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188 Chapter 5 THE DISCRETE FOURIER TRANSFORM
and the symmetry property
W kn+N/2
N
= −W kn
N
of the factor { }
WN
nk .
One algorithm that considers only the periodicity of WN
nk is the
Goertzel algorithm. This algorithm still requires C N = o(N 2 ) multiplications,
but it has certain advantages. This algorithm is described in
Chapter 12. We first begin with an example to illustrate the advantages of
the symmetry and periodicity properties in reducing the number of computations.
We then describe and analyze two specific FFT algorithms that
require C N = o(N log N) operations. They are the decimation-in-time
(DIT-FFT) and decimation-in-frequency (DIF-FFT) algorithms.
□ EXAMPLE 5.20 Let us discuss the computations of a 4-point DFT and develop an efficient
algorithm for its computation.
X(k) =
3∑
n=0
x(n)W nk
4 , 0 ≤ k ≤ 3; W 4 = e −j2π/4 = −j
Solution
These computations can be done in the matrix form
⎡ ⎤ ⎡
X(0) W4 0 W4 0 W4 0 W 0 ⎤ ⎡ ⎤
4 x(0)
X(1)
⎢ ⎥
⎣X(2)
⎦ = W4 ⎢
0 W4 1 W4 2 W4
3 x(1)
⎥ ⎢ ⎥
⎣W 4 0 W4 2 W4 4 W4
6 ⎦ ⎣x(2)
⎦
X(3) W4 0 W4 3 W4 6 W4
9 x(3)
which requires 16 complex multiplications.
Efficient Approach
Using periodicity,
W4 0 = W4 4 =1 ; W4 1 = W4 9 = −j
W4 2 = W4 6 = −1 ;W4 3 = j
and substituting in the above matrix form, we get
⎡ ⎤ ⎡
⎤ ⎡ ⎤
X(0) 1 1 1 1 x(0)
X(1)
⎢ ⎥
⎣X(2)
⎦ = 1 −j −1 j
x(1)
⎢
⎥ ⎢ ⎥
⎣1 −1 1 −1⎦
⎣x(2)
⎦
X(3) 1 j −1 −j x(3)
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