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O n e c a n f a c t o r a c o m m o n m u l t i p l i e r o u t o f t h e s e c o n d s u m i n t h e equation.
It is the two sums are the DFT of the even -indexed part
x
2
m
and the DFT of odd-indexed part
x
2
m
+ 1of the function
xn
. Denote the DFT of the
E
ven-indexed inputs
x
2
m
by
E k
and the DFT of the
O
dd-indexed inputs
x
2
m
+ 1by
Ok
and we obtain:However, these smaller DFTs have a length of
N
/2, so we need compute only
N
/2outputs: thanks to the periodicity properties of the DFT, the outputs for N/2 < k < N froma DFT of length
N
/2 are identical to the outputs for 0< k < N/2. That is,
E k
+
N
/ 2=
E k
and
Ok
+
N
/ 2=
Ok
. The phase factor exp[ − 2π
ik
/
N
] called a twiddle factor whichobeys the relation: exp[ − 2π
i
(
k
+
N
/ 2) /
N
] =
e
− π
i
exp[ − 2π
ik
/
N
] = − exp[ − 2π
ik
/
N
],flipping the sign of the
Ok
+
N
/ 2terms. Thus, the whole DFT can be calculated as follows:This result, expressing the DFT of length
N
recursively in terms of two DFTs of size
N
/2, is the core of the radix-2 DIT fast Fourier transform. The algorithm gains itsspeed by re -
using the results of intermediate computations to compute multiple DFTo utputs. Note that
final outputs are obtained by a +/− combination of
E k
and
Ok
exp( −2π
ik
/
N
), which is simply a size-2 DFT; when this is generalized to larger radices below,the size-2 DFT is replaced by a
larger DFT (which itself can be evaluated with an FFT).
4
T h i s p r o c e s s i s a n e x a m p l e o f t h e g e n e r a l t e c h n i q u e o f d i v i d e a n d
c o n q u e r s a l g o r i t h m s . I n m a n y t r a d i t i o n a l i m p l e m e n t a t i o n s , h o w e v e r , t h e e x p l i c i t
r e c u r s i o n i s avoided, and instead one traverses the computational tree in breadth-first fashion.
Fig 1.1 Decimation In Time FFT
In the DIT algorithm, the twiddle multiplication is performed before the butterflystage whereas for the DIF
algorithm, the twiddle multiplication comes after the Butterflystage.
Fig 1.2 : Decimation In Frequency FFT
The 'Radix 2' algorithms are useful if
N
i s a r e g u l a r p o w e r o f 2 (
N
=2
p
) . I f w e assume that algorithmic complexity provides a direct measure of execution time and thatthe
relevant logarithm base is 2 then as shown in table 1.1, ratio of execution times for the (DFT)
vs. (Radix 2 FFT) increases tremendously with increase in N.
5
T h e t e r m ' F F T ' i s a c t u a l l y s l i g h t l y a m b i g u o u s , b e c a u s e t h e r e a r e
s e v e r a l commonly used 'FFT' algorithms. There are two different Radix 2 algorithms, the so -
called 'Decimation in Time' (DIT) and 'Decimation in Frequency' (DIF) algorithms. Bothof these rely on the
recursive decomposition of an
N
point transform into 2 (
N
/2) pointtransforms. Number of Points, NComplex Multiplicationsin Direct computations, N
2
Complex Multiplicationin FFT Algorithm, (N/2)log
2
NSpeedimprovementFactor 4
1
6
4
4
.
0
8
6
4
1
2
5
.
3 1
6 2
5 6
3 2
8 .
0 3
2 1 0
2 4 8
0 1 2
. 8 6
4 4 0
9 6 1
9 2 2
1 . 3
1 2 8
1 6 3
8 4 4
4 8 3
6 . 6
Table 1.1: Comparison of Execution Times, DFT & Radix – 2 FFT
1.2 BUTTERFLY STRUCTURES FOR FFT
Basically FFT algorithms are developed by means of divide and conquer method,the is depending on the
decomposition of an N point DFT in to smaller DFT’s. If N is factored as N = r
1
,r
2
,r
3
..r
L
where r
1
=r
2
=…=r
L
=r, then r
L
=N. where r is called as Radix of FFFt algorithm.If r= 2, then if is called as radix-2 FFT algorithm,. The basic
DFT is of
a