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