Chap3-Discrete Fourier Transform

8/9/2019 Chap3-Discrete Fourier Transform

1/83

Copyright 2005. Shi Ping CUC

Chapter 3

Discrete 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


2/83


Chapter 3

Discrete 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


3/83



Introduction

Fourier analysis is named afterJean Baptiste Joseph

Fourier(1768-1830), a French mathematician andphysicist.

A signal can be eithercontinuous ordiscrete, and it can

be eitherperiodicoraperiodic. The combination ofthese two features generates the four categories of

Fourier Transform.


4/83



Aperiodic-ContinuousFourier Transform

g

g

;

g

g;

;;!

!;

dejXtx

dtetxjX

tj

tj

)(2

1)(

)()(

T


5/83



Periodic-ContinuousFourier Series

g

g!

;

;

;!

!;

k

tjk

T

T

tjk

ejkXtx

dtetxTjkX

0

0

0

0

)()(

)(

1

)(

0

2

20

0

0

0

22

TF

TT !!;


6/83

g

g!

!

!

T

T

[[

[[

[

T

deeXnx

enxeX

njj

n

njj

)(

2

1)(

)()(



Aperiodic-DiscreteDTFT


7/83



Periodic-DiscreteDFS (DFT)

!

!

!

!

1

0

2

1

0

2

)(1

)(

)()(

N

k

nkN

j

N

n

nkN

j

ekXN

nx

enxkX

T

T


8/83



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

T0

T

0

0

2

T

T!;

Ts

T2

!;

Ts

T2!

0

0

2

T

T!;

return


9/83



Definition

Periodic 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 !

g

g!

!

k

ktT

j

ekXtx

T2

)()(

!

!1

0

2

)(1

)(N

k

knN

j

ekX

N

nx

T

fundamental

frequencyt

Tj

e

T2

nN

j

e

T2fundamentalfrequency


10/83



!

!

!

! elsewhere,0

,1

1

1112

2

1

0

2mNr

e

e

Ne

N rN

j

rNN

jN

n

rnN

j

T

TT

)(1

)(

)(1

)(

1

0

)(1

0

1

0

1

0

1

0

rXeN

kX

eekXN

enx

N

n

nrkN

jN

k

N

n

rnN

jN

k

knN

jN

n

rnN

j

!

-

!

-

!

!

!

!

!

!


11/83



1

0

2

)()(N

n

knN

j

enxkX

T

)()(

)()(

1

0

1

0

)(

kXenx

enxmNkX

N

n

knN

j

N

n

nmNkN

j

!!

!

!

!

Because:

The is a periodic sequence with fundamental

period equal to N

)(kX


12/83



!

!

!!

!!

!

1

0

1

0

~1]~[IDFS~

~]~[DFS~

Let

N

k

kN

N

n

nk

N

j

N

WkN

knx

Wnxnxk

eW NT


13/83



Relation to the z-transform

ee

!elsewhere,0

10, Nnnxnx

kN

jez

zXkX T2|)()(~!!

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

)(~

kX

)(zX

!

!

!!1

0

1

0

))(()(~

,)()(2

N

n

nkjN

n

n NenxkXznxzXT


14/83



Relation to the DTFT

k

j

N

n

nkN

jN

n

njj

NeXkX

enxkXenxeX

T[

[

T

[[

2|)()(

~

)()(~

)()(1

0

21

0

!

!

!

!

!!

ee

!elsewhere,0

10),(~

)(Nnnx

nx

The DFS is obtained by evenly sampling the DTFT at

intervals. It is called frequency resolution and represents the

sampling intervalin the frequency domain.

NT2


15/83



jIm[z]

Re[z]

0!k

k

j

N

ekX T[[

2)()(~

!!

N

T2

N=8frequency resolution


16/83


The properties of DFS


Linearity

)(

~

)(

~

)](~

)(~

[DFS 11 kXbkXanxbnxa

Shift of a sequence

)(~

)(~

)](~[DFS kXekXWmnxmk

Nj

mk

N

T

!!

Modulation

)(~

)](~[DFS lkXnxWlnN

!


17/83



Periodic convolution

!

!

!

!!

!

1

012

1

0

21

21

)(~)(~

)(~)(~)](~

[IDFS)(~then

)(~

)(~

)(~if

N

m

N

m

mnxmx

mnxmxkYny

kXkXkY


18/83


!

!

!

!

!

!

!

!!

-

!

-

!

!!

