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 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]^{8.3}on our example sinusoid. The Blackman window has good (though
suboptimal) characteristics for audio work.

In Octave^{8.4}or the Matlab Signal Processing Toolbox,^{8.5}a Blackman window of length
can be designed very easily:

M = 64; w = blackman(M);Many other standard windows are defined as well, including

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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