Chapter6 sampling

Signals & Systems

Chapter 6

Sampling

INC212 Signals and Systems : 2 / 2554

Chapter 6 Sampling

Overview Sampling theorem Signal reconstruction Interpolation formula Aliasing

INC212 Signals and Systems : 2 / 2554

The sampling theorem

INC212 Signals and Systems : 2 / 2554 Chapter 6 Sampling



Hz2BFs

ssn

s

nT

FT

nFT

F

nTtnTfttftf

22

;)(1

)(

)()()()()(

bandlimited

BF 2for0)(



Te

Tt

Tdtt

Tdtet

Tc

tect

sk

tjkT

T

T

T

T

T

tjkTk

k

tjkkT

s

s

s

2,

1)(

1)1)((

1)(

1

,)(

2

2

2

2

Trigonometric form

0

,3,2,1,2

2

10

k

kk kT

cA

Tc

sssssT FT

tttT

t 22

,cos2coscos211

)(

2

2

)(1

;,)(T

T

tjkk

k

tjkk dtetx

Tctectx ss



)2()2(2cos)(2 sss FFttf F

n

snFT

F )(1

)(

)()(cos)(2 sss FFttf F

ttfttfttftfT

tf sss cos)(22cos)(2cos)(2)(1

)(

)()()( ttftf T

Effect of undersampling and oversampling


)5(sinc)( 2 ttf

20tri2.0)(F



)5(sinc)( 2 ttf

20tri2.0)(F

sec1.0Hz10 TFs



)5(sinc)( 2 ttf

20tri2.0)(F

sec2.0Hz5 TFs



)5(sinc)( 2 ttf

20tri2.0)(F

sec05.0Hz20 TFs



)5(sinc)( 2 ttf

20tri2.0)(F

1.0;Hz10;20

10for0)(

TF

F

ss



Hz2

Hz todBandlimite2for0)(

BF

BBF

s

The minimum sampling rate = 2B The Nyquist rate

The sampling interval = 1/2B The Nyquist interval

Signal Reconstruction Zero-order hold


Signal Reconstruction The Interpolation Formula


Signal Reconstruction The Interpolation Formula


BTH

4rect)(

Btth

BT

BtBTth

2sinc)(

1 2 rate;Nyquist theAssuming

2sinc2)(

n

n

n

n

nBtnTftf

nTtBnTf

nTthnTf

nTtnTfth

tfthtf

)2sinc()()(

))(2sinc()(

)()(

)()(*)(

)(*)()(

Aliasing


Aliasing Amplitude spectrum of time-limited signal


Bs 2

not be bandlimited

Aliasing Anti-aliasing

The sampling frequency may be as large as 10 or 20 times B.


Lowpass filter

Sampling

x (t) x [n]

Chapter6 sampling

Education

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