the discrete fourier transform (dsp) 4

JRTGI 1

The Discrete Fourier Transform

Mr. HIMANSHU DIWAKAR

Mr. HIMANSHU DIWAKARAssistant Professor

JETGI

JRTGI 2Mr. HIMANSHU DIWAKAR

JRTGI 3

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/2j

ez eXzXkX~ kN/2

kX~

1N

0k

knN/2jekX~N1]n[x~


JRTGI 4

Sampling the Fourier Transform Cont’d• 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/2jekX~N1]n[x~ kN/2jeXkX~

mm

1N

0k

mnkN/2j

1N

0k

knN/2j

m

kmN/2j

mnp~mxeN1mx

eemxN1]n[x~

r

1N

0k

mnkN/2j rNmneN1mnp~

rr

rNnxrNnnx]n[x~


JRTGI 5

Sampling the Fourier Transform Cont’d


JRTGI 6

Sampling the Fourier Transform Cont’d• 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


JRTGI 7

The Discrete Fourier Transform• 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 of the DTFT of x[n]• Since x[n] is of length N there is no overlap between terms of x[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]

1Nn0 of outside 0nx

r

rNnxnx~

NkXN mod kXkX~

kX~

else01Nk0kX~kX

NnxN mod nxnx~


JRTGI 8

The Discrete Fourier Transform Cont’d• The DFS pair

• The equations involve only on period so we can write

• The Discrete Fourier Transform

• The DFT pair can also be written as

1N

0k

knN/2jekX~N1]n[x~

1N

0n

knN/2je]n[x~kX~

else01Nk0e]n[x~kX

1N

0n

knN/2j

else01Nk0ekX~N

1]n[x

1N

0k

knN/2j

1N

0n

knN/2je]n[xkX

1N

0k

knN/2jekXN1]n[x

]n[xkX DFT


JRTGI 9

Example• The DFT of a rectangular pulse• x[n] is of length 5• We can consider x[n] of any

length greater than 5• Let’s pick N=5• Calculate the DFS of the

periodic form of x[n]

else0,...10,5,0k5

e1e1

ekX~

5/k2j

k2j

4

0n

n5/k2j


JRTGI 10

Example Cont’d

• If we consider x[n] of length 10

• We get a different set of DFT coefficients

• Still samples of the DTFT but in different places


JRTGI 11

Properties of DFT• Linearity

• Duality

• Circular Shift of a Sequence

kbXkaXnbxnaxkXnxkXnx

21DFT

21

2DFT

2

1DFT

1

mN/k2jDFT

N

DFT

ekX1-Nn0 mnxkXnx

N

DFT

DFT

kNxnXkXnx


JRTGI 12

Example: Duality


JRTGI 13

Symmetry Properties


JRTGI 14

Circular Convolution

• Circular convolution of two finite length sequences

1N

0mN213 mnxmxnx

1N

0mN123 mnxmxnx


JRTGI 15

Example• Circular convolution of two rectangular pulses L=N=6

• DFT of each sequence

• Multiplication of DFTs

• And the inverse DFT

else01Ln01nxnx 21

else00kNekXkX

1N

0n

knN2j

21

else0

0kNkXkXkX2

213

else01Nn0Nnx3


JRTGI 16

Example

• We can augment zeros to each sequence L=2N=12• The DFT of each sequence

• Multiplication of DFTs

N

k2j

NLk2j

21e1e1kXkX

2

Nk2j

NLk2j

3e1e1kX


JRTGI 17

THANK YOU


the discrete fourier transform (dsp) 4

Engineering