Digital Signal Processing, Fall 2006

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Digital Signal Processing, IV, Zheng-Hua Tan, 2006 1

Digital Signal Processing, Fall 2006

Zheng-Hua Tan

Department of Electronic Systems Aalborg University, Denmark

[email protected]

Lecture 4: Sampling and reconstruction

Course at a glance

Discrete-time signals and systems

Fourier-domain representation

DFT/FFT

System structures

Filter structures Filter design Filter

z-transform MM1

MM2

MM9,MM10 MM3

MM6

MM4

MM7 MM8

Sampling and

reconstruction MM5

System analysis System

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Part I: Periodic sampling

Periodic sampling

Frequency domain representation

Reconstruction

Changing the sampling rate using discrete- time processing

Periodic sampling

From continuous-time to discrete-time

Sampling period

Sampling frequency

) (t

x_c ^x^[n^]

∞

<

∞

−

= x nT n

n

x[ ] _c( ),

T T f T

s s

/ 2

/ 1

π

= Ω

=

(3)

Two stages

Mathematically

Impulse train modulator

Conversion of the impulse train to a sequence

∑

∞

−∞

=

∞

−∞

=

∞

−∞

=

−

=

−

=

−

=

n c n c

c s

n

nT t nT x

nT t t x

t s t x t x

nT t t

s

) ( ) (

) ( ) ( ) (

) ( ) (

δ δ δ

τ τ δ

τ t d

x t

x_c()=∫−^∞∞ _c( ) ( − )

∞

<

∞

−

=x nT n

n

x[ ] _c( ),

In practice?

Periodic sampling

Tow-stage representation

Strictly a mathematical representation that is convenient for gaining insight into sampling in both the time and frequency domains.

Physical implementation is different.

a continuous-time signal, an impulse train, zero except at nT

a discrete-time sequence, time normalization, no explicit information about sampling rate

Many-to-many Æ in general not invertible

) (t x_s

] [n x

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Part II: Frequency domain represent.

Periodic sampling

Reconstruction

Frequency-domain representation

From x_c(t)tox_s(t)

The Fourier transform of x_s(t)consists of periodic repetition of the Fourier transform of x_c(t).

∑

∞

−∞

=

∞

−∞

=

∞

−∞

=

∞

−∞

=

∞

−∞

=

−

=

Ω

− Ω

=

−

=

Ω Ω

= Ω

↔

=

Ω

− Ω

= Ω

↔

−

=

n c

k

s c

n c

c c

s

k

s n

nT t nT x

k j T X

nT t t x

j S j X j

X t

s t x t x

T k j S nT

t t

s

) ( ) (

)) (

1 ( ) ( ) (

) (

* ) 2 (

) 1 ( ) ( ) ( ) (

) 2 (

) ( ) ( ) (

s

δ δ

π π δ δ

The Fourier transform of a periodic impulse train is a periodic impulse train.

?

(5)

Frequency-domain

∑

∞

−∞

=

∞

−∞

=

Ω

− Ω

= Ω

Ω

− Ω

= Ω

k

s c

k

s

k j T X j X

T k j S

)) (

1 ( ) (

) 2 (

) (

s

π δ

N s N N

s−Ω >Ω Ω > Ω

Ω or 2

Recovery

Ideal lowpass filter with gain T and cutoff frequency

) ( ) ( )

( _r _s

r jΩ =H jΩ X jΩ

X

) ( ) (

) (

Ω

= Ω

Ω

− Ω

<

Ω

<

Ω

j X j

X_r _c

N s c N

c

(6)

Aliasing distortion

Due to the overlap among the copies of , due to

not recoverable by lowpass filtering )

c(jΩ X

N s ≤2Ω Ω

)

c(jΩ X

Aliasing – an example

t t

x_c()=cosΩ₀

t t

x_r( )=cosΩ₀

t t

x_r()=cos(Ω_s−Ω₀)

a.1

b.2 b.1

a.2

∑^∞

−∞

=

Ω

− Ω

= Ω

k

s

c j k

T X j

X 1 ( ( ))

)

s(

(7)

