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CoreFFT Fast Fourier Transform
Appendix I: Fast Fourier Transform
The FFT is a computationally efficient algorithm for computing a discrete Fourier transform (DFT). The N-point DFT is
defined as
N – 1
X ( k ) ≡
∑ x [ n ] e
( – jnk2 π ) ? N
k = 0, 1, 2, ..., N – 1
n = 0
EQ 7
where N is the transform size, or number of points. The inverse N-point DFT is defined as
N – 1
x ( n ) = ? ---- ?
1
? N ?
∑ X [ k ] e
( jnk2 π ) ? N
k = 1, 2, 3, ..., N – 1
k = 0
EQ 8
It is common practice to call the exponential vector rotating factors above the "twiddle factors." Every twiddle factor
contains real and imaginary parts
P = P r + jP i
Q = Q r + jQ i
W = W r – jW i = e
( – jnk2 π ) ? N
outP = outP r + j × outP i
outQ = outQ r + j × outQ i
EQ 9
-1
W = W r + jW i
Figure 8 ? Radix-2 DIT Butterfly
The butterfly performs the basic FFT computation
according to the following equations:
outP = P + Q × W
By substituting the values from EQ 12 and expressing P
and Q in terms of their real and imaginary parts, EQ 10
and EQ 11 become
outP = P – Q × W
The twiddle factor can be expressed as
W = cos X – j sin X
12
EQ 10
EQ 11
EQ 12
v4.0
outP = P r + jP i + (Q r cos X + Q i sin X) + j(Q i cos X – Q r sin X)
= (P r + Q r cos X + Q i sin X) + j(P i + Q i cos X – Q r sin X)
EQ 13
outQ = P r + jP i – (Q r cos X + Q i sin X) – j(Q i cos X – Q r sin X)
= (P r – Q r cos X – Q i sin X) + j(P i – Q i cos X + Q r sin X)
EQ 14
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