Copyright © 2005. Shi Ping CUC Chapter 3 Discrete Fourier Transform Review Features in common We...

Copyright © 2005. Shi Ping CUC

Chapter 3Discrete Fourier Transform

Review

Features in common

We need a numerically computable transform, that is

Discrete Fourier Transform (DFT)

The DTFT provides the frequency-domain ( ) representation for absolutely summable sequences.

The z-transform provides a generalized frequency-domain ( ) representation for arbitrary sequences.

z

Defined for infinite-length sequences. Functions of continuous variable ( or ). They are not numerically computable transform.

z


Chapter 3Discrete Fourier Transform

Content

The Family of Fourier Transform

The Discrete Fourier Series (DFS)

The Discrete Fourier Transform (DFT)

The Properties of DFT

The Sampling Theorem in Frequency Domain

Approximating to FT (FS) with DFT (DFS)

Summary



Introduction

Fourier analysis is named after Jean Baptiste Joseph Fourier (1768-1830), a French mathematician and physicist.

A signal can be either continuous or discrete, and it can be either periodic or aperiodic. The combination of these two features generates the four categories of Fourier Transform.



Aperiodic-Continuous － Fourier Transform

dejXtx

dtetxjX

tj

tj

)(2

1)(

)()(



Periodic-Continuous － Fourier Series

k

tjk

T

T

tjk

ejkXtx

dtetxT

jkX

0

0

0

0

)()(

)(1

)(

0

2

20

0

00

22

TF



Aperiodic-Discrete － DTFT

deeXnx

enxeX

njj

n

njj

)(2

1)(

)()(



Periodic-Discrete － DFS (DFT)

1

0

2

1

0

2

)(1

)(

)()(

N

k

nkNj

N

n

nkNj

ekXN

nx

enxkX



Summary

Time function Frequency function

Continuous and Aperiodic Aperiodic and Continuous

Continuous and Periodic( ) Aperiodic and Discrete( )

Discrete ( ) and Aperiodic Periodic( ) and Continuous

Discrete ( ) and Periodic ( )

Periodic( )

and Discrete( )

0T

T

T 0T

00

2

T

Ts

2

Ts

2

00

2

T

return



DefinitionPeriodic time functions can be synthesized as a linear combination of complex exponentials whose frequencies are multiples (or harmonics) of the fundamental frequency

Periodic continuous-time function )()( rTtxtx

Periodic discrete-time function )()( rNnxnx

k

ktTjekXtx

2

)()(

1

0

2

)(1

)(N

k

knNjekX

Nnx

fundamental frequency

tTje

2

nNje

2fundamental frequency



elsewhere ,0

,1

1

1112

21

0

2 mNr

e

e

Ne

N rNj

rNNj

N

n

rnNj

)(1

)(

)(1

)(

1

0

)(21

0

1

0

21

0

21

0

2

rXeN

kX

eekXN

enx

N

n

nrkNjN

k

N

n

rnNjN

k

knNjN

n

rnNj



1

0

2

)()(N

n

knNj

enxkX

)()(

)()(

1

0

2

1

0

)(2

kXenx

enxmNkX

N

n

knNj

N

n

nmNkNj

Because:

The is a periodic sequence with fundamental period equal to N

)(kX



1

0

1

0

)(~1

)](~

[IDFS)(~

)(~)](~[DFS)(~

Let2

N

k

nkN

N

n

nkN

jN

WkXN

kXnx

WnxnxkX

eW N



Relation to the z-transform

elsewhere,0

10),(~)(

Nnnxnx

kNjez

zXkX 2|)()(~

The DFS represents N evenly spaced samples of the z-transform around the unit circle.

)(~kX

)(zX

1

0

1

0

))(()(~

, )()(2

N

n

nkjN

n

n NenxkXznxzX



Relation to the DTFT

kj

N

n

nkNjN

n

njj

NeXkX

enxkXenxeX

2|)()(~

)()(~

)()(1

0

21

0

，

elsewhere,0

10),(~)(

Nnnxnx

The DFS is obtained by evenly sampling the DTFT at

intervals. It is called frequency resolution and represents the

sampling interval in the frequency domain.

