Hann Window Spectrum Analysis Results

Finally, the Matlab for computing the DFT of the Hann-windowed complex sinusoid and plotting the results is listed below. To help see the full spectrum, we also compute a heavily interpolated spectrum (via zero padding as before) which we'll draw using solid lines.

% Compute the spectrum and its alternative forms:
Xw = fft(xw);              % FFT of windowed data
fn = [0:1.0/N:1-1.0/N];    % Normalized frequency axis
spec = 20*log10(abs(Xw));  % Spectral magnitude in dB
% Since the nulls can go to minus infinity, clip at -100 dB:
spec = max(spec,-100*ones(1,length(spec)));
phs = angle(Xw);           % Spectral phase in radians
phsu = unwrap(phs);        % Unwrapped spectral phase

% Compute heavily interpolated versions for comparison:
Nzp = 16;                   % Zero-padding factor
Nfft = N*Nzp;               % Increased FFT size
xwi = [xw,zeros(1,Nfft-N)]; % New zero-padded FFT buffer
Xwi = fft(xwi);             % Compute interpolated spectrum
fni = [0:1.0/Nfft:1.0-1.0/Nfft]; % Normalized freq axis
speci = 20*log10(abs(Xwi)); % Interpolated spec mag (dB)
speci = max(speci,-100*ones(1,length(speci))); % clip
phsi = angle(Xwi);          % Interpolated phase
phsiu = unwrap(phsi);       % Unwrapped interpolated phase

figure(1);
subplot(2,1,1);

plot(fn,abs(Xw),'*k'); hold on; 
plot(fni,abs(Xwi),'-k'); hold off;
title('Spectral Magnitude'); 
xlabel('Normalized Frequency (cycles per sample))'); 
ylabel('Amplitude (linear)');

subplot(2,1,2);

% Same thing on a dB scale
plot(fn,spec,'*k'); hold on; plot(fni,speci,'-k'); hold off;
title('Spectral Magnitude (dB)'); 
xlabel('Normalized Frequency (cycles per sample))'); 
ylabel('Magnitude (dB)');

cmd = ['print -deps ', 'specmag.eps']; disp(cmd); eval(cmd);
disp 'pausing for RETURN (check the plot). . .'; pause

figure(1);
subplot(2,1,1);
plot(fn,phs,'*k'); hold on; plot(fni,phsi,'-k'); hold off;
title('Spectral Phase'); 
xlabel('Normalized Frequency (cycles per sample))'); 
ylabel('Phase (rad)'); grid;
subplot(2,1,2);
plot(fn,phsu,'*k'); hold on; plot(fni,phsiu,'-k'); hold off;
title('Unwrapped Spectral Phase'); 
xlabel('Normalized Frequency (cycles per sample))'); 
ylabel('Phase (rad)'); grid;
cmd = ['print -deps ', 'specphs.eps']; disp(cmd); eval(cmd);

Figure 8.8 shows the spectral magnitude and Fig.8.9 the spectral phase.

There are no negative-frequency components in Fig.8.8 because we are analyzing a complex sinusoid , which has frequency only, with no component at .

Notice how difficult it would be to correctly interpret the shape of the ``sidelobes'' without zero padding. The asterisks correspond to a zero-padding factor of 2, already twice as much as needed to preserve all spectral information faithfully, but not enough to clearly outline the sidelobes in a spectral magnitude plot.

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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.