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190 Chapter 5 THE DISCRETE FOURIER TRANSFORM
Second, a smaller 2-point DFT of each column is taken.
[ ] [ ][ ]
x(0) x(1) 1 1 x(0) x(1)
W 2 =
x(2) x(3) 1 −1 x(2) x(3)
[ ]
x(0) + x(2) x(1) + x(3)
=
=
x(0) − x(2) x(1) − x(3)
[ ]
g1 g 2
h 1 h 2
Then each element of the resultant matrix is multiplied by {W pq
4 }, where p is
the row index and q is the column index; that is, the following dot-product is
performed:
[ ] [ ] [ ]
1 1 g1 g 2 g1 g 2
·∗ =
1 −j h 1 h 2 h 1 −jh 2
Finally, two more smaller 2-point DFTs are taken of row vectors.
[ ] [ ] [ ]
g1 g 2 g1 g
1 1 [ ]
2
g1 + g 2 g 1 − g 2
W 2 =
=
h 1 −jh 2 h 1 −jh 2
1 −1
h 1 − jh 2 h 1 + jh 2
[ ]
X(0) X(2)
=
X(1) X(3)
Although this interpretation seems to have more multiplications than the efficient
algorithm, it does suggest a systematic approach of computing a larger
DFT based on smaller DFTs.
□
5.6.1 DIVIDE-AND-COMBINE APPROACH
To reduce the DFT computation’s quadratic dependence on N, one must
choose a composite number N = LM since
L 2 + M 2 ≪ N 2
for large N
Now divide the sequence into M smaller sequences of length L, compute
M smaller L-point DFTs, and then combine these into a larger DFT
using L smaller M-point DFTs. This is the essence of the divide-andcombine
approach. Let N = LM, then the indices n and k in (5.46) can
be written as
n = l + Lm, 0 ≤ l ≤ L − 1, 0 ≤ m ≤ M − 1
k = q + Mp, 0 ≤ p ≤ L − 1, 0 ≤ q ≤ M − 1
(5.48)
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