1

0

12

1

0

21

1

0

1

0

)(21

1

0

2

1

0

1

1

0

2121

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

)(~1)(~

)(~

)(~1

)(~

)(~1

)](~

)(~

[IDFS)(~

N

m

N

m

N

m

N

k

kmnN

N

k

nk

N

N

m

mk

N

N

k

nk

N

mnxmxmnxmx

WkXN

mx

WkXWmxN

WkXkX

N

kXkXny

return


19/83


Introduction


The DFS provided us a mechanism fornumerically

computingthe 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

periodicsignal whoseprimaryshape 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.


20/83


Finite-length sequence & periodic sequence


)(nx Finite-length sequence that has N samples

)(~ nx periodic sequence with the period of N

)()(~)(

,0

10),(~)(

nRnxnxelsewhere

Nnnxnx

N!

ee!

))(()(~

)()(~

N

r

nxnx

rNnxnx

!

! g

g!

Window operation

Periodic extension


21/83


The definition of DFT


10,)(1

)]([ID T)(

10,)()]([D T)(

1

0

1

0

ee

ee

NnWkXN

kXnx

NkWnxnxkX

N

n

nk

N

N

n

nk

N

)()(~)()(1

)(

)()(~)()()(

1

0

1

0

nRnxnRWkXN

nx

kRkXkRWnxkX

N

N

n

N

nk

N

N

N

n

NnkN

!!

!!

!

!

return


22/83



Linearity

)()()]()([DFT 11 kbkanbxnax

N3-point DFT, N3=max(N1,N2)

Circular shift of a sequence

)()]())(([ kXWRmkm

!

)())(()]([ kRlkXWl

!

Circular shift in the frequency domain


23/83



The sum of a sequence

!!

!!

!!1

0

0

1

00

)()()(N

n

N

n

n

NnxWnxX

The first sample of sequence

!

!1

0

)(1

)0(N

k

kXN

x

)())(()]([

)()]([

kRkNNxnXDFT

kXnxDFT

NN!

!


24/83



Circular convolution

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

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

12

1

0

12

1

0

2121

nxnxnRnxmx

nRmnxmxnxnx

N

N

m

N

N

N

m

N

!

-

!

-

!

!

!

N

N

)()()]()([DFT2121

kXkXnxnx !N

)()(1

)]()([DFT 2121 kXkXN

nxnx ! N

Multiplication


25/83



Circular correlation

g

g!

g

g! !! nnxy nymnxmnynxmr )(*)()(*)()(

Linear correlation

Circular correlation

)()(*))((

)())((*)()(

1

0

1

0

mRymx

mRmyxmr

N

N

N

N

N

Nxy

!

!

!

!


26/83



)()(*))((

)())((*)(

)]([IDFT)(

)()()(

1

0

1

0

*

mRnymnx

mRmnynx

kRmrthen

kYkXkRif

N

N

n

N

N

N

n

N

xyxy

xy

!

!

!

!

!

!


27/83



Parsevals 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

NN

n

NN

n

Nnx

N

nxnx

nynx


28/83



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

nNx

nx

nx

nx

nx

nNxnxnxnxnx

!!

!!

It can be proved that

)(

~

)(

~

)(~)(~

*

*

nx

nx

nxnx

oo

ee


29/83



? A

? A )())(())((2

1

)()(~)(

)())(())((2

1

)()(~)(

nRnNn

nRnn

nRnNn

nRnn

NNN

Noop

NNN

Np

!

!

!

!

Circular

conjugate

symmetric

component

Circular

conjugate

antisymmetriccomponent


30/83



)()()( nxnxnxopep

!

)())(()(

)())(()(*

*

nRnNxnx

nRnNxnx

NNopop

NNepep

!

!


31/83



)()()(kXkXkX

opp

!

)())(()(

)())(()(

*

*

kRkNXkX

kRkNXkX

NNopop

NNepep

and)(kXep )(kXop


32/83



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

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

kRkNXkX

kRkNXkX

NNepep

NNepep

!

!

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

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

kRkNXkX

kRkNXkX

NNopop

NNopop

!

!


33/83



)())(()(then

)())(()(if

kRkNXkX

nRnNxnx

NN

NN

!

!

Circular even sequences

Circular odd sequences

)())(()(then)())(()(ifkRkNXkX

nRnNxnx

NN

NN

!

!


34/83



)()())((

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

**

kNXkRkNX

kRkXnx

NN

NN

!!

!

Conjugate sequences

)()]())(([D

)]())(([D

**

*

knRnNx

nRnx

NN

NN

!!


