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 ) from a sampled sinusoid which is continuous 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 [67] on our example sinusoid. The Blackman window has good (though suboptimal) characteristics for audio work.

In Octave or the Matlab Signal Processing Toolbox, a Blackman window of length 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);
figure(1); 
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;
subplot(3,1,3);
plot(fninf,specnf,'-'); axis([-0.5,0.5,-100,10]); grid;
xlabel('Normalized Frequency (cycles per sample))'); 
ylabel('Magnitude (dB)');
text(-.62,20,'c)'); 
cmd = ['print -deps ', '../eps/blackman.eps'];
disp(cmd); eval(cmd);
disp 'pausing for RETURN (check the plot). . .'; pause

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Subsections

• Applying the Blackman Window

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Order a Hardcopy of Mathematics of the DFT

Previous: FFT of a Zero-Padded Sinusoid
Next: Applying the Blackman Window

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written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.