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Parametric Equalizer Design

This example shows how to design parametric equalizer filters. Parametric equalizers are digital filters used in audio for adjusting the frequency content of a sound signal. Parametric equalizers provide capabilities beyond those of graphic equalizers by allowing the adjustment of gain, center frequency, and bandwidth of each filter. In contrast, graphic equalizers only allow for the adjustment of the gain of each filter.

Typically, parametric equalizers are designed as second-order IIR filters. These filters have the drawback that because of their low order, they can present relatively large ripple or transition regions and may overlap with each other when several of them are connected in cascade. The DSP System Toolbox™ provides the capability to design high-order IIR parametric equalizers. Such high-order designs provide much more control over the shape of each filter. In addition, the designs special-case to traditional second-order parametric equalizers if the order of the filter is set to two.

Some Basic Designs

Consider the following two designs of parametric equalizers. The design specifications are the same except for the filter order. The first design is a typical second-order parametric equalizer that boosts the signal around 0.5*pi rad/sample by 6 dB. The second design does the same with a fourth-order filter. Notice how the fourth-order filter is closer to an ideal brickwall filter when compared to the second-order design. Obviously the approximation can be improved by increasing the filter order even further. The price to pay for such improved approximation is increased implementation cost as more multipliers are required.

f = fdesign.parameq('N,F0,BW,Gref,G0,GBW',2,0.5,0.2,0,6,6+10*log10(.5))
h = design(f);
f.FilterOrder = 4;
h1 = design(f);
hfvt = fvtool(h,h1,'Color','white');
legend(hfvt,'2nd-Order Design','4th-Order Design');
 
f =
 
               Response: 'Parametric Equalizer'
          Specification: 'N,F0,BW,Gref,G0,GBW' 
            Description: {6x1 cell}            
    NormalizedFrequency: true                  
            FilterOrder: 2                     
                     F0: 0.5                   
                     BW: 0.2                   
                   Gref: 0                     
                     G0: 6                     
                    GBW: 2.98970004336019      
                                               

One of the design parameters is the filter bandwidth, BW. Note however that you not only specify what the desired bandwidth is, you also specify the reference gain, GBW, at which this bandwidth is defined. In this case, we set GBW to be at half the peak magnitude squared gain of the filter, that is, approximately 3.0103 dB below the 6 dB gain. To see this, we can plot the magnitude squared filter response and verify that the gain at which the filter's bandwidth is 0.2*pi is equal to one half of the peak magnitude squared.

set(hfvt,'Filters',[h h1],'MagnitudeDisplay','Magnitude squared');
legend(hfvt,'2nd-Order Design','4th-Order Design');

Designs Based on Quality Factor

Another common design parameter is the quality factor, Qa. As a first example we design a second order peak filter with a quality factor Qa=5.

N = 2;             % Filter Order
F0 = 0.2;          % Center Frequency
Qa = 5;            % Quality Factor
Gref = 0;          % Reference Gain (dB)
G0 = 10; % Gain at Center Frequency or Boost Gain (dB)

f = fdesign.parameq('N,F0,Qa,Gref,G0',N,F0,Qa,Gref,G0)
h1 = design(f);
 
f =
 
               Response: 'Parametric Equalizer'
          Specification: 'N,F0,Qa,Gref,G0'     
            Description: {5x1 cell}            
    NormalizedFrequency: true                  
            FilterOrder: 2                     
                     F0: 0.2                   
                     Qa: 5                     
                   Gref: 0                     
                     G0: 10                    
                                               

The design may be verified by measuring the quality factor of the filter which is directly obtained from the poles of its second order transfer function. Notice also how the measurement object returns the linear bandwidth BW of 0.037377, referenced to the gain GBW of 5 dB which corresponds to the geometric mean of the magnitude squared values of G0 and Gref.

m = measure(h1)
 
m =
 
Sample Rate            : N/A (normalized frequency)
Center Frequency       : 0.2                       
Bandwidth              : 0.037377                  
Passband Bandwidth     : Unknown                   
Stopband Bandwidth     : Unknown                   
Flow                   : 0.18207                   
Fhigh                  : 0.21944                   
Bandwidth Gain         : 5 dB                      
Low Transition Width   : Unknown                   
High Transition Width  : Unknown                   
Reference Gain         : 0 dB                      
Center Frequency Gain  : 10 dB                     
Passband Gain          : Unknown                   
Stopband Gain          : Unknown                   
Quality Factor (audio) : 5                         
 

We may use the bandwidth defined by Qa for a second order filter to design higher order filters. Higher order designs will yield designs with sharper transition bands while maintaining the same bandwidth referenced to the same gain GBW shown above. As an example, we increase the order of the previously designed peak filter to 4, and 10.

f.FilterOrder = 4;
h2 = design(f);
f.FilterOrder = 10;
h3 = design(f);
set(hfvt,'Filters',[h1 h2 h3],'MagnitudeDisplay','Magnitude (dB)');
axis([0 .5 -0.5 10.5])
legend(hfvt,'N=2','N=4','N=10')

Designs Based on Octave Bandwidth

A desired octave bandwidth can be translated into a quality factor Qa. For instance, we may design two second order peaking filters with 0.5 and 1 octave bandwidths respectively.

