UploadFile_6417

The Fast Fourier Transform 189
Using symmetry, we obtain
X(0) = x(0) + x(1) + x(2) + x(3) = [x(0) + x(2)
} {{ } ]
X(1) = x(0) − jx(1) − x(2) + jx(3) = [x(0) − x(2)
} {{ } ]
X(2) = x(0) − x(1) + x(2) − x(3) = [x(0) + x(2)
} {{ } ]
X(3) = x(0) + jx(1) − x(2) − jx(3) =[x(0) − x(2)
} {{ } ]
g 1
h 1
g 1
h 1
+[x(1) + x(3) ]
} {{ }
g 2
−j[x(1) − x(3) ]
} {{ }
h 2
− [x(1) + x(3) ]
} {{ }
g 2
+ j[x(1) − x(3) ]
} {{ }
h 2
Hence an efficient algorithm is
Step 1
g 1 = x(0) + x(2)
g 2 = x(1) + x(3)
h 1 = x(0) − x(2)
h 2 = x(1) − x(3)
∥
Step 2
X(0) = g 1 + g 2
X(1) = h 1 − jh 2
(5.47)
X(2) = g 1 − g 2
X(3) = h 1 + jh 2
which requires only 2 complex multiplications, which is a considerably smaller
number, even for this simple example. A signal flowgraph structure for this
algorithm is given in Figure 5.19.
An Interpretation This efficient algorithm (5.47) can be interpreted differently.
First, a 4-point sequence x(n) isdivided into two 2-point sequences, which
are arranged into column vectors as shown here.
[[ ] x(0)
x(2)
,
[ ]] x(1)
x(3)
=
[ ] x(0) x(1)
x(2) x(3)
x(0)
g 1
X (0)
x(2)
−1
−j
X (1)
h 1
X (2)
x(1)
g 2
−1
x(3)
−1
h 2
j
X (3)
FIGURE 5.19 Signal flowgraph in Example 5.20
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.