signals and systems prof. h. sameti chapter 7: the concept and representation of periodic sampling...

Signals and SystemsProf. H. Sameti

Chapter 7:• The Concept and Representation of Periodic Sampling of a CT

Signal• Analysis of Sampling in the Frequency Domain• The Sampling Theorem - the Nyquist Rate• In the Time Domain: Interpolation• Undersampling and Aliasing• Review/Examples of Sampling/Aliasing• DT Processing of CT Signals

Book Chapter#: Section#

2

SAMPLING

We live in a continuous-time world: most of the signals we encounter are CT signals, e.g. x(t).

How do we convert them into DT signals x[n]? Sampling, taking snap shots of x(t) every T seconds T sampling period x [n] ≡x (nT), n = ..., -1, 0, 1, 2, ... regularly spaced samples

Applications and Examples Digital Processing of Signals Strobe Images in Newspapers Sampling Oscilloscope …

How do we perform sampling?

Computer Engineering Department, Signal and Systems


3

Why/When Would a Set of Samples Be Adequate?

Observation: Lots of signals have the same samples

By sampling we throw out lots of information (all values of x(t) between sampling points are lost).

Key Question for Sampling: Under what conditions can we reconstruct the original CT signal x(t)

from its samples?Computer Engineering Department, Signal and Systems


4

Impulse Sampling: Multiplying x(t) by the sampling function


( ) ( )n

p t t nT

( ) ( ) ( ) ( ) ( ) ( ) ( )p

n n

x t x t p t x t t nT x nT t nT


5

Analysis of Sampling in the Frequency Domain

Multiplication Property:


( ) ( ) ( )px t x t p t

1( ) ( )* ( )

22

( ) ( )

2

p

sk

s

X j X j P j

P j kT

T

Sampling Frequency

1( ) ( )* ( )p s

k

X j X j kT

1

( ( ))sk

X j kT

Important to note:

1s T


6

Illustration of sampling in the frequency-domain for a band-limited signal (X(jω)=0 for |ω| > ωM)

drawn assuming:


( )pX j

. . 2s M M

s Mi e

No overlap between shifted spectra


7

Reconstruction of x(t) from sampled signals

If there is no overlap between shifted spectra, a LPF can reproduce x(t) from xp(t)



8

Reconstruction of x(t) from sampled signals

Suppose x(t) is band-limited, so that :

Then x(t) is uniquely determined by its samples {x(nT)} if :

where ωs = 2π/T


X(jω)=0 for |ω| > ωM

ωs > 2ωM = The Nyquist rate


9

Observations on Sampling

In practice, we obviously don’t sample with impulses or implement ideal low-pass filters. One practical example: The

Zero-Order Hold



10

Observations (Continued)

Sampling is fundamentally a time varying operation, since we multiply x(t) with a time-varying function p(t). However,

is the identity system (which is TI) for band-limited x(t) satisfying the sampling theorem (ωs > 2ωM ).

What if ωs ≤ 2ωM? Something different: more later.



11

Time-Domain Interpretation of Reconstruction of Sampled Signals: Band-Limited Interpolation

The low-pass filter interpolates the samples assuming x(t) contains no energy at frequencies ≥ ωc


sin( ) ( )* ( ) , ( )

( ( ) ( ) )* ( )

sin[ ( )]( ) ( ) ( )

( )

cr p

n

c

n n

T tx t x t h t where h t

t

x nT t nT h t

T t nTx nT h t nT x nT

t nT


12


OriginalCT signal

After Sampling

After passing the LPF


13

Interpolation Methods

Band-limited Interpolation Zero-Order Hold First-Order Hold : Linear interpolation



14

Undersampling and Aliasing

When ωs ≤ 2ωM => Undersampling



15

Undersampling and Aliasing (continued)

Higher frequencies of x(t) are “folded back” and take on the “aliases” of lower frequencies

Note that at the sample times, xr (nT) = x (nT)Computer Engineering Department, Signal and Systems

Xr (jω) ≠ X(jω)

Distortion because of

aliasing

A Simple Example

X(t) = cos(ωot + )Φ

Picture would be Modified…

Demo: Sampling and reconstruction of cosωot

Sampling Review

Demo: Effect of aliasing on music.

Strobe Demo

> 0, strobed image moves forward, but at a slower paceΔ = 0, strobed image stillΔ < 0, strobed image moves Δ backward.Applications of the strobe effect (aliasing can be useful sometimes):—E.g., Sampling oscilloscope

DT Processing of Band-LimitedCT Signals

Why do this? —Inexpensive, versatile, and higher noise margin.How do we analyze this system? —We will need to do it in the frequency domain in both CT and DT —In order to avoid confusion about notations, specify —CT frequency variableω —DT frequency variable ( = )Ω Ω ωΤStep 1:Find the relation between xc(t) and xd[n], or Xc (j ) and ω Xd (ejΩ)

Time-Domain Interpretation of C/D Conversion

Note: Not full analog/digital (A/D) conversion – not quantizing the x[n] values

Frequency-Domain Interpretation of C/D Conversion

Illustration of C/D Conversion in the Frequency-Domain

D/C Conversion yd[n] →yc(t)Reverse of the process of C/D conversion

Again, = Ω ωΤ

Now the whole picture

Overall system is time-varying if sampling theorem is not satisfied It is LTI if the sampling theorem is satisfied, i.e. for bandlimited inputs xc (t),

with

When the input xc(t) is band-limited (X(j ) = 0 at | | > ω ω ωM)and the sampling theorem is satisfied (ωS> 2ωM), then

Frequency-Domain Illustration of DT Processing of CT Signals

Assuming No Aliasing

In practice, first specify the desired Hc(j ), then designω Hd(ejΩ).