BWoct = 0.5; % Desired Octave Bandwidth
f.Qa = 0.5/( sinh(BWoct*(log(2)/2)*(F0*pi)/sin(F0*pi)));
h1 = design(f);

BWoct = 1; % Desired Octave Bandwidth
f.Qa = 0.5/( sinh(BWoct*(log(2)/2)*(F0*pi)/sin(F0*pi)));
h2 = design(f);

set(hfvt,'Filters',[h1 h2]);
legend(hfvt,'Octave BW=0.5','Octave BW=1')

A Parametric Equalizer That Cuts

The previous design is an example of a parametric equalizer that boosts the signal over a certain frequency band. You can also design equalizers that cut (attenuate) the signal in a given region.

setspecs(f,'N,F0,BW,Gref,G0,GBW,Gp,Gst',6,0.3,0.1,0,-3,-2,-2.5,-0.5);
h = design(f);
set(hfvt,'Filters',h,'legend','off');
axis([0 1 -3 0.5])

Notice that in this case we have specified both a passband gain, Gp, and a stopband gain, Gst. This parameters allow for the filter to ripple in the passband and stopband with the advantage of providing steeper transitions between passband and stopband. For comparison, consider a filter of the same order without ripples. Notice the wider transitions that result as a tradeoff.

setspecs(f,'N,F0,BW,Gref,G0,GBW',6,0.3,0.1,0,-3,-2);
h1 = design(f);
set(hfvt,'Filters',[h h1]);
axis([0 1 -3 0.5])

It is also possible to only specify ripples in the passband or to only specify ripples in the stopband.

Minimum-Order Designs

Returning to the first design, instead of manually increasing the filter order to better approximate a brickwall filter, we can specify the desired shape and design a filter of minimum-order that meets such specifications. To specify the shape, in addition to the bandwidth BW and corresponding gain GBW, we specify a passband bandwidth BWp and corresponding gain Gp. It would also be possible to specify a stopband bandwidth and corresponding gain.

f = fdesign.parameq('F0,BW,BWp,Gref,G0,GBW,Gp',...
    0.5,0.2,0.18,0,6,6+10*log10(.5),5.8);
h = design(f,'butter');

Notice that we specified that we wanted a Butterworth (i.e. a maximally flat) design. If we allow for passband ripples, we can design a Chebyshev Type I filter instead.

h1 = design(f,'cheby1');
set(hfvt,'Filters',[h h1]);
legend(hfvt,'Butterworth Design','Chebyshev Type I Design', ...
    'Location','NorthEast');

In this case, the tradeoff occurs between ripples and filter order.

bord = order(h)  % Order of Butterworth design
cord = order(h1) % Order of Chebyshev Type I design
bord =

    32


cord =

    10

Lowpass and Highpass Shelving Filters

The filter's bandwidth BW is only perfectly centered around the center frequency F0 when such frequency is set to 0.5*pi (half the Nyquist rate). When F0 is closer to 0 or to pi, there is a warping effect that makes a larger portion of the bandwidth to occur at one side of the center frequency. In the edge cases, if the center frequency is set to 0 (pi), the entire bandwidth of the filter occurs to the right (left) of the center frequency. The result is a so-called shelving lowpass (highpass) filter.

f = fdesign.parameq('F0,BW,BWp,Gref,G0,GBW,Gp,Gst',...
    0, 0.3, 0.2, 0, 4, 2, 3.5, 0.5);
f1 = fdesign.parameq('F0,BW,BWp,Gref,G0,GBW,Gp,Gst',...
    1, 0.3, 0.2, 0, 4, 2, 3.5, 0.5);
h = design(f);
h1 = design(f1);
set(hfvt,'Filters',[h h1]);
legend(hfvt,'Lowpass Shelving Filter','Highpass Shelving Filter');

Specifying Low and High Frequencies

Because of the frequency warping mentioned above, in general it can be difficult to control the exact frequency edges at which the bandwidth occurs. To do so, an alternate specification can be used.

f = fdesign.parameq('N,Flow,Fhigh,Gref,G0,GBW,Gst',...
    4,.35,.55,0,-8,-7,-0.5);
h = design(f);
set(hfvt,'Filters',h,'legend','off');

Notice that the gain at 0.35*pi and 0.55*pi rad/sample is exactly -7 dB as specified.

Shelving Filters with a Variable Transition Bandwidth or Slope

One of the characteristics of a shelving filter is the transition bandwidth (sometimes also called transition slope) which may be specified by a shelf slope parameter S. The bandwidth reference gain GBW is always set to half the boost or cut gain of the shelving filter. All other parameters being constant, as S increases the transition bandwidth decreases, (and the slope of the response increases) creating a "slope rotation" around the GBW point as illustrated in the example below.

