N N = 26
>> h = firpm(N,f,m,weights);
>> [db,mag,pha,grd,w] = freqz_m(h,[1]);
>> delta_w = 2*pi/1000; wsi=ws/delta_w; wpi = wp/delta_w;
>> Asd = -max(db(1:1:wsi)) Asd = 49.5918
>> N = N+2;
>> h = firpm(N,f,m,weights);
>> [db,mag,pha,grd,w] = freqz_m(h,[1]);
>> Asd = -max(db(1:1:wsi))
>> Asd = 50.2253
>> M = N+1 M = 29
Note also that we increased the value ofN by two to maintain its even value.
The optimumMwas found to be 29. The time- and the frequency-domain plots
of the designed filter are shown in Figure 7.37.
EXAMPLE 7.26 In this example we will design a “staircase” filter, which has 3 bands with different ideal responses and different tolerances in each band. The design spec- ifications are
Band-1: 0≤ω≤0.3π, Ideal gain = 1, Toleranceδ1= 0.01 Band-2: 0.4π≤ω≤0.7π, Ideal gain = 0.5, Toleranceδ2= 0.005 Band-3: 0.8π≤ω≤π, Ideal gain = 0, Toleranceδ3= 0.001
0 28
−0.4
−0.2 0 0.2 0.4
n
h(n)
Actual Impulse Response
0 0.6 0.75 1
50 0
Magnitude Response in dB
frequency in π units
Decibels
0 0.6 0.75 1
0 1
Amplitude Response
frequency in π units
Hr(w)
0 0.6 0.75 1
−0.0288
−0.00330.00330 0.0288
Error Response
frequency in π units
Hr(w)
FIGURE 7.37 Plots for equiripple highpass FIR filter in Example 7.25
Solution The following MATLAB script describes the design procedure.
>> w1 = 0; w2 = 0.3*pi; delta1 = 0.01;
>> w3 = 0.4*pi; w4 = 0.7*pi; delta2 = 0.005;
>> w5 = 0.8*pi; w6 = pi; delta3 = 0.001;
>> weights = [delta3/delta1 delta3/delta2 1];
>> Dw = min((w3-w2), (w5-w3));
>> M = ceil((-20*log10((delta1*delta2*delta3)^(1/3))-13)/(2.285*Dw)+1)
>> M = 51
>> f = [0 w2/pi w3/pi w4/pi w5/pi 1];
>> m = [1 1 0.5 0.5 0 0];
>> h = firpm(M-1,f,m,weights);
>> [db,mag,pha,grd,w] = freqz_m(h,[1]);
>> delta_w = 2*pi/1000;
>> w1i=floor(w1/delta_w)+1; w2i = floor(w2/delta_w)+1;
>> w3i=floor(w3/delta_w)+1; w4i = floor(w4/delta_w)+1;
>> w5i=floor(w5/delta_w)+1; w6i = floor(w6/delta_w)+1;
>> Asd = -max(db(w5i:w6i)) Asd = 62.0745
>> M = M-1; h = firpm(M-1,f,m,weights);
>> [db,mag,pha,grd,w] = freqz_m(h,[1]);
>> Asd = -max(db(w5i:w6i)) Asd = 60.0299
>> M = M-1; h = firpm(M-1,f,m,weights);
>> [db,mag,pha,grd,w] = freqz_m(h,[1]);
>> Asd = -max(db(w5i:w6i)) Asd = 60.6068
>> M M = 49
The time- and the frequency-domain plots of the designed filter are shown in
Figure 7.38.