Nyquist sampling theorem

Given bandlimited signal with

Then is uniquely determined by its samples If

is called Nyquist frequency is called Nyquist rate

N c j

X ( Ω)=0, for |Ω|≥Ω

ΩN

) (t x_c

∞

<

∞

−

=x nT n

n

x[ ] _c( ),

N

s = T ≥ Ω

Ω 2π 2 )

(t x_c

ΩN

2

Fourier transform of x[n]

From to

By taking continuous-time Fourier transform of x_s(t)

By taking discrete-time Fourier transform of x[n]

) ( )

( ^j^ω

s j X e

X Ω

] [ )

(t xn

x_s

∑

^∞

−∞

=

Ω

= −

Ω

n

Tn j c

s j x nT e

X ( ) ( )

∑^∞

−∞

=

−

=

n c

s t x nT t nT

x () ( )δ( )

∞

<

∞

−

=x nT n

n

x[ ] _c( ),

∑

^∞

−∞

=

= − n

n j

j x n e

e

X( ^ω) ( ) ^ω X_s(jΩ)= X(e^j^ω)|_ω₌_Ω_T= X(e^j^Ω^T) ) )

( )

(

( X j^Ω ⁼

∫

₋^∞_∞x t e⁻^jΩ^tdt

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Fourier transform of x[n]

is simply a frequency-scaled version of with

retains a spacing between samples equal to the sampling period T while always has unity space.

∑

^∞

−∞

=

∞

−∞

=

= −

−

=

k c k

c j

T j k T X

T k j T

T X e

X 2 )

1 ( 2 ))

( 1 (

)

( ω ω π ω π

ΩT ω =

) (

| ) ( )

( ^j _T ^j ^T

s j X e X e

X Ω = ^ω _ω₌_Ω = ^Ω

∑^∞

−∞

=

Ω

− Ω

= Ω

k

s

c j k

T X j

X 1 ( ( ))

)

s(

) (jΩ X_s )

(e^j^ω X

] [n x )

(t x_s

Sampling and reconstruction of Sin Signal

) 4000 (

) ( )

(t ↔X jΩ =πδ Ω− π +πδ Ω+ π

x_c _c

) cos(

) ) 3 / 2 cos((

) 4000 cos(

) ( ] [

aliasing no

12000 /

2 6000

/ 1

4000 )

4000 cos(

) (

0 0

n n

nT nT

x n x

T T

t t

x

c

s c

ω π

π

π π

=

∴

=

= Ω

→

=

= Ω

→

=

T T

j X j

X e

X( ^j^ω)= _s( Ω)|_Ω₌_ω_/_T= _s( ω/ ) with normalizedfrequency ω=Ω

) 16000 cos(

) (

about How

t t

x_c = π

∑^∞

−∞

=

Ω

− Ω

= Ω

k

s

c j k

T X j

X 1 ( ( ))

)

s(

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Part III: Reconstruction

Periodic sampling

Reconstruction

Requirement for reconstruction

On the basis of the sampling theorem, samples represent the signal exactly when:

Bandlimited signal

Enough sampling frequency

+ knowledge of the sampling period Æ recover the signal

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Reconstruction steps

Given x[n] andT,the impulse train is

i.e. the nth sample is associated with the impulse at t=nT.

The impulse train is filtered by an ideal lowpass CT filter with impulse response

∑

^∞

−∞

=

∞

−∞

=

−

=

−

=

n n

c

s t x nT t nT xn t nT

x () ( )δ( ) [ ]δ( )

∑

^∞

−∞

=

−

=

n

r

r t x n h t nT

x () ( ) ( )

(2) (1)

) (t h_r

) ( ) ( )

( _r ^j ^T

r j H j X e

X Ω = Ω ^Ω

) ( Ω

↔H_r j

Ideal lowpass filter

T t

T t t

h_r

/ ) / ) sin(

( π

= π

s T

c /2 /

as frequncy cutoff

choose Commonly

π

= Ω

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Ideal lowpass filter interpolation

CT signal

Modulated impulse train

∑

^∞

−∞

= −

n

r t nT T

T nT n t

x t

x ( )/

) / ) ( ]sin(

[ )

( π

π

Ideal discrete-to-continuous-time converter

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discrete-to-continuous-time converter

From http://en.wikipedia.org/wiki/Digital-to-analog_converter.