35/83



_ a

? A )())(())((21

)()](Re[DFT

* kRkNXkX

kXnx

NNN

ep

!

!

Complex-value sequences

_ a? A )())(())((

2

1)()](Im[DFT

* kRkNXkX

kXnxj

NNN

op

!

!


36/83



!

!

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

1

DFT

)](Re[)]([DFT

*

nnn

kXnp

!

!

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

1DFT

)](Im[)]([DFT

*nRnNxnx

kXjnx

NNN

op


37/83



)())(()(then

sequencevalue-realis)(i*

kRkNXkX

nx

NN!

Real-value sequences

Imaginary-value sequences

)())(()(tpartimagi aryaso ly)(if

*kRkNXkX

nx

NN!


38/83



Summary

)()()(

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

kXkXkX

nxjnxnx

opep !

!

DDD

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

)()()(

kXjkXkX

nxnxnx opep

!

!

DDD

example


39/83



Linear convolution & circular convolution

!

g

g!

!!

!

1

0

2121

211

)()()()(

)()()(N

mm

l

mnxmxmnxmx

nxnxny

Linear convolution

)(1 nx

)(nx

N1 point sequence, 0n N1-1

N2 point sequence, 0n N2-1

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


40/83



Circular convolution

ee

ee!

1,0

10),()(

1

11

1LnN

Nnnn

We have to make both and L-point

sequences by padding an appropriate number of zeros

in order to make L point circular convolution.

)(nx )(2 nx

ee

ee!

,

),()(

LnN

Nnnxnx


41/83



)()(

)()()(

)()()(

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

2

1

0

1

1

0

21

1

0

2121

nRrLny

nRmrLnxmx

nRmrLnxmx

nRmnxmxnxnxny

L

r

l

L

r

L

m

L

L

m r

L

L

m

Lc

-

!

-

!

- !

-

!!

g

g!

g

g!

!

!

g

g!

!

L


42/83



)()()( nRrnynyr

lc

-

!

g

g!

)()()()(isthat

)()(then1if

2121

21

nxnxnxnx

nynyNNL

lc

!

!u

L

return


43/83



Sampling in frequency domain

g

g!!

!!

m

km

NWzWmxzXkX k

N

)(|)()(~

g

g!

g

g!

!

!

g

g!

!

!

-

!

-

!

!!

rm

N

mk

N

N

k

kn

N

m

km

N

N

k

kn

NN

rNnxWN

mx

WWmxN

WkXN

kXnx

)()(

)(

)()]([IDFS)(

)(


44/83



g

g!

!r

NrNnxnx )()(~

Frequency Sampling TheoremForM point finite duration sequence, if the frequency

sampling number N satisfy:

MN uthen

)()()(~)( nxnRnxnxNNN

!!


45/83



Interpolation formula of )(zX

!

!

!

!

!

!

!

!

!

!

!

-

!

-

!

-

!!

1

01

1

01

1

0

1

0

11

0

1

0

1

0

1

0

1

0

1)(1

11)(1

)(1

)(1

)(1

)()(

N

kk

N

NN

kk

N

NNk

N

N

k

N

n

nk

N

N

k

N

n

nnk

N

N

n

n

N

k

nk

N

N

n

n

zWkX

N

zzWzWkX

N

zWkXN

zWkXN

zWkXN

znxzX


46/83



1

1

0

1

01

1

11)(

)()(1

)(1)(

!

!

!*

*!

!

zW

z

Nz

zkXzW

kX

N

zzX

k

N

N

k

N

k

k

N

kk

N

N

Interpolation

function


47/83



!

!

*!*!1

0

1

0

)2

()()()()(N

k

N

k

j

K

jwk

NkXekXeX

T

[[

2

1

2

2

sin

sin)(

!*

NjN

eN

[

[

[

[

Interpolation

function

Interpolation formula of )( [jeX

return


48/83



Approximating to FT of continuous-time aperiodic

signal with DFT

g

g;

g

g

;

;;!

!;

dejXtx

dtetxjX

tj

tj

)(21)(

)()(

T

CTFT


49/83


g

g!

g

gppp

n

TdtTdtnTt ,,

Sampling in time domain

g

g!

;g

g

; }!;n

nTjtjenTxTdtetxjX )()()(

;;

g

g

;

;;}

;;!S

dejXntx

dejXtx

nTj

tj

0)(

2

1)(

)(2

1

)(

T

T

Tfss

T

T

22 !!;



50/83


Truncation in time domain

)1~0(:,,)~0(:00 ! NNTTTt

!

