Introduction to Digital Signal and System Analysis - Tutorsindia

Introduction to Digital Signal and System Analysis
Discrete Fourier Transform
The number of direct calculation of its DFT will be 8 2 = 64, approximately. However, it can be divided into 2 length
N=4 sequences:
3
3
rk k
rk
X[ k]
= x[2r]
W4
+ W8
x[2r
+ 1] W4
r=
0
r=
0
Further, they can be divided in to 4 length N=2 sequences:
X[
k]
=
1
s=
0
x[2(2s)]
W
+ W
k
8
1
s=
0
sk
2
+ W
1
k
4
r=
0
x[2(2s)
+ 1] W
x[2
sk
2
( 2s
+ )
+ W
1 ] W
1
k
4
s=
0
sk
2
x[2(2s
+ 1) + 1] W
sk
2
Explicitly, from the above,
X[
k]
= x[0]
W
+ W
k
8
0k
1k
k
0k
1k
2
+ x[4]
W2
+ W4
( x[2]
W2
+ x[6]
W2
)
0k
1k
k
0k
1
( x[1]
W + x[5]
W + W ( x[3]
W + x[7]
W
k
)
2
2
4
2
2
0
1
where we know W = 1, W = 1, therefore,
2 2
−
1k
k
1k
k
1k
( x[2]
+ x[6]
W ) + W ( x[1]
+ x[5]
W + W ( x[3]
x[7
W
)
X[ k]
= x[0]
+ x[4]
W
+
1k
k
2
+ W4
2 8
2 4
]
2
where only
k
k
W 4
and W 8
are actually complex numbers, there are as many as only 3 + 7 = 12.
In original DFT, there are approximately
2
N multiplications in
W kn
N
2πnk
= exp
− j
N (there are some unities when k
or n=0). However, in the FFT algorithm, redundant computation in multiplying W are reduced by re-arranging samples
kn kn kn 0 1
to shorter sequences to enable multiplication by much fewer distinct W , WN
/ 2 , WN
/ 4 ,... W2
and W in a butterfly shaped
N 2
flow chart. Figure 6.4 illustrates
2
N multiplications in a length N=8 DFT.
kn
N
95
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