Sampling

1 Sampling: an introduction Filters for sampling Representation of a CT Signal by Its Samples: Sampling Converting to a sequenceThe Effect of Under-sampling: Aliasing Reconstruction of a Signal from Its Samples: Interpolation Examples Discrete-Time Processing of Continuous-Time Signals Sampling with Zero-Order Hold Sampling of Discrete-Time Signals (not in the exam) Summary

ELEC364: Signals And Systems Sampling and reconstruction

Dr. Aishy AmerConcordia UniversityElectrical and Computer Engineering

Figures, examples and some text in these course slides are taken from the following sources: A. Oppenheim, A.S. Willsky and S.H. Nawab, Signals and Systems, 2nd Edition, Prentice-Hall, 1997 Signals and Systems: Analysis Using Transform Methods and MATLAB, M. J. Roberts, McGraw Hill, 2004 Yao Wang, EE3414 --- Multimedia Communication Systems I, http://eeweb.poly.edu/~yao/EE3414/Slides by Robert Akl, http://www.cse.unt.edu/~rakl/

2 A continuous-time signal: music, speech, image, video, the fluctuation of air pressure at the entrance of the

ear canal Sampling is

An operation that transforms a CT signal into a DT signal (a sequence of numbers)

The DT signal gives the values of the CT signal read at intervals of T seconds

The reciprocal of the sampling interval is called sampling rate fs

Sampling: introduction

3Sampling: introduction

Sampling A crucial step in converting CT signals x(t), to DT

signals x[n] With x[n] we can take advantage of the

advanced discrete time systems technologies to process them

How do we perform sampling? Taking snap shots of x(t) every T second T: Sampling period x(nT), n=,-1,0,1,Samples


Key Question for Sampling: How do we determine T?

The frequency range of the signal. Can we reconstruct the original CT

signal x(t) from its samples, under certain conditions?

Nyquist sampling theorem

6Outline

Introduction to sampling Filters for sampling Representation of a CT Signal by Its Samples: Sampling Converting to a sequence The Effect of Under-sampling: Aliasingo Reconstruction of a Signal from Its Samples: Interpolationo Discrete-Time Processing of Continuous-Time Signalso Sampling with Zero-Order Holdo Sampling of Discrete-Time Signalso Summary

7Frequency content in signals

Importance of sinusoidal signals:An arbitrary signal can be expressed as a sum of many sinusoidal signals

with different frequencies, amplitudes and phases

Phase: how much the max. of the sinusoidal signal is shifted away from t=0Phase is translation in time

=pi

pi 2

)(21][ deeXnx njj

8Frequency content in signalsMusic notes are essentially sinusoids at different frequencies

9Frequency content in signals Sinusoidal signals have a distinct (unique) frequency An arbitrary signal x(t) does not have a unique frequency

x(t) can be decomposed into many sinusoidal signals with differentfrequencies, each with different magnitude and phase

Fourier transform: given a value of , the FT gives back a complex number It is magnitude and phase (translation in time) of the sinusoidal component at

that frequency

Fourier analysis: find frequency spectrum for signals

=pi

pi 2

)(21][ deeXnx njj

=

=

n

njj enxeX ][)(

)( jeX

10

Frequency content in signals

=

+

=

+

=

pi

pi

pi

pi 2

2

2

)(21][

DT PCT :TransformFourier DT Inverse

][)(

PCTDT :TransformFourier DT

deeXnx

enxeX

njj

n

njj

Note: the function e^j is periodic with N=2

11


12


=

j ez

Continuous-time analog signal

x(t)

Continuous-time analog signal

x(t)

Discrete-time analog sequence

x [n]

Discrete-time analog sequence

x [n]

Sample in timeSampling period = Ts

=2pif = Ts,scale amplitude by 1/Ts

Sample in frequency, = 2pin/N,N = Length

of sequence

ContinuousFourier Transform

X(f)

ContinuousFourier Transform

X(f)

f-

dt e x(t) ft2 j- pi

Discrete Fourier Transform

X(k)

Discrete Fourier Transform

X(k)

10

e [n]x 1

0 =n

Nnk2j-

Nk

N pi

Discrete-Time Fourier Transform

X()