N2



jIm[z]

Re[z]

0k

kj

NeXkX

2|)()(~

N

2

N=8 frequency resolution


The properties of DFS


Linearity

)(~

)(~

)](~)(~[DFS 2121 kXbkXanxbnxa

Shift of a sequence

)(~

)(~

)](~[DFS2

kXekXWmnxmk

Nj

mkN

＋

Modulation

)(~

)](~[DFS lkXnxW lnN



Periodic convolution

1

012

1

021

21

)(~)(~

)(~)(~)](~

[IDFS)(~ then

)(~

)(~

)(~

if

N

m

N

m

mnxmx

mnxmxkYny

kXkXkY


1

012

1

021

1

0

1

0

)(21

1

02

1

01

1

02121

)(~)(~)(~)(~

)(~1

)(~

)(~

)(~1

)(~

)(~1

)](~

)(~

[IDFS)(~

N

m

N

m

N

m

N

k

kmnN

N

k

nkN

N

m

mkN

N

k

nkN

mnxmxmnxmx

WkXN

mx

WkXWmxN

WkXkXN

kXkXny

return


Introduction


The DFS provided us a mechanism for numerically computing the discrete-time Fourier transform. But most of the signals in practice are not periodic. They are likely to be of finite length.

Theoretically, we can take care of this problem by defining a periodic signal whose primary shape is that of the finite length signal and then using the DFS on this periodic signal.

Practically, we define a new transform called the Discrete Fourier Transform, which is the primary period of the DFS.

This DFT is the ultimate numerically computable Fourier transform for arbitrary finite length sequences.


Finite-length sequence & periodic sequence


)(nx Finite-length sequence that has N samples

)(~ nx periodic sequence with the period of N

)()(~)(

,0

10 ),(~)(

nRnxnx

elsewhere

Nnnxnx

N

))(()(~

)()(~

N

r

nxnx

rNnxnx

Window operation

Periodic extension


The definition of DFT


10 ,)(1

)]([IDFT)(

10 ,)()]([DFT)(

1

0

1

0

NnWkXN

kXnx

NkWnxnxkX

N

n

nkN

N

n

nkN

)()(~)()(1

)(

)()(~

)()()(

1

0

1

0

nRnxnRWkXN

nx

kRkXkRWnxkX

N

N

nN

nkN

N

N

nN

nkN

return



Linearity

)()()]()([DFT 2121 kbXkaXnbxnax N3-point DFT, N3=max(N1,N2)

