122 8. FREQUENCY DOMAIN TRANSFORMS
N
s
T
s
t
t
Figure 8.1: Periodicity condition of sampling.
to the other bands of the transform. is is evident in Figure 8.2b by the diminished peak at
250 Hz and increased amplitude of the bands to either side of the peak.
8.3 WINDOWING
Leakage can be reduced by applying a windowing function to the incoming signal. In the time
domain, a windowing function is shaped such that when it is applied to a signal, the beginning
and end taper toward zero. One such window is the Hanning window, shown in Figure 8.3.
An example C code which generates the Hanning window appears below:
float* Hanning(int N) {
float* window = (float*)malloc(N*sizeof(float));
for(i=0; i<N; i++) {
window[i] = (float)((1.0-cos(2.0*M_PI*(i+1)/(N+1)))*0.5);
}
return window;
}
8.4. OVERLAP PROCESSING 123
Frequency (kHz)
(a)
Amplitude (dB)Amplitude (dB)
0
0
0.1 0.2 0.3 0.4 0.5
Frequency (kHz)
(b)
0.1 0.2 0.3 0.4 0.5
0
-50
-100
-150
-200
-250
-300
-350
0
-10
-20
-30
-40
-50
-60
-70
-80
-90
-100
Due to limited resolution
of numerical calculation
Due to improper sampling
Figure 8.2: FFTs of (a) 250 Hz and (b) 251 Hz sinusoids.
8.4 OVERLAP PROCESSING
It should be noted that windowing introduces distortion of the signal information. Overlap
processing is often used to reduce this distortion. Overlap processing is a method by which
the time-domain resolution of the Fourier transform is increased. In overlap processing, in-
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