Example:Digital Differentiator Applications: Edge Enhancement

Courtesy of Jason Oppenheim. Used

with permission.

Courtesy of Jason Oppenheim. Used with permission.

Construction of Digital Differentiator

Bandlimited Differentiator Desired:

Set Assume (Nyquist rate met)

Choice for Hd(ejΩ):

Band-Limited Digital Differentiator (continued)

Changing the Sampling Rate

30

Downsampling

n -3 -2 –1 0 1 2

)(nx

a

bc d

ef

n -1 0 1

)(nxdb

d f

2

Downsample by a factor of 2

• Downsample by a factor of N: Keep one sample, throw away (N-1) samples

• Advantage?

31Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

Downsampling)(txc

-3T -2T –T 0 T 2T 3T

:)(txc A continuous-time signal

Suppose we now sample at two rates:)(txc

)(txc C/D

T

)()( nTxnx c

(1))(txc C/D

MT

)(

)(

nMTx

nx

c

d (2)

•Q: Relationship between the DTFT’s of these two signals?


Downsampling

)(txc C/D

T

)()( nTxnx c(1)

)(txc C/D

MT

)(

)(

nMTx

nx

c

d (2)

kc T

kX

TnxDTFTX )

2(

1)}({)(

rcdd MT

rX

MTnxDTFTX )

2(

1)}({)(

• Change of variable: kMir 10

Mi

k


Downsampling

kc T

kX

TnxDTFTX )

2(

1)}({)(

rcdd MT

rX

MTnxDTFTX )

2(

1)}({)(


Mi

k

10:10:

;0

Mr

Mi

k

12:10:

;1

MMr

Mi

k

132:10:

;2

MMr

Mi

k


Downsampling

kc T

kX

TnxDTFTX )

2(

1)}({)(

rcdd MT

rX

MTnxDTFTX )

2(

1)}({)(


Mi

k

1

0)

22(

11)(

M

i kcd MT

i

T

k

MTX

TMX

1

0)

22(

11)(

M

i kcd T

k

MT

iX

TMX

)2

(M

iX

35

Downsampling

1

0)

22(

11)(

M

i kcd T

k

MT

iX

TMX

)2

(M

iX

)2

(1

)(1

0

M

id M

iX

MX

If M=2, )2

2(

2

1)(

1

0

id

iXX

)]2

2()

2([

2

1)(

XXX d

rcdd MT

rX

MTnxDTFTX )

2(

1)}({)(

kMir

10

Mi

k

36

Downsampling (example) )]2

()2

([2

1)( XXX d

)(X

2

222

)2

(

X

44

)2

2(

X

2 62 3

37

Downsampling (example) )]2

()2

([2

1)( XXX d

)2

(

X

44

)2

2(

X

2 62

44

)(dX

2 62

•What could go wrong here? 38

Downsampling with aliasing (illustration)


Downsampling (preventing aliasing)


Downsampling with aliasing (illustration)


Upsampling

• Application?

• It is the process of increasing the sampling rate by an integer factor.

)(txc C/D

T

)()( nTxnx c(1)

)(txc C/D

T/L

)()(L

nTxnx ci (2)


Upsampling

n -3 -2 –1 0 1 2

)(nxia

b c d

ef

)(nxbd f


Upsampling)(cX

c

)(X

c

22)(iX

L

22

L

44

Upsampling

Question: How can we obtain from ?)(nxi )(nx

• Proposed Solution:

)(nx L)(nxe

Low-pass filter withgain L and cut-off frequency

L

)(nxi

,...2,,00

)( LLnOtherwise

L

nx

nxe


Upsampling

n -3 -2 –1 0 1 2

)(nxia

b c d

ef

)(nxbd f

,...2,,00

)( LLnOtherwise

L

nx

nxe

n -3 -2 –1 0 1 2

)(nxed f


Upsampling•Q: Relationship between the DTFT’s of these three signals?

,...2,,00

)( LLnOtherwise

L

nx

nxe

k

e kLnkxnx )()()(

nj

n ke ekLnkxX

)()()(

k n

nje ekLnkxX )()()(

Lkje Shifting property: 47

Upsampling

k n

nje ekLnkxX )()()(

Lkje

k

Lkje ekxX )()(

k

kjekxnxDTFTX )()}({)(On the other hand:

(Eq.2)

(Eq.1)

(Eq.1) and (Eq.2) )()( LXX e


Upsampling)(X

22)(Xe

L

L

L

2

In order to get from we thus need an ideal low-pass filter.

)(eX)(iX

L

L

)(H

L

49

Upsampling)(Xe

L

L

L

2

L

L

)(H

L

)(Xi

L

L

2

50

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