Circular shift of a sequence

)()]())(([DFT kXWnRmnx kmNNN

)())(()]([DFT kRlkXnxW NNnl

N

Circular shift in the frequency domain



The sum of a sequence

1

00

1

00

)()()(N

nk

N

n

nkNk

nxWnxkX

The first sample of sequence

1

0

)(1

)0(N

k

kXN

x

)())(()]([

)()]([

kRkNNxnXDFT

kXnxDFT

NN



Circular convolution

)( )()())(()(

)())(()()( )(

12

1

012

1

02121

nxnxnRmnxmx

nRmnxmxnxnx

N

N

mN

N

N

mN

N

N

)()()]( )([DFT 2121 kXkXnxnx N

)( )(1

)]()([DFT 2121 kXkXN

nxnx N

Multiplication



Circular correlation

nn

xy nymnxmnynxmr )(*)()(*)()(

Linear correlation

Circular correlation

)()(*))((

)())((*)()(

1

0

1

0

mRnymnx

mRmnynxmr

N

N

nN

N

N

nNxy



)()(*))((

)())((*)(

)]([IDFT)(

)()()(

1

0

1

0

*

mRnymnx

mRmnynx

kRmrthen

kYkXkRif

N

N

nN

N

N

nN

xyxy

xy



Parseval’s theorem

1

0

*1

0

* )()(1

)()(N

k

N

n

kYkXN

nynx

1

0

21

0

2

1

0

*1

0

*

)(1

)(

)()(1

)()( then

)()( let

N

k

N

n

N

k

N

n

kXN

nx

kXkXN

nxnx

nynx



Conjugate symmetry properties of DFT

and)(nxep )(nxop

Let be a N-point sequence)(nx Nnxnx ))(()(~

]))(())(([2

1)](~)(~[

2

1)(~

]))(())(([2

1)](~)(~[

2

1)(~

NNo

NNe

nNxnxnxnxnx

nNxnxnxnxnx

It can be proved that

)(~)(~)(~)(~

*

*

nxnx

nxnx

oo

ee



)())(())((2

1

)()(~)(

)())(())((2

1

)()(~)(

nRnNxnx

nRnxnx

nRnNxnx

nRnxnx

NNN

Noop

NNN

Neep

Circular conjugate symmetric

component

Circular conjugate

antisymmetriccomponent



)()()( nxnxnx opep

)())(()(

)())(()(*

*

nRnNxnx

nRnNxnx

NNopop

NNepep



)()()( kXkXkX opep

)())(()(

)())(()(*

*

kRkNXkX

kRkNXkX

NNopop

NNepep

and)(kX ep )(kX op



)]())((Im[)](Im[

)]())((Re[)](Re[

kRkNXkX

kRkNXkX

NNepep

NNepep

)]())((Im[)](Im[

)]())((Re[)](Re[

kRkNXkX

kRkNXkX

NNopop

NNopop



)())(()( then

)())(()( if

kRkNXkX

nRnNxnx

NN

NN

Circular even sequences

Circular odd sequences

)())(()( then

)())(()( if

kRkNXkX

nRnNxnx

NN

NN



)()())((

)())(()]([DFT**

**

kNXkRkNX

kRkXnx

NN

NN

Conjugate sequences

)()]())(([DFT

)]())(([DFT**

*

kXnRnNx

nRnx

NN

NN



)())(())((2

1

)()](Re[DFT

* kRkNXkX

kXnx

NNN

ep

Complex-value sequences

)())(())((2

1

)()](Im[DFT

* kRkNXkX

kXnxj

NNN

op



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

1DFT

)](Re[)]([DFT

* nRnNxnx

kXnx

NNN

ep

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

1DFT

)](Im[)]([DFT

* nRnNxnx

kXjnx

NNN

op



)())(()( then

sequence value-real is )( if* kRkNXkX

nx

NN

Real-value sequences

Imaginary-value sequences

)())(()( then

part imaginary has only )( if* kRkNXkX

nx

NN



Summary

)( )( )(

)](Im[ )](Re[)(

kXkXkX

nxjnxnx

opep

)](Im[ )](Re[)(

)( )( )(

kXjkXkX

nxnxnx opep

example



Linear convolution & circular convolution

1

02121

21

1

)()()()(

)()()(N

mm

l

mnxmxmnxmx

nxnxny

Linear convolution

)(1 nx

)(2 nx

N1 point sequence, 0≤n≤ N1-1

N2 point sequence, 0≤n≤ N2-1

)(nyl L point sequence, L= N1+N2-1



Circular convolution

1 ,0

10 ),()(

1

111 LnN

Nnnxnx

We have to make both and L-point

sequences by padding an appropriate number of zeros

in order to make L point circular convolution.

)(1 nx )(2 nx

1 ,0

10 ),()(

2

222 LnN

Nnnxnx



)()(

)()()(

)()()(

)())(()()( )()(

2

1

01

1

021

1

02121

nRrLny

nRmrLnxmx

nRmrLnxmx

nRmnxmxnxnxny

Lr

l

Lr

L

m

L

L

m r

L

L

mLc

L



)()()( nRrLnyny Lr

lc

)()()( )( isthat

)()( then

1 if

2121

21

nxnxnxnx

nyny

NNL

lc

L

return



Sampling in frequency domain

m

kmNWz

WmxzXkX kN

)(|)()(~

rm

N

k

nmkN

N

k

knN

m

kmN

N

k

knNN

rNnxWN

mx

WWmxN

WkXN

kXnx

)(1

)(

)(1

)(~1

)](~

[IDFS)(~

1

0

)(

1

0

1

0



r

N rNnxnx )()(~

Frequency Sampling TheoremFor M point finite duration sequence, if the frequency sampling number N satisfy:

