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 [70]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)$.

Applying the Blackman Window

Now let's apply the Blackman window to the sampled sinusoid and look at the effect on the spectrum analysis:

% Windowed, zero-padded data:
n = [0:M-1];          % discrete time axis
f = 0.25 + 0.5/M;     % frequency
xw = [w .* cos(2*pi*n*f),zeros(1,(zpf-1)*M)];

% Smoothed, interpolated spectrum:
X = fft(xw);

% Plot time data:
title('Windowed, Zero-Padded, Sampled Sinusoid');
xlabel('Time (samples)');

% Plot spectral magnitude:
spec = 10*log10(conj(X).*X);  % Spectral magnitude in dB
spec = max(spec,-60*ones(1,nfft)); % clip to -60 dB
title('Smoothed, Interpolated, Spectral Magnitude (dB)');
xlabel('Normalized Frequency (cycles per sample))');
ylabel('Magnitude (dB)'); grid;
Figure 8.6 plots the zero-padded, Blackman-windowed sinusoid, along with its magnitude spectrum on a dB scale. Note that the first sidelobe (near $ -40$ dB) is nearly 60 dB below the spectral peak (near $ +20$ dB). This is why the Blackman window is considered adequate for many audio applications. From the dual of the convolution theorem discussed in §7.4.6, we know that windowing in the time domain corresponds to smoothing in the frequency domain. Specifically, the complex spectrum with magnitude displayed in Fig.8.4b (p. [*]) has been convolved with the Blackman window transform (dB magnitude shown in Fig.8.5c). Thus, the Blackman window Fourier transform has been applied as a smoothing kernel to the Fourier transform of the rectangularly windowed sinusoid to produce the smoothed result in Fig.8.6b. This topic is pursued in detail at the outset of Book IV in the music signal processing series [70].

Figure 8.6: Effect of the Blackman window on the sinusoidal data.

Next Section:
Hann-Windowed Complex Sinusoid
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FFT of a Zero-Padded Sinusoid