EXAMPLE 7.27 In this example we will design a digital differentiator with different slopes in each band. The specifications are
Band-1: 0≤ω≤0.2π, Slope = 1 sam/cycle Band-2: 0.4π≤ω≤0.6π, Slope = 2 sam/cycle Band-3: 0.8π≤ω≤π, Slope = 3 sam/cycle
Solution We need desired magnitude response values in each band. These can be ob- tained by multiplying band-edge frequencies in cycles/sam by the slope values in sam/cycle
0 48
−0.1 0 0.1 0.2 0.3 0.4 0.5 0.6
n
h(n)
Actual Impulse Response
0 0.3 0.4 0.7 0.8 1
60 0
Magnitude Response in dB
frequency in π units
Decibels
0 0.3 0.4 0.7 0.8 1
0 0.5 1
Amplitude Response
frequency in π units
Hr(w)
0 0.3 0.4 0.7 0.8 1
−1 0
1× 10−3 Weighted Error
frequency in π units
Hr(w)
FIGURE 7.38 Plots for equiripple staircase FIR filter in Example 7.26
Band-1: 0≤f≤0.1,Slope = 1 sam/cycle⇒0.0≤ |H| ≤0.1 Band-2: 0.2≤f≤0.3,Slope = 2 sam/cycle⇒0.4≤ |H| ≤0.6 Band-3: 0.4≤f≤0.5,Slope = 3 sam/cycle⇒1.2≤ |H| ≤1.5 Let the weights be equal in all bands. The MATLAB script is:
>> f = [0 0.2 0.4 0.6 0.8 1]; % in w/pi unis
>> m = [0,0.1,0.4,0.6,1.2,1.5]; % magnitude values
>> h = firpm(25,f,m,’differentiator’);
>> [db,mag,pha,grd,w] = freqz_m(h,[1]);
>> subplot(2,1,1); stem([0:25],h); title(’Impulse Response’);
>> xlabel(’n’); ylabel(’h(n)’); axis([0,25,-0.6,0.6])
>> set(gca,’XTickMode’,’manual’,’XTick’,[0,25])
>> set(gca,’YTickMode’,’manual’,’YTick’,[-0.6:0.2:0.6]);
>> subplot(2,1,2); plot(w/(2*pi),mag); title(’Magnitude Response’)
>> xlabel(’Normalized frequency f’); ylabel(’|H|’)
>> set(gca,’XTickMode’,’manual’,’XTick’,f/2)
>> set(gca,’YTickMode’,’manual’,’YTick’,[0,0.1,0.4,0.6,1.2,1.5]); grid The frequency-domain response is shown in Figure 7.39.
0 25
−0.6
−0.4
−0.2 0 0.2 0.4 0.6
n
h(n)
Impulse Response
0 0.1 0.2 0.3 0.4 0.5
0.10 0.4 0.6 1.2 1.5
Magnitude Response
Normalized frequency f
|H|
FIGURE 7.39 Plots of the differentiator in Example 7.27
EXAMPLE 7.28 Finally, we design a Hilbert transformer over the band 0.05π≤ω≤0.95π.
Solution Since this is a wideband Hilbert transformer, we will choose an odd length for our filter (i.e., a Type-3 filter). Let us chooseM = 51. The MATLAB script is:
>> f = [0.05,0.95]; m = [1 1]; h = firpm(50,f,m,’hilbert’);
>> [db,mag,pha,grd,w] = freqz_m(h,[1]);
>> subplot(2,1,1); stem([0:50],h); title(’Impulse Response’);
>> xlabel(’n’); ylabel(’h(n)’); axis([0,50,-0.8,0.8])
>> set(gca,’XTickMode’,’manual’,’XTick’,[0,50])
>> set(gca,’YTickMode’,’manual’,’YTick’,[-0.8:0.2:0.8]);
>> subplot(2,1,2); plot(w/pi,mag); title(’Magnitude Response’)
>> xlabel(’frequency in pi units’); ylabel(’|H|’)
>> set(gca,’XTickMode’,’manual’,’XTick’,[0,f,1])
>> set(gca,’YTickMode’,’manual’,’YTick’,[0,1]);grid
The plots of this Hilbert transformer are shown in Figure 7.40.
0 50
−0.8
−0.6
−0.4
−0.2 0 0.2 0.4 0.6 0.8
n
h(n)
Impulse Response
0 0.05 0.95 1
0 1
Magnitude Response
frequency in π units
|H|
FIGURE 7.40 Plots of the Hilbert transformer in Example 7.28