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192 Chapter 5 THE DISCRETE FOURIER TRANSFORM
□ EXAMPLE 5.21 Develop the divide-and-combine FFT algorithm for N = 15.
Solution
Let L =3and M =5.Then, from (5.48), we have
n = l +3M, 0 ≤ l ≤ 2, 0 ≤ m ≤ 4
k = q +5p, 0 ≤ p ≤ 2, 0 ≤ q ≤ 4
(5.54)
Hence (5.49) becomes
X(p, q) =
{
2∑
l=0
W lq
15
[ 4∑
m=0
x(l, m)W mq
5
]}
W lp
3 (5.55)
To implement (5.55), we arrange the given sequence x(n) inthe form of
an array {x(l, m)} using a column-wise ordering as
x(0) x(3) x(6) x(9) x(12)
x(1) x(4) x(7) x(10) x(13)
x(2) x(5) x(8) x(11) x(14)
(5.56)
The first step is to compute 5-point DFTs F (l, q) for each of the three
rows and arrange them back in the same array formation
F (0, 0) F (0, 1) F (0, 2) F (0, 3) F (0, 4)
F (1, 0) F (1, 1) F (1, 2) F (1, 3) F (1, 4)
F (2, 0) F (2, 1) F (2, 2) F (2, 3) F (2, 4)
(5.57)
which requires a total of 3 × 5 2 =75complex operations. The second step
is to modify F (l, q) toobtain the array G(l, q) using the twiddle factors
W lq
15
G(0, 0) G(0, 1) G(0, 2) G(0, 3) G(0, 4)
G(1, 0) G(1, 1) G(1, 2) G(1, 3) G(1, 4)
G(2, 0) G(2, 1) G(2, 2) G(2, 3) G(2, 4)
(5.58)
which requires 15 complex operations. The last step is to perform 3-point
DFTs X(p, q) for each of the five columns to obtain
X(0, 0) X(0, 1) X(0, 2) X(0, 3) X(0, 4)
X(1, 0) X(1, 1) X(1, 2) X(1, 3) X(1, 4)
X(2, 0) X(2, 1) X(2, 2) X(2, 3) X(2, 4)
(5.59)
using a total of 5 × 3 2 =45complex operations. According to (5.54) the
array in (5.59) is a rearrangement of X(k) as
X(0) X(1) X(2) X(3) X(4)
X(5) X(6) X(7) X(8) X(9)
X(10) X(11) X(12) X(13) X(14)
(5.60)
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