Introduction to Digital Signal and System Analysis - Tutorsindia
Introduction to Digital Signal and System Analysis - Tutorsindia
Introduction to Digital Signal and System Analysis - Tutorsindia
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<strong>Introduction</strong> <strong>to</strong> <strong>Digital</strong> <strong>Signal</strong> <strong>and</strong> <strong>System</strong> <strong>Analysis</strong><br />
Discrete Fourier Transform<br />
<br />
exp<br />
−<br />
<br />
2πkn<br />
2πkn<br />
2πkn<br />
j = cos − j sin<br />
N N N<br />
where k is the frequency of the sinusoidal function which runs through all possibilities from 0 (direct current) <strong>to</strong> N-1.<br />
The following figure shows the first few sinusoidal components.<br />
2<br />
k=0<br />
0<br />
-2<br />
0<br />
2<br />
5 10 15<br />
k=1<br />
0<br />
-2<br />
0<br />
2<br />
5 10 15<br />
k=2<br />
0<br />
-2<br />
0<br />
2<br />
5 10 15<br />
k=3<br />
0<br />
-2<br />
0 5 10 15<br />
2<br />
0<br />
-2<br />
0 5 10 15<br />
2<br />
0<br />
-2<br />
0 5 10 15<br />
2<br />
0<br />
-2<br />
0 5 10 15<br />
2<br />
0<br />
-2<br />
0 5 10 15<br />
2πkn<br />
2πkn<br />
cos − sin<br />
N<br />
N<br />
Figure 6.3 Decomposition of a periodic digital in<strong>to</strong> cosine <strong>and</strong> sine waves.<br />
6.2 Properties of DFT<br />
1. Periodicity<br />
In the time domain,<br />
x [ n ± rN ] = x[<br />
n]<br />
(6.3)<br />
<strong>and</strong> in the frequency domain<br />
X [ k ± rN ] = X [ k]<br />
(6.4)<br />
where r is an arbitrary integer <strong>and</strong> N is the period. This property says that the shape of the signal stays the same when it<br />
is shifted <strong>to</strong> left or right by integer number of N samples.<br />
90<br />
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