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FFTDecimation in Time

Spring 2009

© Ammar Abu-Hudrouss -Islamic University Gaza

Slide ٢Digital Signal Processing

Decimation in Time

12/

0

12/

0

)12(2 )12()2()(N

m

N

m

mkN

mkN WmxWmxkX

12/

0

12/

02/2/ )12()2()(

N

m

N

m

kmN

kN

mkN WmxWWmxkX

In this method we split x (n) into the even indexed x (2m) and the odd indexed x (2m + 1) each N/2 long. DFT can be expressed as:

mkN

NmkjNmkjmkN WeeW 2/

)2//(2/222

(1)

K = 0,1,2,3,4,……………..,N-1

We have used the fact

Slide ٣Digital Signal Processing

Decimation in Time

12/,....,2,1,0)2()(12/

02/

NkWmxkGN

m

mkN

Define new functions as

12/,...,3,2,1,0)12()(12/

02/

NkWmxkHN

m

kmN

G (K) is a DFT for the even samples of x (n), or

H (K) is a DFT for the odd samples of x(n), or

Then equation (1) for the first N/2 points can be expressed as

12/,....,2,1,0)()()( NkkHWkGkX kN

Slide ٤Digital Signal Processing

Decimation in Time

kN

kNN WW )2/(

NNNkkHWkGNkX kN ,....,12/,2/)()()2/(

But

Then

12/,...,2,1,0),2/()( NkkNGkG

12/,....,2,1,0)2/()( NkNkHkH

This can be expressed in the following diagram

Slide ٥Digital Signal Processing

Decimation in Time

28W

18W

38W

4-pointDFT

X(0)

X(1)

X(2)

X(3)

G(0)

G(1)

G(2)

G(3)

x(0)

x(2)

x(4)

x(6)

4-pointDFT

X(4)

X(5)

X(6)

X(7)

H(0)

H(1)

H(2)

H(3)

x(1)

x(3)

x(5)

x(7)

08W

-1

-1

-1

-1

Slide ٦Digital Signal Processing

Decimation in Time

28W

18W

38W

08W

08W

28W

-1

-1

08W

28W

-1

-1

2-point DFT

2-point DFT

2-point DFT

2-point DFT

x(0)

x(4)

x(2)

x(6)

x(1)

x(5)

x(3)

x(7)

X(0)

X(1)

X(2)

X(3)

X(4)

X(5)

X(6)

X(7)-1

-1

-1

-1

Slide ٧Digital Signal Processing

Decimation in Time

08W

28W

08W

28W

-1

-1

-1

-1

08W

08W

08W

08W

28W

18W

38W

08W

-1

-1

-1

-1

X(0)

X(1)

X(2)

X(3)

X(4)

X(5)

X(6)

X(7)-1

-1

-1

-1

x(0)

x(4)

x(2)

x(6)

x(1)

x(5)

x(3)

x(7)

Slide ٨Digital Signal Processing

Decimation in Time

Example Given a sequence x(n) where x(0) = 1, x(1) = 2, x(2) = 3, x(3) = 4 and x(n) = 0 elsewhere ,find DFT for the first four points

SSolutionolution

104 W

jW 14

-1

-1

-1

-1

x(0)=1

x(2)=3

x(1)=2

x(3)=410

4 W

104 W

4

-2

6

-2

X(0 )= 10

X(1)= -2+2j

X(2)= -2

X(3)= -2-2j

Slide ٩Digital Signal Processing

Inverse Fourier Transform

The inverse discrete Fourier can be calculated using the same method but after changing the variable WN and multiplying the result by 1/NExample Given a sequence X(n) given in the previous example. Find the IFFT using decimation in time method

Solution

x(0) = 1

x(1) = 3

x(2) = 2

x(3) = 4

X(0) =10

X(2) =-2

X(1) = -2+2j

X(3) = -2-2j

1/4

1/4

1/4

1/4

1~04 W

jW ~14

-1

-1

-1

-11

~04 W

1~04 W

8

12

-4

j4

4

8

12

16