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Use of a Blackman Window

As Fig.8.4a suggests, the previous example can be interpreted as using a rectangular window to select a finite segment (of length $ N$ ) from a sampled sinusoid that continues for all time. In practical spectrum analysis, such excerpts are normally analyzed using a window that is tapered more gracefully to zero on the left and right. In this section, we will look at using a Blackman window [73]8.3on our example sinusoid. The Blackman window has good (though suboptimal) characteristics for audio work.

In Octave8.4or the Matlab Signal Processing Toolbox,8.5a Blackman window of length $ M=64$ can be designed very easily:

M = 64;
w = blackman(M);
Many other standard windows are defined as well, including hamming, hanning, and bartlett windows.

In Matlab without the Signal Processing Toolbox, the Blackman window is readily computed from its mathematical definition:

w = .42 - .5*cos(2*pi*(0:M-1)/(M-1)) ...
       + .08*cos(4*pi*(0:M-1)/(M-1));

Figure 8.5 shows the Blackman window and its magnitude spectrum on a dB scale. Fig.8.5c uses the more ``physical'' frequency axis in which the upper half of the FFT bin numbers are interpreted as negative frequencies. Here is the complete Matlab script for Fig.8.5:

M = 64;  
w = blackman(M);
subplot(3,1,1); plot(w,'*'); title('Blackman Window');
xlabel('Time (samples)'); ylabel('Amplitude'); text(-8,1,'a)'); 

% Also show the window transform:
zpf = 8;                      % zero-padding factor
xw = [w',zeros(1,(zpf-1)*M)]; % zero-padded window
Xw = fft(xw);                 % Blackman window transform
spec = 20*log10(abs(Xw));     % Spectral magnitude in dB
spec = spec - max(spec);      % Normalize to 0 db max
nfft = zpf*M;
spec = max(spec,-100*ones(1,nfft)); % clip to -100 dB
fni = [0:1.0/nfft:1-1.0/nfft];   % Normalized frequency axis
subplot(3,1,2); plot(fni,spec,'-'); axis([0,1,-100,10]); 
xlabel('Normalized Frequency (cycles per sample))'); 
ylabel('Magnitude (dB)'); grid; text(-.12,20,'b)'); 

% Replot interpreting upper bin numbers as frequencies<0:
nh = nfft/2;
specnf = [spec(nh+1:nfft),spec(1:nh)];  % see fftshift()
fninf = fni - 0.5;
plot(fninf,specnf,'-'); axis([-0.5,0.5,-100,10]); grid;
xlabel('Normalized Frequency (cycles per sample))'); 
ylabel('Magnitude (dB)');
cmd = ['print -deps ', '../eps/blackman.eps'];
disp(cmd); eval(cmd);
disp 'pausing for RETURN (check the plot). . .'; pause

Figure 8.5: The Blackman window: a) window itself in the time domain, b) dB magnitude spectrum plotted over normalized frequencies $ [0,1)$ , and c) same thing plotted over $ [-0.5,0.5)$ .

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``Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition'', by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8
Copyright © 2024-04-02 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University