Discrete-Time Fourier Transform

X()

pi20

e [n]x - =n

j-

n

LaplaceTransform

X(s)s = +j

LaplaceTransform

X(s)s = +j

s-

dt e x(t) st

z-TransformX(z)

z-TransformX(z)

z =n

n- z [n]x

s = j=2pif

C CC

C

C D

D

=

j ez

13


14

Frequency content in signals A constant signal:

only zero frequency component (DC component) A sinusoid signal:

contain only a single frequency component Periodic signals :

contain the fundamental frequency and harmonics : Line spectrum Slowly varying signal: contain low frequency only Fast varying signal: contain very high frequency Sharp transition signal: contain from low to high frequency Real signals such as music, speech,

contain both slowly varying and fast varying components, wide bandwidth

15

Why frequency representation? Clearly shows the frequency composition a signal Can change the magnitude of any frequency

component arbitrarily by a filtering operation A filter blocks some frequency content from a signal

Can shift the central frequency by modulation A core technique for communication, which uses

modulation to multiplex many signals into a singlecomposite signal, to be carried over the same physical medium

Processing of signals (e.g. speech and music)speech and music)

16

Filtering

Filters separate what is desired from what is not desired

A filter blocks some frequency content from a signal It may change the shape of the signal

Lowpass -> smoothing, noise removal Highpass -> edge/transition detection High emphasis -> edge enhancement

A filter can be seen as a transfer function H(f) Y(f) = H(f)X(f) or y[n]=h[n]*x[n]

17

Filtering

An ideal filter passes all signal power in its passband without distortion and completelyblocks signal power outside its passband

Distortion means that the signal shape is changed after the filtering

A distortion-less filter has an impulse response of the form h[n]= A (n-m) H( f ) = Ae(-j2fm) This is because a filter can multiply by a constant

or shift in time without distortion

18

Ideal Filters Lowpass -> smoothing, noise removal Highpass -> edge/transition detection Bandpass -> Retain only a certain frequency range

Bandstop -> most frequencies unaltered, attenuates those in a specific range to very low levels

19

Typical Filters

sinc functions

20

Filtering

All the impulse responses of ideal filters are sinc functions, or related functions, which are infinite in extent

Two-sided impulse responses, i.e., all ideal filter impulse responses begin before time, t = 0

This makes ideal filters non-causal Ideal filters cannot be physically

realized, but closely approximated

21

Low Pass Filtering(Remove high freq, make signal smoother)

22

High Pass Filtering(remove low freq, detect edges)

23

Filtering in Temporal Domain(Convolution)

24

Real filters

25

Noise filter Noise is present in most signals Noise is high frequency content If the noise band is much wider than the signal band a large

improvement in signal fidelity is possible

26

Outline

Introduction Filters for sampling Representation of a CT Signal by Its Samples: Sampling Converting to a sequence The Effect of Under-sampling: Aliasingo Reconstruction of a Signal from Its Samples: Interpolationo Exampleso Discrete-Time Processing of Continuous-Time Signalso Sampling with Zero-Order Holdo Sampling of Discrete-Time Signalso Summary

27

Observation: x1(t), x2(t), x3(t) have the same samples By doing sampling, we lose a lot of information (the

values of x(t) between the sampling points)

Fig. 7.1

Representation of a CT Signal by Its Samples: Sampling

28

Sampling methods

Impulse trainZero-order hold

29

Impulse-Train Sampling

o Use a periodic impulse train multiplied by the continuous-time signal x(t)

)()()( tptxtx p =

+

=

=

n

nTttp )()( sampling period

sampling function

(7.1)

(7.2)

Ts /2pi =

sampling frequency

30

x(t)p(t)

xp(t)

t 0

x(t)L L

0 t

)(tp1 T

t 0

xp(t)T x(0) x(T)

+

=

=

n

p nTtnTxtx )(][)(

Fig. 7.2

Impulse-Train Sampling

(7.3)

31

)( Tx

0

)(tx

t

0

)(tx p)(tx

)(tp

TT2T3 T3T2 t

1

)(tp

T

0

)(tx p

t

)0(x

32

Analysis of Sampling

The Fourier Transform X() of a DT signal x[n] is a function of the continuous variable , and it is periodic with period 2 Given a value of , the Fourier transform

gives back a complex number that can be interpreted as magnitude and phase (translation in time) of the sinusoidal component at that frequency

