Filter Design Toolbox 4.6

Octave-Band and Fractional Octave-Band Filters

Octave-band and fractional-octave-band filters are commonly used in acoustics, for example, in noise control to perform spectral analysis. Acousticians prefer to work with octave or fractional (often 1/3) octave filter banks because it gives them a more meaningful measure of the noise power in different frequency bands.

Contents

Design of a Full Octave-Band and a 1/3-Octave-Band Filter Banks

An octave is the interval between two frequencies having a ratio of 2:1. An octave-band or fractional-octave-band filter is a bandpass filter determined by its center frequency and its order. The magnitude attenuation limits are defined in the ANSI® S1.11-2004 standard for three classes of filters: class 0, class 1 and class 2. Class 0 allow only +/-.15 dB of ripples in the passband while class 1 filters allow +/-.3 dB and class 2 filters allow +/-.5 dB. Levels of stopband attenuation vary from 60 to 75dB depending on the class of the filter.

Design a full octave-band filter bank:

BandsPerOctave = 1;
N = 6;           % Filter Order
F0 = 1000;       % Center Frequency (Hz)
Fs = 48000;      % Sampling Frequency (Hz)
f = fdesign.octave(BandsPerOctave,'Class 1','N,F0',N,F0,Fs)

f =

               Response: 'Octave and Fractional Octave'
         BandsPerOctave: 1
                   Mask: 'Class 1'
          Specification: 'N,F0'
            Description: {'Filter Order';'Exact Midband Frequency'}
    NormalizedFrequency: false
                     Fs: 48000
            FilterOrder: 6
                     F0: 1000

Get all the valid center frequencies in the audio range to design the filter bank:

F0 = validfrequencies(f);
Nfc = length(F0);
for i=1:Nfc,
    f.F0 = F0(i);
    Hd(i) = design(f,'butter');
end

Now design a 1/3-octave-band filter bank. Increase the order of each filter to 8:

f.BandsPerOctave = 3;
f.FilterOrder = 8;
F0 = validfrequencies(f);
Nfc = length(F0);
for i=1:Nfc,
    f.F0 = F0(i);
    Hd3(i) = design(f,'butter');
end

Visualize the magnitude response of the two filter banks. The 1/3-octave filter bank will provide a finer spectral analysis but at an increased cost since it requires 30 filters versus 10 for the full octave filter bank to cover the audio range [20 20000 Hz].

hfvt = fvtool(Hd,'FrequencyScale','log','color','white');
axis([0.01 24 -90 5])
title('Octave-Band Filter Bank')
hfvt = fvtool(Hd3,'FrequencyScale','log','color','white');
axis([0.01 24 -90 5])
title('1/3-Octave-Band Filter Bank')

Spectral Analysis of White Noise

The human ear interprets loudness of sound on a scale much closer to a logarithmic scale than a linear one but a DFT-based frequency analysis leads to linear frequency scale. Compute the (DFT-based) mean-square spectrum of a white noise signal using Welch's method:

rand('state',0); Nx = 100000;
xw = randn(Nx,1);
hp = spectrum.welch;
msPyyw = msspectrum(hp,xw,'Fs',Fs);

Now filter the white noise signal with the 1/3-octave filter bank and compute the average power at the output of each filter:

yw = zeros(Nx,Nfc);
for i=1:Nfc,
    yw(:,i) = filter(Hd3(i),xw);
    Pyyw(i) = avgpower(psd(hp,yw(:,i),'Fs',Fs));
end

While the mean-square spectrum of a white noise signal is flat, the high frequencies are perceived louder. The 1/3-octave spectrum paints a picture that is closer to the human ear perception. It shows a spectrum where the power level rise 3dB per octave because each band (i.e. filter) has twice the frequency range of the preceding octave.

figure('Color','white')
semilogx(msPyyw.Frequencies,10*log10(msPyyw.Data),'o')
axis([20 20000 -60 0])
title('Welch Mean-Square Spectrum Estimate of White Noise')
xlabel('Frequency (Hz)');ylabel('Power (dB)')
figure('Color','white')
semilogx(F0,10*log10(Pyyw),'o')
axis([20 20000 -60 0])
title('1/3-Octave Spectrum of White Noise')
xlabel('Frequency (Hz)');ylabel('Power (dB)')

Spectral Analysis of Pink Noise

While a white noise signal has the same distribution of power for all frequencies, a pink noise signal has the same distribution of power for each octave, so the power between 0.5 Hz and 1 Hz is the same as between 5,000 Hz and 10,000 Hz. Here again, compute the (DFT-based) mean-square spectrum of a pink noise signal using Welch's method:

load pinknoise;
msPyy = msspectrum(hp,x,'Fs',Fs);

Now filter the pink noise signal with the 1/3-octave filter bank and compute the average power at the output of each filter:

for i=1:Nfc,
    y(:,i) = filter(Hd3(i),x);
    Pyy(i) = avgpower(psd(hp,y(:,i),'Fs',Fs));
end

The power of the pink noise signal decline at higher frequencies at the rate of about -3dB per octave as show by the Welch mean-square spectrum estimate. However it sounds "constant" to the human hear and 1/3 octave-band spectrum shows flat at the output of the filter bank.

figure('Color','white')
semilogx(msPyy.Frequencies,10*log10(msPyy.Data),'o')
axis([20 24000 -80 -20])
title('Welch Mean-Square Spectrum Estimate of Pink Noise')
xlabel('Frequency (Hz)');ylabel('Power (dB)')
figure('Color','white')
semilogx(F0,10*log10(Pyy),'o')
axis([20 24000 -80 -20])
title('1/3-Octave Spectrum of Pink Noise')
xlabel('Frequency (Hz)');ylabel('Power (dB)')