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 />
where<br />
W k<br />
8<br />
⎛ 2p<br />
k<br />
= exp⎜<br />
− j<br />
⎝ 8<br />
⎞<br />
⎟<br />
⎠<br />
<strong>and</strong><br />
W k<br />
8<br />
2π<br />
k<br />
= exp<br />
− j<br />
8<br />
<br />
<br />
<br />
In the following Figure 6.5, two boxes represents 2 length N=4 DFTs. The solid lines represent moves <strong>and</strong> the doted lines<br />
represent complex multiplications.<br />
x[n]<br />
X[k]<br />
0<br />
<br />
0<br />
<br />
0<br />
⊕ ⊕<br />
0<br />
⊕<br />
1<br />
<br />
2<br />
<br />
3<br />
<br />
2<br />
<br />
4<br />
<br />
6<br />
<br />
4<br />
⊕ ⊕<br />
2<br />
⊕ ⊕<br />
6<br />
⊕ W ⊕<br />
1<br />
W 8<br />
2<br />
W<br />
8<br />
1<br />
⊕<br />
2<br />
⊕<br />
3<br />
⊕<br />
4<br />
<br />
1<br />
<br />
W 4<br />
W 8<br />
1<br />
⊕ ⊕<br />
3<br />
W 8<br />
4<br />
⊕<br />
5<br />
<br />
3<br />
<br />
5<br />
⊕ ⊕<br />
5<br />
W 8<br />
5<br />
⊕<br />
6<br />
<br />
5<br />
<br />
W<br />
3<br />
⊕ ⊕<br />
6<br />
W 8<br />
6<br />
⊕<br />
7<br />
<br />
7<br />
<br />
7<br />
⊕ ⊕<br />
7<br />
W 8<br />
7<br />
⊕<br />
Figure 6.5 1 N=8 point DFT are changed <strong>to</strong> 2 N=4 point DFTs<br />
In the two boxes, 4 N=2 DFTs can be obtained as:<br />
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