prepared by: deepak kumar rout1 chapter 5 prepared by: –deepak kumar rout. –synergy

Prepared by: Deepak Kumar Rout

1

Chapter 5

• Prepared by:– Deepak Kumar Rout.– Synergy


2

Chapter 8. The Discrete Fourier Transform

8.1 Laplace, z-, and Fourier Transforms

8.2 Fourier Transform

8.3 Fourier Series

8.4 Discrete Fourier Transform (DFT)

8.5 Properties of DFS/DFT

8.6 DFT and z-Transform

8.7 Linear Convolution vs. Circular Convolution

8.8 Discrete Cosine Transform(DCT)

BGL/SNU


3

-Analog systems (continuous time)

H(s)

stesH )( ste

-Digital Systems (discrete time)

H(z)

nzzH )( nz

dehsH

esHdeeh

dtxhx(t)h(t) y(t)

sr

ststs

)()(

)( )(

)()(

k

k

nnk

k

k

zkhzH

zzH zzkh

knxkhx(n)h(n) y(n)

)()(

)( )(

)()(

BGL/SNU

1. Laplace, z-, Fourier Transforms


4

planes planez j

je

1

1

1

12

z

z

Ts

d

-Laplace transform -z-transform

),( ),(

LHP inside u.c

jssHjH )()(

jez

j zHeH

)()(

Fouier transformsBGL/SNU


5

dejXtx

dtetxjX

tj

tj

)(2

1 )(

)()(

(1) continuous aperiodic signals

conti aper

aper conti

elsewherettxeq

,0 2/|| ,1 )( )(

)2/sin(2

)(1

)(

2/2/

2/

2/

jj

tj

eej

dtejX

2/

x(t)

2/

1

t

)( jX

2

BGL/SNU

2. Fourier Transform


6

deeXnx

enxeX

njj

n

njj

)(2

1 ][

][)(

(2) Discrete aperiodic signals

conti per

aper discr

elsewhere

ntxeq

,0

|| ,1 )( )(

2sin

)21

(sin

1

1)(

22

)2

1()

2

1(

)12(

jj

jj

j

jj

n

njj

ee

ee

e

eeeeX

x(n)

1

t

12

2

)( jeX12

2ω


7

k

tjkk

T

tjkk

Teatx

dtttxT

a

2 , )(

)(1

00

0

(1) continuous periodic signals

discrete aper

per conti

T)x(tx(t)Tt

ttxeq

,2/|| 2/ ,0

2/|| ,1 )( )(

0

000

1(note)

)2/sin(

)(111

0

2/2/

0

2/

2/

kΩΩk

jkjktjkk

)X(jT

ak

k

eejkT

dteT

a

BGL/SNU

3. Fourier Series


8

2/

X(t)

2/

1

T

t

T

T

2

2

k

(2) discrete periodic signals (*Discrete Fourier Series)

N

2 , ][

1 ][

][][

00

0

Nk

njk

Nn

njk

ekXN

nx

enxkX discrete per

per discre

ka

BGL/SNU


9

0

0

0

00

)X(eX[k] note)

2sin

)21

(sin

1

1][

j

0

0)12(

k

jk

jkjk

n

njk

k

k

e

eeekX

x[n]

1

n

N

][kX

12

kN

][][ || ,0

|| ,1 ][ )(

NnxnxNn

nnxeq

BGL/SNU


10

-For a numerical evaluation of Fourier transform and its inversion, (i.e,computer-aided computation), we need discrete expression of of both the time and the transform domain data.

-For this,take the advantage of discrete Fourier series(DFS, on page 4), in which the data for both domain are discrete and periodic.

periodic) : (~ ][~

][~ kXnx

discrete periodic periodic discrete

-Therefore, given a time sequence x[n], which is aperiodic and discrete, take the following approach.

BGL/SNU

4. Discrete Fourier Transform (DFT)


11

X[k] ][~

[n]x~ ][ kXnx

Mip

MipTop

Top][][

~nRkX N

][][~ nRnx N DFS

DFT

][ , :

]))[(( , :

nRperiodonetakeTop

nxperiodicitmakeMip

N

N

Reminding that, in DFS

][~1

][~

][~][~

1

0

2

1

0

2

N

k

nN

jk

N

n

nN

jk

ekXN

nx

enxkX

BGL/SNU


12

Define DFT as

otherwise 0,

1-N 0,1,...,n , ][1

][

otherwise ,0

1N0,1,...,k ,][ ][

1

0

2

1

0

2

N

k

nN

jk

N

n

nN

jk

ekXN

nx

enxkX

][][~][

][][~

][ (note)

n Rnxn x

kRkXkX

N

N

(eq)X[k] x[n]