MN then

)()()(~)( nxnRnxnx NNN



Interpolation formula of )(zX

1

01

1

01

1

0

1

0

11

0

1

0

1

0

1

0

1

0

1

)(1

1

1)(

1

)(1

)(1

)(1

)()(

N

kk

N

NN

kk

N

NNkN

N

k

N

n

nkN

N

k

N

n

nnkN

N

n

nN

k

nkN

N

n

n

zW

kX

N

z

zW

zWkX

N

zWkXN

zWkXN

zWkXN

znxzX



1

1

0

1

01

1

11)(

)()(1

)(1)(

zW

z

Nz

zkXzW

kX

N

zzX

kN

N

k

N

kk

N

kk

N

N

Interpolation function



1

0

1

0

)2

()()()()(N

k

N

k

jK

jw kN

kXekXeX

2

1

2

2

sin

sin)(

Nj

N

eN

Interpolation function

Interpolation formula of )( jeX

return



Approximating to FT of continuous-time aperiodic signal with DFT

dejXtx

dtetxjX

tj

tj

)(2

1)(

)()(

CTFT


n

TdtTdtnTt ,,

Sampling in time domain

n

nTjtj enTxTdtetxjX )()()(

S

dejXntx

dejXtx

nTj

tj

0)(

2

1)(

)(2

1)(

Tf ss

22



Truncation in time domain

)1~0(:,,)~0(: 00 NnNTTTt

1

0

)()(N

n

nTjenTxTjX

S

dejXnTx nTj

0)(

2

1)(



Sampling in frequency domain

NT

TT

TFNT

f

N

FT

ddk

s

N

n

s

22

22,

1

,,

00

000

00

1

00000

)]([DFT

)()()(1

0

21

00

0

nxT

enxTenTxTjkXN

n

nkNjN

n

nTjk



)]([IDFT1

)]([IDFT

)(1

)(1

)(2

)(

00

1

0

2

0

1

0

2

00

1

00

0 0

jkXT

jkXf

ejkXN

f

ejkXN

NF

ejkXnTx

s

N

k

nkNj

s

N

k

nkNj

N

k

nTjk

demo



Approximating to FS of continuous-time periodic signal with DFS

k

tjk

T tjk

ejkXtx

dtetxT

jkX

0

00

)()(

)(1

)(

0

00

0

000

22

TF



1

000

0

,,N

n

TTdtNTTTdtnTt

Sampling in time domain

)]([DFS1

)(1

)()(1

0

21

000

0

nxN

enxN

enTxT

TjkX

N

n

nkNjN

n

nTjk



Truncating in frequency domain

)1,0(: let, , 00 NkNFfNTT s

)]([IDFS)(1

)()()(

)()(

0

1

0

2

0

1

0

2

0

1

00

0

0

0

jkXNejkXN

N

ejkXejkXnTx

ejkXtx

N

k

nkNj

N

k

nkNjN

k

nTjk

k

tjk



Some problems Aliasing

Otherwise, the aliasing will occur in frequency domainhs

hs ffTff

2

11,2 Sampling in time domain:

Sampling in frequency domain:0

0

1

FT

Period in time domain0T Frequency resolution 0F

NT

T

F

f s 0

0

andis contradictory

hf 0F



Spectrum leakage

sequence length-finite ),()()(

sequence length-infinite ),(

12

1

nRnxnx

nx

N

)()()( 12 j

Rjj eWeXeX

Spectrum extension (leakage)

Spectrum aliasing


demo


Fence effect

N

fF

f

F

fNs

ss

000

0 ,22

Frequency resolution

00

11

TNTN

fF s

demo



Comments

return

demo

Zero-padding is an operation in which more zeros are appended to the original sequence. It can provides closely spaced samples of the DFT of the original sequence.

The zero-padding gives us a high-density spectrum and provides a better displayed version for plotting. But it does not give us a high-resolution spectrum because no new information is added.

To get a high-resolution spectrum, one has to obtain more data from the experiment or observation.

example



Summary

return

The frequency representations of x(n)

)(nx

)(kX

)(zX

)( jeX

Time sequence

z-transform of x(n)

Complex frequency

domain

DTFT of x(n)Frequency

domainDFT of x(n)

Discrete frequency domain

ZT

DTFT

DFTk

N

2

jez kNjez

2

interpolation

interpolation


Illustration of the four Fourier transforms

Discrete Fourier SeriesSignals that are discrete and periodic

DTFTSignals that are discrete and aperiodic

Fourier SeriesSignals that are continuous and periodic

Fourier TransformSignals that are continuous and aperiodic


)(~ ny

n0 1 2 3 4 5 6

0n m

)(~1 mx

0

m

)(~2 mnx

0

1n2n3n4n5n6n

return


n

)(nxep

0 5

)())(()( * nRnNxnx NNepep

n

Nep nNx ))((

0 5

)(nRN


0 1 2 3 4 5 6 7 8 9 10

0

5

10

Original sequence

n

x(n

)

0 1 2 3 4 5 6 7 8 9 10

0

5

10

Circular conjugate symmetric component

n

xe

p(n

)

