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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:
subplot(2,1,1); 
plot(xw);
title('Windowed, Zero-Padded, Sampled Sinusoid');
xlabel('Time (samples)'); 
ylabel('Amplitude');
text(-50,1,'a)'); 

% Plot spectral magnitude:
spec = 10*log10(conj(X).*X);  % Spectral magnitude in dB
spec = max(spec,-60*ones(1,nfft)); % clip to -60 dB
subplot(2,1,2);
plot(fninf,fftshift(spec),'-'); 
axis([-0.5,0.5,-60,40]); 
title('Smoothed, Interpolated, Spectral Magnitude (dB)'); 
xlabel('Normalized Frequency (cycles per sample))'); 
ylabel('Magnitude (dB)'); grid;
text(-.6,40,'b)');
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 [73].

Figure 8.6: Effect of the Blackman window on the sinusoidal data.
\includegraphics[width=\twidth]{eps/xw}


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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 © 2014-10-23 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
CCRMA