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Hann-Windowed Complex Sinusoid

In this example, we'll perform spectrum analysis on a complex sinusoid having only a single positive frequency. We'll use the Hann window (also known as the Hanning window) which does not have as much sidelobe suppression as the Blackman window, but its main lobe is narrower. Its sidelobes ``roll off'' very quickly versus frequency. Compare with the Blackman window results to see if you can see these differences.

The Matlab script for synthesizing and plotting the Hann-windowed sinusoid is given below:

% Analysis parameters:
M = 31;         % Window length 
N = 64;         % FFT length (zero padding factor near 2)

% Signal parameters:
wxT = 2*pi/4;   % Sinusoid frequency (rad/sample)
A = 1;          % Sinusoid amplitude
phix = 0;       % Sinusoid phase

% Compute the signal x:
n = [0:N-1];    % time indices for sinusoid and FFT
x = A * exp(j*wxT*n+phix); % complex sine [1,j,-1,-j...]

% Compute Hann window:
nm = [0:M-1];   % time indices for window computation
% Hann window = "raised cosine", normalization (1/M)
% chosen to give spectral peak magnitude at 1/2:
w = (1/M) * (cos((pi/M)*(nm-(M-1)/2))).^2;  

wzp = [w,zeros(1,N-M)]; % zero-pad out to the length of x
xw = x .* wzp;          % apply the window w to signal x


% Display real part of windowed signal and Hann window
plot(n,wzp,'-k'); hold on; plot(n,real(xw),'*k'); hold off;
title(['Hann Window and Windowed, Zero-Padded, ',...
       'Sinusoid (Real Part)']); 
xlabel('Time (samples)'); ylabel('Amplitude');
The resulting plot of the Hann window and its use on sinusoidal data are shown in Fig.8.7.

Figure 8.7: A length 31 Hann window (``raised cosine'') overlaid with the real part of the Hann-windowed complex sinusoid. Zero-padding is also shown. The sampled sinusoid is plotted using `*' with no connecting interpolation lines. You must now imagine the continuous real sinusoid (windowed) threading through the asterisks.

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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 © 2024-04-02 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University