N Nnk

1

BGL/SNU


13

Graphical Development of DFT

)(th

t0T

)( jH

)(1 ts )(1 jS

t

t0T

T T

1

T

1

T

1

)()()( 1 jSjHjH S)()()( 1 tsthths

T

1

T

1


14

][~

nh

t0T

][~

kH

)(2 ts )(2 jS

t

n

0T

N

T

1T

1

)()()( 1 jSjHjH S)()()( 2 tsthth sd

0T

0

1

T

0k

N0

DFSDFS

BGL/SNU


15

][nh ][kH

nN0

kN0

DFTDFT

BGL/SNU


16

5. Property of DFS/DFT (8.2 , 8.6)(1) Linearity

)seq. (finite][][)seq. periodic(][

~][

~

][][:

][~][~:

kbYkaXkYbkXa

nbynaxDFT

nybnxaDFS

(2) Time shift

][]))[((:

][~

][~:

kXWmnxDFT

kXWmnxDFSkm

NN

kmN

]))[((~

][:

][~

][~:

Nnl

N

nlN

lkXnxWDFT

lkXnxWDFS

(3) Frequency shift

BGL/SNU


17

(4) Periodic/circular convolution in time

(5) Periodic/circular convolution in frequency

][][]))[((][:

][~

][~

][~][~:

1

0

1

0

kYkXmnymxDFT

kYkXmnymxDFS

N

mN

N

m

1

0

1

0

]))[((][1

][][:

][~

][~1

][~][~:

N

lN

N

l

lkYlXN

nynxDFT

lkYlXN

nynxDFS

BGL/SNU


18

]}[Im{][]}[~

Im{][~

]}[Re{][]}[~

Re{][~

][~

]}[Im{][~

]}[~Im{

][~

]}[Re{][~

]}[~Re{

][~

]))[((~][~

][~]))[((

~][][

~][~

****

****

kXjnxkXjnx

kXnxkXnx

kXnxjkXnxj

kXnxkXnx

kXnxkXnx

kXnxkXnx

opo

epe

opo

epe

N

N

(6) Symmetry

BGL/SNU

DFS DFT


19

6. DFT and Z-Transform(1) Evaluation of from][

~kX )(zX

1

0

2

][~][~

][)(

N

n

nN

jk

n

n

enxkX

znxzX

①If length limited in time, (I.e., x[n]=0, n<0, n>=N)

then Njk

ezzXkX 2)(][

~

)2

at samplingby )(

from obtained is ][~

ly,(equivaant

)1,,2,1,2

,at

samplingby X(z) from obtained is ][~

(

NkeX

kX

NkN

kez

kX

kjω

kj k

BGL/SNU


20

② What if x[n] is not length-limited?

then aliasing unavoidable.

r

N

k

mnN

jk

m

N

k

nN

jk

m

mN

jk

N

k

nN

jk

m

mN

jk

ez

rNnx

eN

mx

eemxN

ekXN

nx

emxzXkX Njk

][

1][

][1

][~1

][~then

][)(][ take weIf

1

0

)(2

1

0

22

1

0

2

22

][nx ][~ nx

n0 1N 12 N 13 N

… … …

n0 1N 12 N 13 N

… … …


21

(2) Recovery of [or ] from

(in the length-limited case)

