The Hamming window described over - (N-1)/2 ≤ n ≤ (N-1)/2 is
%Hamming window defined over n = -(N-1)/2 to (N-1)/2
% w(n) = 0.54 + 0.46 cos(2*pi*n/(N-1))
N = 31; n = -(N-1)/2: (N-1)/2; wn = 0.54 + 0.46 * cos(2*pi*n/(N-1)), stem (n, wn); xlabel('n'), ylabel('w(n)'); grid; title ('Hamming window')
n = -5 -4 -3 -2 -1 0 1 2 3 4 5
w(n) = {0.0800 0.0901 0.1198 0.1679 0.2322 0.3100 0.3979 0.4919 0.5881
0.6821 0.7700 0.8478 0.9121 0.9602 0.9899 1.0000 0.9899 0.9602 0.9121
0.8478 0.7700 0.6821 0.5881 0.4919 0.3979 0.3100 0.2322 0.1679 0.1198
0.0901 0.0800}
The Fourier transform (spectrum) of the window is
The magnitude at dc is
That is computed below for N = 11, 21, 31 and 41:
%Magnitude at DC N = 11:10:41,
WdcN = 0.54*N + 0.46* sin(pi*N./(N-1))./sin(pi./(N-1))
N = 11 21 31 41
WdcN = 5.4800 10.8800 16.2800 21.6800
The width of the main lobe is given as the separation between the zero crossings on both side of ω = 0. This is calculated by setting W (ejw) = 0 and computing for ω; it is provided as
Width of main lobe (Hamming) = 8π/N
twice that of the rectangular window.
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