Sampling the CT signal x(t) with interval T, we get the DT signal x[n]=x[nT] which is a function of the discrete variable n

33

pi

djPjXjX p ))(()(21)( =

+

Ts /2pi =

sampling frequency

(7.4)

+

=

=

kskT

jP )(2)( pi (7.5)

Multiplication Property


+

=

=

ksp kjXTjX ))((

1)( (7.6)

+

=

=

n

p nTtnTxtx )(][)(

34


2 , no overlap between shifted replicas of ( )s M x j >

35

x(t) H(j) )(txr)(txp)( jXp

+

=

=

n

nTttp )()(

Reconstruction of x(t) from sampled signals

Fig 7.4 (a)

Exact Recovery by an Ideal Lowpass Filter (LPF):

36

)( jX

0 MM

1)( jXp

0 MM ss

T1

Ms 2>

)( jXr

0 MM

1)( jH

0 cc

T )( MscM

37

Let x(t) be a band-limited signal with X(j)=0 for

Then x(t) is uniquely determined by its samples x(nT), n=0,1, 2, , if

The Sampling Theorem

M >||

Ms 2> Tspi

2

=

38

Given the samples x(nT), we can reconstruct x(t) by generating a periodic impulse train in which successive impulses have amplitudes that are successive sample values

This impulse train is then processed through an ideal lowpass filter with gain T and cutoff frequency greater than and less than

The resulting output signal x(t) will exactly equal x(t)


Ms M

N yq u is t ra te = 2N yq u is t freq u en cy = N yq u is t ra te / 2 =

M

M

t 0

xp(t)T x(0) x(T)

39


A continuous-time signal x(t), whose spectral content is limited to frequencies smaller than wm (i.e., it is band-limited to )

can be perfectly recovered from its sampled version x[n], if the sampling rate is larger than twice the bandwidth (i.e., if )

Physical interpretation: must get at least two samples within each cycle

M >||

Ms 2>

40

Sampling: Applications Audio sampling:

Human hearing: 2020,000 Hz range Sampling rate is at

44.1 kHz (CD), 48 kHz (professional audio), or 96kHz The sampling rate is a consequence of the Nyquist theorem

Speech sampling: The energy of human speech: 5Hz - 4 kHz range Sampling rate: 8 kHz

(Used by nearly all telephony systems) Video sampling:

Standard-definition television (SDTV): 720x480 pixels (US) or 704x576 pixels(UE)

High-definition television (HDTV): 1440x1080 Sampling-rate conversion: Given a digital signal, change its sampling rate

Necessary for image display when original image size differs from the display size Necessary for converting speech/audio/image/video from one format to another Sometimes we reduce sample rate to reduce the data rate

Down-sampling: reduce the sampling rate Up-Sampling: increase the sampling rate

41

Outline


42

Estimate sampling frequency from x(t) Find the shortest ripple in x(t) In the shortest ripple, there should be

at least two samples The inverse of its length (Tmin) is

approximately the maximum frequency (fmax) of the signal

Need at least two samples in this interval (ripple), in order not to miss the rise and fall pattern

43

Converting to a sequence(C/D Conversion)

Fig. 7.21

44


Illustration of C/D Conversion in the Frequency-Domain

Fig. 7.22

45

CT

(7.18)


46


DT

(7.19, 7.20)

(7.21)

47

Outline


48

When undersampling2s M The Effect of Undersampling: Aliasing

Fig. 7.3(a, b, d)

49

o Aliasing: overlapping in frequency domain

The Effect of Undersampling: Aliasing

( ) ( )r

X j X j

50


)( jX

0 00

)( jX p

0 0 s)( 0 s

s

2s

ttx 0cos)( =

)(cos)( 0 txttxr ==

60s =

An example:

Fig. 7.15

51

)( jX p

0 0 s

)( 0 ss

2s

Aliasing

64

0s = )()cos()( 0 txttx sr =


52


Fig. 7.16

0 6s =

02

6s =

53

04

6s =


05

6s =

Fig. 7.16

54


Aliasing is the presence of unwanted components in the reconstructed signal

These components were not present when the original signal was sampled

In addition, some of the frequencies in the original signal may be lost in the reconstructed signal