)( jeX ][~

kX

1

0 12

1

0

1

0

12

1

0

1

0

2

1

0

1

1][

~1

][~1

][~1

][)(X

N

k Njk

N

N

k

N

n

n

Njk

N

n

nN

k

nN

jk

N

n

n

ze

zkX

N

zekXN

zekXN

znxz

BGL/SNU

)(zX


22

2sin

2sin

1

11)(

)2

(][~

1

1][

~1

1

)(1][

~1)(X

2

1

1

0

1

0 )2

(

)2

(

1

0 12

12

N

N

ee

e

Nwhere

NkkX

e

ekX

N

ze

zekX

Ne

Nj

j

Nj

N

k

N

k Nkj

NN

kj

N

k

ezN

jk

NNjk

j

j

2

)0(~X

)(

)2

(N

)1(

~X

BGL/SNU


23

7. Linear Convolution vs. Circular Convolution

(1) Definition

NlengthNn

knhkxnhnx

NnNlengthnh

NnNlengthnx

N

k

2,12,,1,0

,][][][][

1,,1,0,],[

1,,1,0,],[

1

0

① Linear convolution

BGL/SNU


24

② Circular convolution

NlengthNn

nRknhkxnhnx N

N

kNN

,1,,1,0

][]))[((]))[((][][1

0

N

Rectangular window of length

N

]))[((][

]))[((][

NMip

NMip

nhnh

nxnx

Periodic

convolution

1

0

]))[((]))[((N

kNN knhkx

][][ nhnx N

TopNperiodWith

nhnh N ][~

]))[((

BGL/SNU


25

(2) Comparison

][][ 21 nxnx

][][ 21 nxnx

0

1

N

0

N

N2

][][ nhnx N

N

0

N

N2][][ nhnx 2N

H[n]][nx ][][][ nhnxny

)2(

][

N

kX

)2(

][

N

kH][][][ kHkXkY 2N

)2( N

) period zero N (

2Nlength a of ],[ ,][for 21 nxnx

Omit chap. 8.7


26

- Effects of Energy compaction

8. Discrete cosine transform (DCT)Definition

otherwise

kke

NnN

knkXkenx

NkN

knnxke

NkX

N

k

N

n

,1

0,2

1

][

1,,1,0,2

)12(cos][][][

1,,1,0,2

)12(cos][][

2][

1

0

1

0

BGL/SNUTest signal for computing DFT and DCT


27(a) Real part of N-point DFT; (b) Imaginary part of N-point DFT; (c) N-point DCT-2 of the test signal

BGL/SNU


28

Comparison of truncation errors for DFT and DCT-2

BGL/SNU


29

Appendix: Illustration of DFTs for Derived Signals

]1[][1 nNxng

][)1(][2 nxng n

otherwise

NnNNnx

Nnnx

ng

,0

12],[

10],[

][3

n1N

n

n1N

1N 12 N

A

A

A

A

][1 ng

][2 ng

][3 ng

BGL/SNU


30

otherwise

NnNnxnxng

,0

12/0],2/[][][4

otherwise

NnN

Nnnx

ng

,0

12,0

10],[

][5

odd ,0

even ],2

[][6

n

nn

xng

n1

2

N

A2

][4 ng

][5 ng

A

1Nn

n

][6 ng

A

12 N

BGL/SNU


31

]2[][7 nxng

][7 ng

A

n1

2

N

)(][

)(][

)(][

)(][

)]()([5.0][

)(][

oddk ,0

evenk ),(][

)(][

)(][

/29

)2/)(/2(8

/2/27

/46

/)2/(2/25

)12/(24

/2

3

2/22

/21

Nkj

NkNj

NkjNkj

Nkj

NNkjNkj

Nkj

Nkj

Nkj

Nkj

eXkH

eXkH

eXekH

eXkH

eXeXkH

eXkH

eXkH

eXkH

eXkH

BGL/SNU


32

][)( ][

][ ]1[][

7

2221

0

2

1

0

)1(1

01

kHeXeeemx

WmxWnNxkG

kN

jN

kjmk

NjN

m

N

kj

N

m

mNkN

N

n

knN

][)( ][

][ ][)1(][

8

)2

(21

0

)2

(2

1

0

2

1

02

kHeXenx

WWnxWnxkG

Nk

NjN

n

nN

kN

j

N

n

knN

Nn

N

N

n

knN

n

][)(])1(1[ ]1[][

]][[ ][ ][][

32

21

022

)(2

1

02

12

2

1

023

kHeXWWnx

WWnxWNnxWnxkG

N

kj

kN

n

NkN

nkN

knNN

N

n

nkN

N

Nn

knN

N

n

knN

BGL/SNU


33

][)( ][

][][])2

[][(][

6

41

0 2

1

2

)2

(

2

12

0 2

12

0 2

4

kHeXWnx

WnxWnxWN

nxnxkG

NjN

n

nkN

N

Nn

kN

nN

N

n

knN

N

n

knN

][)(X ][][ 2

12

025 kHeWnxkG N

kjN

n

knN

][)(X ][][ 1

21

0

226 kHeWnxkG N

kjN

n

knN

][))()([2

1 ]][[

2

1

]2

)1(1][[ ]2[][

5

)2

(221

0

)2

(

1

0

2

2

12

0 2

7

kHeXeXWWnx

WnxWnxkG

Nk

N

kj

N

kjN

n

Nkn

Nnk

N

N

n

nk

N

nN

n

knN

BGL/SNU

prepared by: deepak kumar rout1 chapter 5 prepared by: –deepak kumar rout. –synergy

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