;};1

0

)()(N

n

nTjenTxTjX

S

dejXnTxnTj

0)(

21)(T



51/83


Sampling in frequency domain

NT

TT

TFNT

fN

FT

ddk

s

N

n

s

TT

TT

22

22,1

,,

0

0

0

00

0

0

1

0

00

00

!!;

!!;!!!

;p;;p;;p;

!

;

)]([T

)()()(1

0

21

0

0

0

nxT

enxTenTxTjkXN

n

nkN

jN

n

nTjk

!

!};

!

!

;T



52/83


)]([I T1

)]([I T

)(1

)(1

)(2

)(

00

1

0

2

0

1

0

2

00

1

0

0

0 0

;!;!

;!

;!

;;

}

!

!

!

;

jkXT

jkXf

ejkXN

f

ejkXN

NF

ejkXnTx

s

N

k

nkN

j

s

N

k

nkN

j

N

k

nTjk

T

T

T

demo



53/83


Approximating to FS of continuous-time periodic

signal with DFS

g

g!

;

;

;!

!;

k

tjk

Ttjk

ejkXtx

dtetxT

jkX

0

00

)()(

)(1

)(

0

00

0

0

00

22

TF

TT !!;



54/83


!

!1

00

0

0

,,N

n

T

TdNTTTdnTt

Sampling in time domain

)]([1

)(1

)()(1

0

21

00

00

nxN

enxN

enTxT

TjkX

N

n

nkN

jN

n

nTjk

!

!};

!

!

;T



55/83


Truncating in frequency domain

),(:let,, !@! NkNFfNTT s3

)]([IDFS)(1

)()()(

)()(

0

1

0

2

0

1

0

2

0

1

0

0

0

0

0

;!;!

;!;}

;!

!

!

!

;

g

g!

;

jkXNejkXN

N

ejkXejkXnTx

ejkXtx

N

k

nkN

j

N

k

nkN

jN

k

nTjk

k

tjk

T

T



56/83

Some problems

Aliasing

Otherwise, the aliasing will occur in frequency domainhs

hs

ffff

2,2 !"Sampling in time domain:

Sampling in frequency domain:

0

0

1

FT !

Period in time domain0T Frequency resolution0F

NT

T

F

fs!!

0

0

and

is contradictoryhf 0F



57/83

Spectrum leakage

sequencelength-finite),()()(

sequencelength-infinite),(

12

1

nRnxnx

nx

N!

)()()(j

R

jjeWeXeX !

Spectrum extension (leakage)

Spectrum aliasing


demo


58/83


Fence effect

N

fF

f

F

fN

s

ss

!!;

!! 000

0 ,22 TT

[

Frequency resolution

00

11

TT

fF

s!!!

demo



59/83


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 obtainmore data from the experiment or observation.

example



60/83


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 frequencydomain

ZT

DTFTDFT

kN

T2!

[jz! kjz

2

!

interpolation

interpolation


61/83


Illustration of the four Fourier transforms

Discrete Fourier SeriesSignals that are discrete and

periodic

DTFTSignals that are discrete andaperiodic

Fourier SeriesSignals that are continuousand periodic

FourierTransformSignals that are continuousand aperiodic

~


62/83


)(~ ny

n0 1 2 3 4 5 6

0!n m

)(~1 mx

0

m

)(~2 mnx

0

1!n2!n

3!n4!n5!n

6!n

return

? A1


63/83

Copyright 2005. Shi Ping CUCreturn

? A)())(()(2

1)( nRnNxnxnx NNep !

n

)(nx

0

n

N

nNx

))((

0 55

n

)(nxep

05


64/83


n

)(nxep

0 5

)())(()( * nRnNxnx NNepep !

n

epn))((

0 5

)(nRN


65/83


0 1 2 3 4 5 6 7 8 9 10

0

5

10

O r ig i n a l s e q u e n c e

n

x

(n

)

0 1 2 3 4 5 6 7 8 9 10

0

5

10

C i r c u l ar c o n j u g a t e s y m m e t r ic c o m p o n e n t

n

xep(n

)

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

-2

0

2

4C i r c u l ar c o n j u g a t e an t is y m m e t r ic c o m p o n e n t

n

xo

p

(n

)

return

)()8.(1 11 nRn

v


66/83


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

0

5

1 0

Ci

l

v

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

0

2 0

4 0

T h e D FT

f

k

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

0

2 0

4 0

k

return

)(kX

)())(( nkNX NN


67/83


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

-2

0

2

4Ci

l

d d

e q u e n c e

n )

n

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

-1 0

0

1 0

T h e i !

in a r y "

a rt

f D FT [x (n ) ]

k

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

-1 0

0

1 0

k

return

)(kXd

)())(( nRkNXNN

d


68/83


)())(()(*

kkNXkXNN

!