Ideally sampled signal. Piecewise constant signal typical of a practical DAC output.

“Practical DACs do not output a sequence of dirac impulses (that, if ideally low-pass filtered, result in the original signal before sampling) but instead output a sequence of piecewise constant values or rectangular pulses”

Applications

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Part IV: Changing the sampling rate

Periodic sampling

Reconstruction

Downsampling

By reconstruction & re-sampling though not desirable Using DT processing only:

Sampling rate reduction by an integer factor – downsampling by “sampling” it

) ( ] [ ]

[n xnM x nMT

x_d = = _c

) ' ( ] [ '

) ( ] [

nT x n x

c c

=

(14)

Frequency domain

DT Fourier transform x_d[n]= x[nM]=x_c(nMT)

∞

<

∞

−

≤

∞

<

∞

− +

=i kM k i M r

r , , 0 1,

∑

∞

−∞

=

∞

−∞

=

∞

−∞

=

−

=

−

=

−

=

r c r

c j

d

k c j

MT r j MT MT X

T r j T T X

e X

T k j T T X

e X

2 )) ( 1 (

')) 2 ( ' ' (

) 1 (

2 )) ( 1 (

) (

π ω

ω ω

∑

∑ ∑

−

=

−

∞

−∞

=

−

=

∞

−∞

=

→

−

=

−

=

1

0

/ ) 2 ( /

) 2 (

1

0

) 1 (

) (

2 )) ( 2

1 ( ) (

Since

2 ))]

( 2 1 (

1 [ ) (

M

i

M i j j

d

k c M

i j

M

i k

c j

d

e M X e X

T k MT

i j MT T X

e X

MT i T

k j MT

T X e M

X

π ω ω

π π ω

Similar to the Eq. above!

Frequency domain - an example

Sampling results in copies at

Same, downsampling generates M copies of . with frequency scaled by M and shifted.

Aliasing can be avoided if is bandlimited

T n nΩ_s =2 π/

) (e^j^ω X

N N j

M e

X

ω π

π ω ω

ω

2 / 2 and

2

|

| , 0 ) (

≥

≤

=

) (e^j^ω X

This example N

s = Ω

Ω 4

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Frequency domain - an example

Downsampling factor is too large, causing aliasing, resulting in the need for DT ideal lowpass filter with cutoff frequency pi/M To avoid aliasing in

downsampling by a factor of M requires that

π ω_NM <

T' N N=Ω ω

A general downsampling system

The system is also called decimator

The process is called decimation

(16)

Increasing sampling rate – upsampling

DownsamplingÆ analogous to sampling a CT signal

UpsamplingÆ analogous to D/C conversion

,...

2 , , 0 ), / ( ] / [ ]

[n x n L x nT L n L L

x_i = = _c = ± ±

L T T nT

x n x

nT x n x

c i

c

=

' ), ' ( ] [

) ( ] [

Expander

∑^∞

−∞

=

−

=

⎩⎨

⎧ = ± ±

=

k e e

kL n k x n x

L L n L n n x x

] [ ] [ ] [

otherwise , 0

...

, , , 0 ], / ] [ [

δ

Fourier domain

Fourier transform of the output of expander

Which is s frequency scaled version, wis replaced by wLso

' ΩT ω=

) ( ]

[

) ] [ ] [ ( ) (

] [ ] [ ] [

L j k

Lk j

n j

n k

j e

k e

e X e

k x

e kL n k x e

X

kL n k x n x

ω ω

ω

ω δ

δ

=

−

=

−

=

∑

∑ ∑

∑

∞

−∞

=

−

∞ −

−∞

=

∞

−∞

=

∞

−∞

=

(17)

An example

DTFT of

System: interpolator Process: interpolation

) ( ]

[n x nT

x = _c

) ( )

( ^j ^j ^L

e e X e

X ^ω = ^ω

Summary

Periodic sampling

Reconstruction

Changing the sampling rate using discrete-

time processing

(18)

Course at a glance

Discrete-time signals and systems

Fourier-domain representation

DFT/FFT

System structures

Filter structures Filter design Filter

z-transform MM1

MM2

MM9,MM10 MM3

MM6

MM4

MM7 MM8

Sampling and

reconstruction MM5

System analysis System