0 1 2 3 4 5 6 7 8 9 10-4

-2

0

2

4Circular conjugate antisymmetric component

n

xo

p(n

)

return

)()8.0(10 11 nRn


0 1 2 3 4 5 6 7 8 9 10

0

5

10

Circular even sequence x(n)

n

0 1 2 3 4 5 6 7 8 9 100

20

40

The DFT of x(n)

k

0 1 2 3 4 5 6 7 8 9 100

20

40

k

return

)(kX

)())(( nRkNX NN


0 1 2 3 4 5 6 7 8 9 10-4

-2

0

2

4Circular odd sequence x(n)

n

0 1 2 3 4 5 6 7 8 9 10

-10

0

10

The imaginary part of DFT[x(n)]

k

0 1 2 3 4 5 6 7 8 9 10

-10

0

10

k

return

)(kX

)())(( nRkNX NN


)())(()( * kRkNXkX NN

number real a is )0(

)0()())(()0( *

0

*

X

XkRkNXXkNN

number real a is )2

(

)2

()())(()2

(

even is if

*

2

*

NX

NXkRkNX

NX

N

NkNN


)())(()( * kRkNXkX NN

number imaginary an is )0(

)0()())(()0( *

0

*

X

XkRkNXXkNN

number imginary an is )2

(

)2

()())(()2

(

even is if

*

2

*

NX

NXkRkNX

NX

N

NkNN


)( , )( 21 nxnx N-point real-value sequences

)]([DFT)( )],([DFT)( 2211 nxkXnxkX

)()()]([DFT)]([DFT

)]()([DFT)]([DFT)(

)()()(

2121

21

21

kjXkXnxjnx

njxnxnykY

njxnxny

)())(()(2

1)()](Re[DFT)(1 kRkNYkYkYnykX NNep

)())(()(2

1)(

1)](Im[DFT)(2 kRkNYkY

jkY

jnykX NNop


0 1 2 3 4 5 6 7 8 9

0

2

4

6

8

10

12

Linear convolution

n 0 1 2 3 4 5 6 7 8 9

0

2

4

6

8

10

12

Circular convolution N = 6

n

0 1 2 3 4 5 6 7 8 9

0

2

4

6

8

10

12


n 0 1 2 3 4 5 6 7 8 9

0

2

4

6

8

10

12


n

return

],2,3,2,1[)( ],2,2,1[)( 21 nxnx


-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

Magnitude Response, N = 8

frequency in pi units

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1

-0.5

0

0.5

1Phase Response


pi

)(

return

N

2N

4


0 1 2 3 4 5 6 70

1

2

3

4

5

6

X(k),N = 8

k

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0

2

4


return

)()0( X )2

()1(N

X

)4

()2(N

X

)6

()3(N

X


0 5 10 15 20 250

2

4

6

8

10

t-1 -0.5 0 0.5 10

10

20

30

40

50

rad

0 5 10 15 20 250

2

4

6

8

10

n-2 -1 0 1 2

0

10

20

30

40

50

pi

ta tx )8.0(10)( )( jX a

FT

DTFT

)(nx )( jeX


-10 -5 0 5 100

2

4

6

8

10

n-2 -1 0 1 2

0

10

20

30

40

50

pi

-10 0 100

2

4

6

8

10

n-10 0 10

0

10

20

30

40

50

k

)()( 11 nRnx )()( jj eReX

)(~ nxN )(~kX N

DTFT

DFS


-10 0 100

2

4

6

8

10

n-10 0 10

0

10

20

30

40

50

k

return

)(nxN )(kX N

DFT


0

)( jeR

0

)(2jeX

n

)(nRN

0

n

)(2 nx

0

n

)(1 nx

0 0

)(1jeX


0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

2

4

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

2

4

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

2

4

pi

pi

pi

DTFT DFT

DTFT DFT

DTFT DFT

return

]1,1,1,1[)( nx

]0,0,0,0,1,1,1,1[)( nx

]0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1[)( nx


0 2 4 6 8 10 12 14 16 18 20-2

-1

0

1

2signal x(n), 0<=n<=19

n

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

5

10

15

20

pi

)(20 kX

)52.0cos()48.0cos()( nnnx


0 10 20 30 40 50 60 70 80 90 100-2

-1

0

1

2signal x(n), 0<=n<=19+80 zeros

n

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

5

10

15

20

pi

)(100 kX


0 10 20 30 40 50 60 70 80 90 100-2

-1

0

1

2signal x(n), 0<=n<=99

n

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

10

20

30

40

50

pi

)(100 kX


0 50 100 150 200 250 300 350 400-2

-1

0

1

2signal x(n), 0<=n<=99+300 zeros

n

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

20

40

60

pi

return

)(400 kX


Suppose

kHz 4 Hz, 100 hfF

Determine

N , ,0 TT

Solution sF

T 1.010

11

00

msff

Ths

125.01042

1

2

113

102422

80010125.0

1.0

10

30

mN

T

TN

Copyright © 2005. Shi Ping CUC Chapter 3 Discrete Fourier Transform Review Features in common We...

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