F0 = 0;  % F0=0 designs a lowpass filter, F0=1 designs a highpass filter
Fc = .2; % Cutoff Frequency
G0 = 10;
S = 1.5;
f = fdesign.parameq('N,F0,Fc,S,G0',N,F0,Fc,S,G0);
h1 = design(f);

f.S = 2.5;
h2 = design(f);

f.S = 4;
h3 = design(f);

set(hfvt,'Filters',[h1 h2 h3]);
legend(hfvt,'S=1.5','S=2.5','S=4');

The transition bandwidth and the bandwidth gain corresponding to each value of S can be obtained using the MEASURE method. We verify that the bandwidth reference gain GBW is the same for the three designs and we quantify by how much the transition width decreases when S increases.

m = measure([h1 h2 h3]);
get(m,'GBW')
ans = 

    [5]
    [5]
    [5]

get(m,'HighTransitionWidth')
ans = 

    [0.2083]
    [0.0959]
    [0.0539]

As the shelf slope parameter S increases, the ripple of the filters also increases. We can increase the filter order to reduce the ripple while maintaining the desired transition bandwidth.

h1 = h3;

f.FilterOrder = 3;
h2 = design(f);

f.FilterOrder = 4;
h3 = design(f);

set(hfvt,'Filters',[h1 h2 h3]);
legend(hfvt,'N=2','N=3','N=4');
hold on;
m = measure(h1);
plot(m.BWpass,m.G0,'k*','markersize',10)
plot(m.BW,m.GBW,'k*','markersize',10)
plot(m.BWstop,m.Gref,'k*','markersize',10)

The three responses intercept at the three points (marked with asterisks on the above figure) that respectively define the passband bandwidth, the bandwidth reference gain, and the stopband bandwidth. Hence, all three filters have the same transition width and bandwidth. Note however that the higher order filters have considerably less ripple.

Shelving Filters with a Prescribed Quality Factor

The quality factor Qa may be used instead of the shelf slope parameter S to design shelving filters with variable transition bandwidths.

F0 = 1;    % Highpass Shelving Filter
Fc = .3;
Qa = 0.48;
f = fdesign.parameq('N,F0,Fc,Qa,G0',N,F0,Fc,Qa,G0);
h1 = design(f);

f.Qa = 1/sqrt(2);
h2 = design(f);

f.Qa = 2.0222;
h3 = design(f);

close(hfvt);
hfvt = fvtool([h1 h2 h3],'Color','white');
legend(hfvt,'Qa=0.48','Qa=0.7071','Qa=2.0222');

Cascading Parametric Equalizers

Parametric equalizers are usually connected in cascade (in series) so that several are used simultaneously to equalize an audio signal. To connect several equalizers in this way, we use the CASCADE function.

f1 = fdesign.parameq('N,F0,BW,Gref,G0,GBW',2,.4,.2,0,5,5+10*log10(.5));
f2 = fdesign.parameq('N,F0,BW,Gref,G0,GBW',2,.6,.15,0,-5,-5-10*log10(.5));
h1 = design(f1);
h2 = design(f2);
hc = cascade(h1,h2);
set(hfvt,'Filters',[h1 h2 hc]);
legend(hfvt,'Second-Order Boost Filter','Second-Order Cut Filter',...
    'Cascade of the Two Filters');
axis([0 1 -5.2 5.2]);

Low-order designs such as the second-order filters above can interfere with each other if their center frequencies are closely spaced. Higher-order designs are less prone to such interference.

f1.FilterOrder = 8;
f2.FilterOrder = 8;
h3   = design(f1);
h4   = design(f2);
hc2  = cascade(h3,h4);
set(hfvt,'Filters',[h3 h4 hc2]);
legend(hfvt,'Eighth-Order Boost Filter','Eighth-Order Cut Filter',...
    'Cascade of the Two Filters','Location','NorthEast');

Complementary Peak and Notch Parametric Equalizers

Next we design a complementary pair of second order peak and notch parametric equalizers. Notice that because these filters are complementary, the cascaded response corresponds to an all-pass filter.

N  = 2;
F0 = 0.3;
Qa = 5;
Gref = 0;
G0 = 10;    % Boost Gain (dB)
f = fdesign.parameq('N,F0,Qa,Gref,G0',N,F0,Qa,Gref,G0);
h1 = design(f);

f.G0 = -10; % Cut Gain (dB)
h2 = design(f);
h3 = cascade(h1,h2);
set(hfvt,'Filters',[h1 h2 h3]);
axis([0 1 -10 10]);
legend(hfvt,'Peak parametric equalizer G0=10 dB', ...
            'Notch parametric equalizer G0=-10 dB','Cascaded responses')

Designing Traditional Filters

Traditional bandpass filters can be designed by setting the reference gain to negative infinity (dB), i.e. zero in absolute units. This example shows a minimum-order lowpass elliptic design with a given 3-dB point.

f = fdesign.parameq('F0,BW,BWp,Gref,G0,GBW,Gp,Gst',...
    0, 0.2, 0.19, -Inf, 0, 10*log10(0.5),-0.5, -85);
h = design(f);
set(hfvt,'Filters',h,'legend','off');

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