? Anumberrealais)0(

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

*

X

XkRkNXXk

NN

@

!!!

? A

numberrealais)2

(

)

2

()())(()

2

(

evenisif

*

2

*

NX

NXRNX

NX

N

NNN

@

!!!


69/83


)())(()(*

kRkNXkXNN

!

numb rimaginaryanis)0(

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

*

X

XRNXXNN

@

!!!

? A

numberimginaryanis)2

(

)

2

()())(()

2

(

evenisif

*

2

*

NX

NXkRkNX

NX

N

Nk

NN

@


70/83


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

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

nxnx !!

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

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

kjkj

jykYjy

!!

!!!

_ a )())(()(2

1)()](Re[DFT)(1 kRkNYkYkYykX NNep !!!

_ a ? A )())(()(2

1)(

1)](Im[DFT)(2 kRkNYkY

jkY

jnykX NNop !!!


71/83


0 1 2 3 4 5 6 7 8 9

0

2

4

68

1 0

1 2

L in e a r c o n vo lut io n

n 0 1 2 3 4 5 6 7 8 9

0

2

4

68

1 0

1 2

Circ ula rc o n vo lut io n N = 6

n

0 1 2 3 4 5 6 7 8 9

0

2

46

8

1 0

1 2


n 0 1 2 3 4 5 6 7 8 9

0

2

46

8

1 0

1 2


n

return

],2,3,2,1[)(],2,2,1[)( 21 !! nn

)([*


72/83


-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

M a g n i tude R e s p o n s e , N = 8

fr e q u e n c y in p iu n i ts

-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

1P h a s e R e s p o n s e


p

i

)([*

return

N

T2N

T4


73/83


0 1 2 3 4 5 6 7 0

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 )()(N

X T[*

)4

()2(N

T[ *

)6()3(N

X T[ *

ttx )80(0)( v )( ;jX


74/83


0 5 1 0 1 5 2 0 2 5 0

2

4

6

8

1 0

t

-1 -0 .5 0 0 .5 10

1 0

2 0

3 0

4 0

5 0

ra d

0 5 1 0 1 5 2 0 2 5 0

2

4

6

8

1 0

n

-2 -1 0 1 20

1 0

2 0

3 0

4 0

5 0

p i

atx )8.0(0)( v! )( ;jXa

FT

DTFT

)(nx )([jeX

)()( R )()([[ jj

eReX


75/83


-1 0 -5 0 5 1 0 0

2

4

6

8

1 0

n

-2 -1 0 1 20

1 0

2 0

3 0

4 0

5 0

p i

-1 0 0 1 0 0

2

4

6

8

1 0

n

-1 0 0 1 0 0

1 0

2 0

3 0

4 0

5 0

k

)()( nRnx )()(jj

eReX

)(~ nxN )(~

kN

DTFT

DF

)( )(kX


76/83


-1 0 0 1 0 0

2

4

6

8

1 0

n

-1 0 0 1 0 0

1 0

2 0

3 0

4 0

5 0

k

return

)(nxN

)(kXN

DFT

)(1 nx )([j

eX


77/83


[0

)([j

eR

[0

)(2[j

X

n

)(nR

0

n

)(nx

0

n

)(1

0[0

)(eX


78/83


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

p i

p i

p i

DTFTDFT

DTFT

DFT

DTFT

DFT

return

],,,[)( !nx

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

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

s ig n a l (n) 0 < n < 1 9

)50cos()480cos()( TT


79/83


0 2 4 6 8 1 0 1 2 1 4 1 6 1 8 2 0 -2

-1

0

1

2s ig n a lx(n),0


80/83


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

-1

0

1

2s ig n a lx(n),0


81/83


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

-1

0

1

2s ig n a lx(n),0


82/83


0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0 3 5 0 4 0 0 -2

-1

0

1

2s ig n a lx(n),0


83/83

Suppose kHz4Hz,100 ee hfF

Determine N,,0

TT

Solutions

FT .!u!

msThs

125.01042

1

2

113

!vv

!e!

102422

80010125.0

1.0

10

3

0

!!!

!vu!

mN

T

TN

Documents

Chap3-Discrete Fourier Transform