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The Fast Fourier Transform 191
and write sequences x(n) and X(k) asarrays x(l, m) and X(p, q), respectively.
Then (5.46) can be written as
L−1
∑
X(p, q) =
M−1
∑
l=0 m=0
{
L−1
∑
=
l=0
L−1
∑
⎪⎨
=
l=0
⎧
⎪⎩
W lq
N
W lq
N
x(l, m)W (l+Lm)(q+Mp)
N
[ M−1
]}
∑
x(l, m)W Lmq
N
W Mlp
N
m=0
[ M−1
]
∑
x(l, m)W mq
M
m=0
} {{ }
M-point DFT
⎫
⎪⎬
⎪⎭
W lp
L
} {{ }
L-point DFT
(5.49)
Hence (5.49) can be implemented as a three-step procedure:
1. First, we compute the M-point DFT array
M−1
∑
F (l, q)△ x(l, m)W mq
M ; 0 ≤ q ≤ M − 1 (5.50)
m=0
for each of the rows l =0,...,L− 1.
2. Second, we modify F (l, q) toobtain another array.
G(l, q) =W lq
N F (l, q), 0 ≤ l ≤ L − 1
0 ≤ q ≤ M − 1
(5.51)
The factor W lq
N
is called a twiddle factor.
3. Finally, we compute the L-point DFTs
L−1
∑
X(p, q) = G(l, q)W lp
L
0 ≤ p ≤ L − 1 (5.52)
l=0
for each of the columns q =0,...,M − 1.
The total number of complex multiplications for this approach can now
be given by
C N = LM 2 + N + ML 2