Chapter4 Sampling of Continuous-Time Signals

8/13/2019 Chapter4 Sampling of Continuous-Time Signals

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12014/1/25 1 Zhongguo Liu_Biomedical Engineering_Shandong Univ

Biomedical Signal processingChapter 4 Sampling of Continuous-

Time SignalsZhongguo Liu

Biomedical EngineeringSchool of Control Science and Engineering, Shandong University


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Chapter 4: Sampling ofContinuous-Time Signals

4.0 Introduction 4.1 Periodic Sampling

4.2 Frequency-Domain Representation ofSampling 4.3 Reconstruction of a Bandlimited Signal

from its Samples 4.4 Discrete-Time Processing of

Continuous-Time signals


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4.0 Introduction Continuous-time signal processing can be

implemented through a process ofsampling , discrete-time processing , andthe subsequent reconstruction of a

continuous-time signal.

f=1/T: sampling frequency ,c x n x nT n srad T s /,2

T: sampling period


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n

nT t )( 4.1 Periodic

Sampling

Continuous-time signal

T:samplingperiod


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4.2 Frequency-Domain Representation of Sampling

[ ] ( ) | ( )c t nT c x n x t x nT

T sample period; fs=1/T :sample rates=2/T : sample rate

n

nT t t s

n

s c cn

c x t t nT x t x t s t x nT t nT

2 sk

S j k T

1 c sk

X j k T

1( )

21 2 1

1*

2

( ) ( )

2

c

s c s

k

s c

c

k

S j X j d

k X j

X j X j S j

d k X j d

T T

Representation of

in terms of jwe X j X s


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T sample period; fs=1/T :sample rate; s=2/T :sample rate

n

nT t t s

2 sk

S j k T

2 sk

S j k T

s jk t k

n

a e s(t) T

1 s jk t

n

eT

2 ( ) s jk t F se k -T

1

t T0

( ) s t

/2

/2

1 1( ) s

T jk t k T

a t e dt T T

0

( )S j

2T

2T

2T


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s c c cn n

x t x t s t x t t nT x nT t nT

ck

j Tn x nT e

( ) j t s cn

X j x nT t nT e dt

[ ] ( )c x n x nT

( ) j cn

j n X e x nT e DTFT

Representation of in terms of , jwe X j X s c X j

( ) j T X e

T rad

, rad/s

2 s T

1

s c sk X j X j k T

, rad/s


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( ) ( ) j j T X e X e

1 2( )

ck

j k X X jT T T

e

0,cif X jT

1( ) c

jthen X X jT T

e

1 s c sk

X j X j k T

DTFT

Representation of in terms of , jwe X j X s c X j

Continuous FT/ T


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Nyquist Sampling Theorem Let be a bandlimited signal

with . Then isuniquely determined by itssamples ,if

The frequency is commonly referred asthe Nyquist frequency .

The frequency is called the Nyquist rate .

t X c N c for j X 0 t X c

,2,1,0, nnT xn x c N s T

22

N

N 2


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N N s

N N s

aliasing frequencyT

2/ s

2

No aliasing

aliasing

k sc s k j X

T j X ))((

1)(

k c

T s j

T k j X T

j X e X

)/)2((1

|)()( /

frequency spectrum of idealsample signal


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4.3 Reconstruction of a Bandlimited Signalfrom its Samples

n

r r nT t hn xt x

T t T t

t hr sin

sin

n

t nT T x n

t nT T

T jr r e X j H j X

Gain: T


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4.4 Discrete-Time Processing ofContinuous-Time signals

nT xn x c

1 2 jw

ck

X e w k X j

T T T

nr T nT t T nT t

n yt y

sin

otherwiseT

eTY eY j H jY

T j

T j

r r

,0

,

jw H e


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C/D Converter Output of C/D Converter

nT xn x c

k c

jw

T k

T w

j X T

e X 21


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D/C Converter Output of D/C Converter

nr

T nT t

T nT t n yt y

sin

otherwise

T eTY

eY j H jY

T j

T jr r

,0

,

,0,

r T H j T

otherwise


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,

0,

j T c H e X j

T

T

4.4.1 Linear Time-InvariantDiscrete-Time Systems

j X c jwe X jY r jweY jwe H

jw jw jw

e X e H eY 1 2

j T j T r r

j T r c

k

Y j H j H e X ek

H j H e X j

T T


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Linear and Time-Invariant Linear and time-invariant system behavior

depends on two factors : First, the discrete-time system must be

linear and time invariant . Second, the input signal must be

bandlimited , and the sampling rate mustbe high enough to satisfy NyquistSampling Theorem .


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k c

T j

r

T jT jr r

T

k

j X T e H j H

e X e H j H jY

21

0 ,c If X j for T

T

T j X e H

jY c

T j

r

,0

,

j X j H jY ceff r

T

T e H

j H

T j

eff

,0

,

,0,

r T H j T

otherwise


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4.4.2 Impulse InvarianceGiven:

jw H eDesign:

,

0,

j T

c eff

H eT H j H j

T

ch n Th nT

h n ch nT

j X c jwe X jY r

jw

eY jwe H

impulse-invariant version of the continuous-time system

,c H j


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4.4.2 Impulse InvarianceTwo constraints

, j

c H e H j T

T j H that suchchosenisT

c ,0

1.2.

ch n T T h nThe discrete-time system is called an impulse-invariant version of the continuous-time system

ch n h nT ( ) 1 c j X j T T X e

ch n T T h n

( ) c j

X X jTe

/C T

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Chapter4 Sampling of Continuous-Time Signals