Discret FT

7/28/2019 Discret FT

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The Discrete Fourier Transform

Content and Figures are from Discrete-Time Signal Processing, 2e by Oppenheim, Shafer, and Buck, 1999-2000 Prentice HallInc.


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Spring 2008 2

Sampling the Fourier Transform

Consider an aperiodic sequence with a Fourier transform

Assume that a sequence is obtained by sampling the DTFT

Since the DTFT is periodic resulting sequence is also periodic

We can also write it in terms of the z-transform

The sampling points are shown in figure

could be the DFS of a sequence

Write the corresponding sequence

kN/2jkN/2

j eXeXkX~

jDTFT eX]n[x

kN/2jez eXzXkX~

kN/2

kX~

1N

0k

knN/2jekX~

N

1]n[x

~


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Spring 2008 3

Sampling the Fourier Transform Contd

The only assumption made on the sequence is that DTFT exist

Combine equation to get

Term in the parenthesis is

So we get

m

mjj emxeX

1N

0k

knN/2j

ekX

~

N

1

]n[x

~

kN/2jeXkX

~

mm

1N

0k

mnkN/2j

1N

0k

knN/2j

m

kmN/2j

mnp

~

mxeN

1

mx

eemxN

1]n[x

~

r

1N

0k

mnkN/2j rNmneN

1mnp

~

rr

rNnxrNnnx]n[x~


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Spring 2008 4



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Spring 2008 5


Samples of the DTFT of an aperiodic sequence

can be thought of as DFS coefficients

of a periodic sequence

obtained through summing periodic replicas of original sequence

If the original sequence

is of finite length

and we take sufficient number of samples of its DTFT

the original sequence can be recovered by

It is not necessary to know the DTFT at all frequencies

To recover the discrete-time sequence in time domain Discrete Fourier Transform

Representing a finite length sequence by samples of DTFT

else0

1Nn0nx~nx


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Spring 2008 6


Consider a finite length sequence x[n] of length N

For given length-N sequence associate a periodic sequence

The DFS coefficients of the periodic sequence are samples ofthe DTFT of x[n]

Since x[n] is of length N there is no overlap between terms ofx[n-rN] and we can write the periodic sequence as

To maintain duality between time and frequency

We choose one period of as the Fourier transform of x[n]

1Nn0ofoutside0nx

r

rNnxnx~

NkXNmodkXkX~

kX~

else0

1Nk0kX~

kX

NnxNmodnxnx~


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Spring 2008 7

The Discrete Fourier Transform Contd

The DFS pair

The equations involve only on period so we can write


The DFT pair can also be written as

1N

0k

knN/2jekX~

N

1]n[x

~

1N

0n

knN/2je]n[x~kX~

else0

1Nk0e]n[x~

kX

1N

0n

knN/2j

else0

1Nk0ekX~N1

]n[x

1N

0k

knN/2j

1N

0n

knN/2je]n[xkX

1N

0k

knN/2jekX

N

1]n[x

]n[xkX DFT


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Spring 2008 8

Example

The DFT of a rectangular pulse

x[n] is of length 5

We can consider x[n] of anylength greater than 5

Lets pick N=5

Calculate the DFS of theperiodic form of x[n]

else0

,...10,5,0k5

e1

e1

ekX~

5/k2j

k2j

4

0n

n5/k2j


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Spring 2008 9

Example Contd

If we consider x[n] of length 10

We get a different set of DFTcoefficients

Still samples of the DTFT but indifferent places


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Spring 2008 10

Properties of DFT

Linearity

Duality

Circular Shift of a Sequence

kbXkaXnbxnax

kXnx

kXnx

21DFT

21

2DFT

2

1DFT

1

mN/k2jDFTN

DFT

ekX1-Nn0mnx

kXnx

N

DFT

DFT

kNxnX

kXnx


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Spring 2008 11

Example: Duality


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Spring 2008 12

Symmetry Properties


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Spring 2008 13

Circular Convolution

Circular convolution of of two finitelength sequences

1N

0mN213 mnxmxnx

1N

0m

N123 mnxmxnx


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Spring 2008 14

Example

Circular convolution of two rectangular pulses L=N=6

DFT of each sequence

Multiplication of DFTs

And the inverse DFT

else0

1Ln01nxnx 21

else0

0kNekXkX

1N

0n

knN

2j

21

else0

0kNkXkXkX

2

213

else0

1Nn0Nnx3


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Spring 2008 15

Example

We can augment zeros toeach sequence L=2N=12

The DFT of each sequence

Multiplication of DFTs

N

k2j

N

Lk2j

21

e1

e1kXkX

2

N

k2j

N

Lk2j

3

e1

e1kX

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Discret FT