meiling chen signals & systems 1

Lecture #05

Fourier representationfor discrete-time signals

And Sampling Theorem

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2

2

0)(1

T

T

dtetfT

C tjnn

0

0)(n

tjnneCtf

1

0

1

0

0

0

][1

][

][][

N

n

njk

N

k

njk

enxN

kX

ekXnx

n

njj

njj

enxeX

deeXnx

][)(

)(2

1][

Fourier series for periodic discrete signals

Fourier Transform for nonperiodic discrete signals

dtetfjF

dejFtf

tj

tj

)()(

)(2

1)(

00

TN

tn

tn

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Example 3.2

525 0NT

)}sin(1{5

1}

2

11

2

1{5

1

}]2[]1[]0[]1[]2[{5

1

][5

1][

1][

52

)2()1()0()1()2(

2

2

1

0

52

52

52

52

52

52

52

52

0

kjkjk

jkjkjkjkjk

n

njkN

n

njk

jee

exexexexex

enxenxN

kX

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Example 3.5 Inverse DTFS

1)cos(4)cos(2

22

2

]3[]2[]0[

]2[]3[

][][

929

][][

394

32

96

32

)2()3(

4

4

0

1

0

96

32

94

3

94

396

32

92

92

92

92

92

0

nn

jjjjj

jjjj

jj

jj

k

kj

N

k

njk

nn

nn

nn

nn

n

eeeee

eeee

eXeXX

eXeX

ekXnx

NT

ekXnx

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Example 3.17 Find DTFT of the sequence

][][ nunx n

00

)(

][][][

n

nj

n

njn

n

njn

n

njj

ee

enuenxeX

jj

j

eeX

eX

1

1][,1

][,1

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Example 3.18 Find DTFT of the sequence

Mn

Mnnx

,0

,1][

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,4,2,0,12

,4,2,0,1

1

][

1][][

)12(

2

0

2

0

)(

Me

ee

ee

eeX

Mnmlet

eenxeX

j

MjMj

M

m

mjMj

M

m

Mmjj

M

Mn

nj

n

njj

Mn

Mnnx

,0

,1][

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Example 3.18 Find Inverse DTFT of a unit impulse spectrum

Example 3.17 Find DTFT of a unit impulse spectrum

][][ nnx

1][][][ 0

j

n

nj

n

njj eenenxeX

2

1

2

1

)(2

1][

),(][

0

j

nj

j

e

denx

eX

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][][ nnx 1][ jeX

),(][ jeX 21

][ nx

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Sampling

• Sampling is a process of converting a signal into a numeric sequence (a function of discrete time or space).

• The sampling theorem states that exact reconstruction of a continuous time baseband signal from its samples is possible if the signal is bandlimited and the sample frequency is greater than twice the signal bandwidth.

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)( nTt

)(txc

)(txs

n

scs nTttxtx )()()(

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)]()([2

1)()( 2121

jFjFtftf

ks

sc

cs

kT

jX

tpFjXjX

)(2

)(2

1

)}({)(2

1)(

Take Fourier transform

k

scs

s kjXT

jX )]([1

)(

)()()()()( tpnTxnTttxtx scn

scs

n

snTttp )()(

Fourier transform

)( jXC

For example

k

sk )(

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TdtetT

kP

TnTttp

T

T

stjk

sn

s

1)(1

][

2,)()(

2

2

Example 4.2

Fourier series of p(t)

dtetpjP

ekPtpeCtf

tj

k

tjk

n

tjnn

)()(

][)()( 00

0 dtetfjF

dejFtf

tj

tj

)()(

)(2

1)(

Fourier transform of p(t)

}1{)( 0tjkeT

FjP

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kk

tjk kkXjXekXtx )(][2)(][)( 00

k

sk

kT

kT

jP )(2

)(2

)( 0

(v) Frequency shifting modulation)]([)( 00 jFetf tj

)(21

}1{)( 0tjkeT

FjP

Fourier series Fourier Transform

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)( jX c

nn s

Case I: ns 2

s

s

Case II: ns 2Aliasing

k

scs

s kjXT

jX )]([1

)(

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Sampling theorem : Let represent a band-limited signal,

so that for . If , where

Is the sampling frequency, then is uniquely determined by its

samples

)()( jXtx

0)( jX m ms 2

ss T

2

,2,1,0),( nnTx s

)(tx

m2 The minimum sampling frequency, Nyquist sampling rate.

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)( jX)(tx

4

1sT

1sT

5.1sT