Sampling 7

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7

Sampling

Dept. of Electronics Eng.

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Sample  Continuous-time signal : not unique

* For a Band-limited signal  unique restoration

7.1 Representation of a continuous-time signal by its samples

: The sampling theorem

Dept. of Electronics Eng.

DH26029 Signals and Systems

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) ( ) ( )

( t x t p t x _p 



^









n

nT t

t

p ( )  ( )



^









n

p

t x nT t nT

x ( ) ( )  ( )

 ( ) ( ) 

2 ) 1

(  

  X j P j j

X

_p

 

Continuous-time Fourier Transform

7.1.1 Impulse-train sampling

Dept. of Electronics Eng.

DH26029 Signals and Systems

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

^









k

s

p

X j k

j T

X 1 ( ( ))

)

(   

 ^









k

k s

j T

P 2 ( )

)

(     

 ( ) ( ) 

2 ) 1

(  

  X j P j

j

X

_p

 



^









n

nT t

t

p ( )  ( )

Continuous-time Fourier Transform

s T

 ²

Dept. of Electronics Eng.

DH26029 Signals and Systems

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T

n nT x t

x

j X t

x

s

M s

M

 











2 where

, 2

if

, , 1 , 0 ),

( samples its

by determined uniquely

is ) ( Then,

. for

0 ) ( with signal

limited -

band a

be ) ( Let

: Theorem Sampling















•

Dept. of Electronics Eng.

DH26029 Signals and Systems

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Exact recovery of a continuous-time signal from its samples

Dept. of Electronics Eng.

DH26029 Signals and Systems

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 

 



^



 )

^

^{2 sin(}  ^{/ 2 )}

(

^/2

0

e T j

H

^{j T}

7.1.2 Sampling with a zero-order hold

| ) (

|H₀ j



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Zero-order hold with a reconstruction filter



 



^

) 2 / sin(

2

) ) (

(

2 /

T j H j e

H

T j

r





 

 



^



 )

^

^{2 sin(}  ^{/ 2 )}

(

^/2

0

e T j

H

^{j T}

| ) (

|H₀ j

) ( ) ( )

( j  H

₀

j  H j 

H 

_r

Dept. of Electronics Eng.

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• Linear interpolation

• Interpolation by ideal low-pass filtering

 ^









n

r t x nT h t nT

x ( ) ( ) ( )

t t t T

h

c c c





 sin( ) )

( 

) (

)) (

) sin(

( )

( t nT

nT t

nT T x

t x

c c n

c

r 

  ^ 



 







The fitting of a continuous signal to a set of sample values

 

 



  2

s c

 

7.2 Reconstruction of a signal from its samples using interpolation

• One simple interpolation procedure: zero-order hold

• More complicated interpolation formulas: higher order polynomials

or other mathematical functions

Dept. of Electronics Eng.

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 

 



^



 )

^

^{2 sin(}  ^{/ 2 )}

(

^/2

0

e T j

H

^{j T}

Interpolation by ideal low-pass filtering Band-limited interpolation

Dept. of Electronics Eng.

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• Zero-order hold: discontinuous signal

• First-order hold (linear interpolation): continuous signal, discontinuous derivative

• Second-order hold: continuous signal, continuous first derivative, discontinuous second derivative

First-order hold

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First-order hold

2

2 /

) 2 / sin(

) 1

(  

 

 

  ^T

j T

H

Dept. of Electronics Eng.

DH26029 Signals and Systems

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) 2 ( 

_s

 

_M

7.3 The effect of undersampling : aliasing













j j

e e t

t x







: line dashed of

amplitude -

: line solid of

amplitude -

) cos(

) (

If

₀

 

 

1 ) ( f)

cos )

( 6 / 5 e)

cos )

( 6 / 4 d)

cos )

( 6 / 2 c)

cos )

( 6 / b)

cos )

(

0

0 0

0 0

0 0

0 0

0



























t x

t t

x

t t

x

t t

x

t t

x t t

x

r s

s r

s

s r

s

r s

r s



































    













t t

x

_{r s}

s

) (

cos )

( then

, 6 / 4 if

0 0

Dept. of Electronics Eng.

DH26029 Signals and Systems

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Dept. of Electronics Eng.

DH26029 Signals and Systems

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• Examples of the effect of undersampling (aliasing)

 The wheels of a stagecoach in Western movies

 Strobe effect (p. 533, Fig. 7.18)

• Note) Irrespecti ve of aliasing, x

_r

( nT )  x ( nT ), n  0 ,  1 ,  2 , 

Dept. of Electronics Eng.

DH26029 Signals and Systems

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) (

]

[ n x nT

x _d  _c y _d [ n ]  y _c ( nT )

< Notation for continuous-to-discrete-time conversion and discrete-to-continuous-time conversion>

7.4 Discrete-time processing of continuous-time signals

Dept. of Electronics Eng.

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Sampling with a periodic impulse train followed by conversion to a discrete-time sequence

Normalization in time (scaling in time)

cf.) Fig. 1.12 at p.9

Dept. of Electronics Eng.

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 ^









n

c

p t x nT t nT

x ( ) ( )  ( )  ^





  n

nT j c

p j x nT e

X (  ) ( ) ^



 ^







 









  

n

n j c

n

n j d

j

d e x n e x nT e

X ( ) [ ] ( )

variable frequency

time -

continuous

 :

variable frequency

time -

discrete

 :

 T



) 

/ (

)

( e X j T

X _d ^{j }  _p 

, )) (

1 ( )

(

Since 

^









k

s c

p

X j k

j T

X   



^







  

k

c j

d

X j k T

e T

X 1 ( ( 2 ) / )

)

( 

CTFT

] [n x

_d

DTFT

Compare

Dept. of Electronics Eng.

DH26029 Signals and Systems

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 ^





   

k

c j

d X j k T

e T

X 1 ( ( 2 ) / )

)

( 