#### 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.

**Next Section:**

Spectral Phase

**Previous Section:**

Applying the Blackman Window