Aliasing occurs because signal frequencies can overlap if the sampling frequency is too low

Frequencies "fold" around half the sampling frequency - which is why this frequency is often referred to as the folding frequency

Sometimes the highest frequency components of a signal are simply noise, or do not contain useful information

To prevent aliasing of these frequencies, we can filter out these components before sampling the signal using ANTI-Aliasing filter (a low-pass filter that filters out high frequency components and lets lower frequency components through)

55

Demo: Aliasing

The Effect of Under-sampling: AliasingRun applet under

http://www2.egr.uh.edu/~glover/applets/Sampling/Sampling.html

56

Outline

Introduction Filters for sampling Representation of a CT Signal by Its Samples: Sampling The Effect of Under-sampling: Aliasingo Reconstruction of a Signal from Its Samples: Interpolationo Exampleso Discrete-Time Processing of Continuous-Time Signalso Sampling with Zero-Order Holdo Sampling of Discrete-Time Signalso Summary

57

Reconstruction of a Signal from Its Samples: Interpolation Methods Interpolation: connecting samples using interpolation

kernels Band-limited (ideal) Interpolation:

Time-domain Interpretation of Reconstruction of Sampled Signals

Zero-Order Hold: e.g. scanned images

First-Order Hold: Linear interpolation: commonly used in plotting

Common practical pre-filter: averaging within one sampling interval

58

Band-limited Interpolation

+

=

==n

pr nTthnTxthtxtx )()()()()(

t

tTth

c

cc

pi

)sin()( =

(7.9)

(7.10) sinc function

59


)())(sin()()(

nTtnTtT

nTxtxc

c

n

cr

= +

=

pi

(7.11)

C.T. FT

60

)(tx

t)(tx p

t

)(txr

t

Fig. 7.10


Graphic Illustration of Time-domain Interpolation

Band limited signal x(t)

Impulse Train of Samples of x(t) Ideal band-limited interpolation in which the impulse train is replaced by the superposition of sincfunctions [(7.11)]

61

Zero-Order Hold InterpolationZero-order hold filter converts a DT signal to a CT signal by holding each sample value for one sample interval

62

Ideal interpolating filter

|)(| jH rT

s 2s

s2

s 0

Zero-order hold

Fig 7.11

Zero-Order Hold Interpolation

Transfer functions of the zero-order hold and of the ideal interpolating filter

63

First-Order Hold: Linear interpolation

Fig. 7.9

Fig. 7.13

Impulse-train sampling followed by convolution with a triangular impulse response

64

First-order versus zero-order hold filters First-order hold filter: the signal is

reconstructed as a piecewise linear approximation to the original signal that was sampled

Zero-order hold filter converts a discrete-time signal to a continuous-time signal by holding each sample value for one sample interval

65

Reconstruction of a sampled signal with a zero-order hold

66

Comparison of frequency responses of ideal lowpass and zero-order hold reconstruction filters

67

Reconstruction of a sampled signal with a first-order hold

68

Comparison of frequency responses of ideal lowpass, zero-order hold, and first-order hold reconstruction filters

69

Reconstruction of a sampled signal with ideal lowpass filter

70

Sampling and Interpolation of Images

Fig. 7.12

71

Sampling and Interpolation of Images

Fig. 7.12 & Fig 7.14

72

Outline

Introduction Filters for sampling Representation of a CT Signal by Its Samples: Sampling The Effect of Under-sampling: Aliasingo Reconstruction of a Signal from Its Samples: Interpolation o Exampleso Discrete-Time Processing of Continuous-Time Signalso Sampling with Zero-Order Holdo Sampling of Discrete-Time Signalso Summary

73

Example 1

For the following system

find the FT of the output signal if

Suppose

)(tx c

=

n

nTt )(

)(txpConversion to

a sequence

)(][ nTxnx c=

>

74

Example 1Solution: according to the diagram given:

TnjX

TjX

n

sscppi

2)),((1)( ==

=

=

==

n

cpj

TnjX

TTjXeX ))2((1)()( pi

75

Example 1

The Fourier transform of x[n] is

1/T

-2pi-wmT -2pi -2pi+wmT 0-wmT wmT 2pi-wmT 2pi 2pi+wmT

76

Problem 7.39

A signal ( ) is obtained through impulse train sampling of a sinusoidal signal ( ) whose frequenceis equal to half the sampling frequence . s

s( ) = cos( ) and ( ) ( ) ( ),2

x tpx t

x t t x t x nT t nTp

+ = 2

where

n

Ts

pi

+

=

=

Example 2

77

s(a) Find ( ) such that ( ) =cos( )cos( )+ ( )2

Using Trigonometric identities,

s s scos( )=cos( )cos( ) - sin( )sin( )2 2 2

s( ) -sin( )sin( ) (1)2

g t x t t g t

t t t

g t t

+

=

Example 2

78

(b) Show that ( ) = 0, for n=0, 1, 2,...

2By replacing with , and by in the equation (1), we gets2( ) = -sin( )sin( )= -sin( )sin( ), the right hand side of the2

equation is equal to

g nT

t nTT

g nT nT nT

pi

pi pi

zero for n=0, 1, 2,...

Example 2

79

(c) Using the results of the previous two parts, show that if ( ) is applied as the input to an ideal lowpass

sfilter with cutoff frequence , the resulting output is2

sy( ) =cos( )cos( ).2

x tp

t t

Example 2

80

F r o m p a r t s ( a ) a n d ( b ) , w e g e t ( ) ( ) ( )

s( ) c o s ( ) c o s ( ) + ( )2

s( )c o s ( ) c o s ( ) . 2

W h e n t h e s y s t e m i s p a s s e d t h r o u g h a l o w p a s s f i l t e r ,w e a r e p e r f o r m i n g

x t x n T t n Tpn

t n T n T g N Tn

t n T n Tn

+ =

=

+ =

= +

= =

a b a n d - l i m i t e d i n t e r p o l a t i o n , t h e

sr e s u l t i s t h e s i g n a l ( ) = c o s ( ) c o s ( ) .2

y t t

Example 2

81

Example 7.1

Consider Sinusoidal signal s( ) = cos( )2

x t t +

Suppose that this signal is sampled, using impulse sampling, at exactly twice the frequency of the sinusoid, i.e., at sampling frequency S

As shown in Problem 7.39, if this impulse-sampled signal is applied as the input to an ideal lowpass filter with cut frequency S/2., the resulting output is:

s( )=cos( )cos( )2r

x t t

Example 3

82

As a consequence, we see that perfect reconstruction of x(t) occurs only in the case in which the phase is zero (or an integer multiple of 2. Otherwise, the signal xr(t)does not equal x(t).

As an extreme example, consider the case in which

= - /2, so that

s( )=sin( )2

x t t

Example 3

83

The values of the signal at integer multiples of the sampling period 2 / S are zero.

Consequently. sampling at this rate produces a signal that is identically zero, and when this zero input is applied to the ideal lowpass filter, the resulting output xr(t) is also identically zero.

Fig. 7.17

Example 3

84

Example 4 A system uses the sampling frequency fs=20 kHz to

process audio signal that is frequency limited at 10 kHz, but the lowpass filter still allows frequencies up to 30 khz pass through even at small amplitudes. What signal will we get back from the samples?

Solution: for sampling rate fs=20 kHz, the Nyquist interval is [-10kHz, 10kHz] the audio frequency 0 10 kHz will be recovered as is The audio frequency from 10 20 kHz will be aliased into the frequency range 10 0 kHz, and the audio frequency from 20 30 kHz will be aliased into the frequency range 0 10 kHzThe resulting audio will be distorted due to the superposition of the 3 frequency bands

85

It is important to note that the sampling theorem explicitly requires that the sampling frequency be greater than twice the highest frequency in the signal, rather than greater than or equal to twice the highest frequency

The next example illustrates that sampling a sinusoidal signal at exactly twice its frequency (i.e., exactly two samples per cycle) is not sufficient

Example 5: Strobe Effect

86


Fig. 7.18

Stroboscopic effect: higher frequencies are reflected into lower frequencies

A disc rotating at a constant rate with a single radial line marked on the disc

The flashing strobe illuminates the disc for extremely brief time intervals at a periodic rate

The flashing strobe acts as a sampling system

87

When the strobe frequency is much higher than the rotational speed of the disc, the speed of rotation of the disc is perceived correctly

When the strobe frequency becomes less than twice the rotational frequency, the rotation appears to be at a lower frequency than is actually the case

If we track the position of a fixed line on the disc at successive samples, then when 0 < s

88

At one flash per revolution, corresponding to s = 0 , the radial line appears stationary (i.e., the rotational frequency of the disc and its harmonies have been aliased to zero frequency)

Similar effect observed in western movies The wheels of a stagecoach appear to be rotating more slowly

sometimes in the wrong direction. In this case, the sampling process corresponds to fact that moving pictures are a sequence of individual frames with a rate (usually between 18 and 24 frames per second) corresponding to the sampling frequency


89


Practical Application of Aliasing : Sampling Oscilloscope

Displaying on an oscilloscope screen waveforms having very shorttime structures, e.g. thousandths of nanoseconds. The idea is tosample the fast waveform x(t) once each period, at successively later points in successive periods

Fig. P7.38(a)

90

The increment should be an appropriately chosen sampling interval in relation to the bandwidth of x(t)

If the resulting impulse train is then passed through an appropriate interpolating lowpass filter, the output y(t) will be proportional to the original fast waveform slowed down or stretched out in time [i.e., y(t) is proportional to x(at), where a < 1 ]

Fig. P7.38(b)


91

Example 6

Consider the following sinusoidal signal with the fundamental frequency f = 4kHz:

g(t) = 5cos(2ft) = 5cos(8000t). The sinusoidal signal is sampled at a sampling rate

fs = 6000 samples/second and reconstructed withanideas low-pass filter (LPF) with the following transfer function:

H1(jw) = 1/6000 : |w|

92

Examples

95

Outline

Introduction Filters for sampling Representation of a CT Signal by Its Samples: Sampling The Effect of Under-sampling: Aliasingo Reconstruction of a Signal from Its Samples: Interpolationo Exampleso Discrete-Time Processing of Continuous-Time Signalso Sampling with Zero-Order Holdo Sampling of Discrete-Time Signalso Summary

96

DT LTI systems The impulse response h[n] completely characterizes

an LTI system

n Convolutio ][][][ ][][][ nhnxnyknhkxnyk

==

DT LTI systems are described mathematically by difference equations ][]2[]1[2][3 nxnynyny =++

97

DT Processing of CT signals using a DT system

Fig. 7.24

98

DT Processing of CT Signals

DT-S )(tyc)(txc C/DConversionD/C

Conversion)(][ nTxnx cd = )(][ nTyny cd =

T T

)(][ nTxnx cd = )(][ nTyny cd =Fig. 7.20

Reason for this: We can take advantage of the vast variety of general- or special-purpose discrete time signal processing devices

99

C/D Conversion

Two steps: sampling in time and quantization the amplitude

Sampling x[n] = x(nT) Quantization: map amplitude values

into a set of discrete values +-pQ(with an quantization interval) x[n] = Q(x[n])

100

C/D Conversion

Fig. 7.21

101

C/D ConversionIllustration of C/D Conversion in the Frequency-Domain

Fig. 7.22

102

CT

(7.18)

C/D Conversion

103

C/D Conversion

DT

(7.19, 7.20)

(7.21)

104

D/C Conversion

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

Fig. 7.23

105

DT Processing of CT Signals: Frequency-domain Illustration

Fig. 7.25

106

DT Processing of CT SignalsAssuming No Aliasing

(7.24)

(7.25)

107

DT Processing of CT Signals

( ) 2( )

0 2

j T sd

Cs

H eH j

( ) ( ) ( )C C CY j X j H j =

Fig. 7.26( 7.25)

108

Example: Problem 7.29Solution on page 15-16 in elec364/assign/Solution/SolutionELEC364Chap7.pdf

109

Example: Digital DifferentiatorConstruction of Band-limited Digital Differentiator

Desired: (7.27)

(7.28)

110

Example: Digital Differentiator( ) ( ), jdH e j T pi

=

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Sampling