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class notes, M. Rodwell, copyrighted 2013
ECE 2C, notes set 12: Sampling and Analog-Digital Conversion
Mark Rodwell
University of California, Santa Barbara
[email protected] 805-893-3244, 805-893-3262 fax
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class notes, M. Rodwell, copyrighted 2013
Goals of this note set
coversion digital-analog
and
signals time-sampled totime-continuous from conversion
withinvolved
ipsrelationshfrequency and Temporal
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class notes, M. Rodwell, copyrighted 2013
Sampling: Turning Signals into Numbers
? drive hardour on store must we s#many How
). 20kHz-20Hz~ :ears (good
20kHz-DC as hearing of rangeour eapproximat usLet
long. seconds) 40 minutes, (16 seconds 1000 is music The
:computerour on file music a store want toWe
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class notes, M. Rodwell, copyrighted 2013
Remember the Fourier Sine/Cosine Series:
)( tscoefficien lcosinsoida 012 need also We
)( tscoefficien sinsoidal 012)Hz(20kHz/10 need We
20kHz ..., , Hz012 Hz,011 Hz,0at wavescosine :signalOur
20kHz ..., , Hz012 Hz,011 Hz,0at wavessine :signalOur
long. seconds 10 is rmOur wavefo
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1
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class notes, M. Rodwell, copyrighted 2013
Nyquist's Sampling Theorem: 1st viewpoint
signal thedescribe to
nts)(measureme numbers 2 need Then we
. and DCbetween limited is bandwidth its If
.duration timeof signal a have weIf
TfN
f
T
highsample
high
signal thedescribe to
unit timeper samples 2 need Then we
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signal a have weIf
-So-
highsample
high
ff
f
.componentsFourier countingby proven theoremsNyquist'
s.25every samples 40KHz;at samplemust wesignal, kHz 20-DC aFor
![Page 6: ECE 2C, notes set 12: Sampling and Analog-Digital ConversionECE 2C, notes set 12: Sampling and Analog-Digital Conversion Mark Rodwell University of California, Santa Barbara rodwell@ece.ucsb.edu](https://reader033.vdocuments.net/reader033/viewer/2022060912/60a6ff34459e7b6d080aba8e/html5/thumbnails/6.jpg)
class notes, M. Rodwell, copyrighted 2013
Sampling in the time domain.
)( signal time-continuous analog some start with We tVC
)( signal producing gate, hold-sample ait with sample then We , tV RS
gate hold-sample idealizedan is This
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class notes, M. Rodwell, copyrighted 2013
This is a Real Sample-Hold Gate
sequencedesign IC level-senior :about thislearn To
Daneshgar Saeid
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class notes, M. Rodwell, copyrighted 2013
Real vs Ideal Sampling (1)
pulses square :Sampling Real
easier)math the(makes impulses :Sampling Ideal
? theserelate wedo How
![Page 9: ECE 2C, notes set 12: Sampling and Analog-Digital ConversionECE 2C, notes set 12: Sampling and Analog-Digital Conversion Mark Rodwell University of California, Santa Barbara rodwell@ece.ucsb.edu](https://reader033.vdocuments.net/reader033/viewer/2022060912/60a6ff34459e7b6d080aba8e/html5/thumbnails/9.jpg)
class notes, M. Rodwell, copyrighted 2013
Real vs Ideal Sampling (2)
.duration of pulse square a is response impulse whose
),(filter a have weSuppose
th
)2/(
)2/sin(1)(
)exp(1
)exp(1
)exp()()(
:FunctionTransfer
)/(rect1
)( :response Impulse
2/2/
2/
2/
2/
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tth
![Page 10: ECE 2C, notes set 12: Sampling and Analog-Digital ConversionECE 2C, notes set 12: Sampling and Analog-Digital Conversion Mark Rodwell University of California, Santa Barbara rodwell@ece.ucsb.edu](https://reader033.vdocuments.net/reader033/viewer/2022060912/60a6ff34459e7b6d080aba8e/html5/thumbnails/10.jpg)
class notes, M. Rodwell, copyrighted 2013
Real vs Ideal Sampling (3)
:Filter
)2/(
)2/sin()( :FunctionTransfer
)/(rect1
)( :response Impulse
jH
tth
http://en.wikipedia.org/wiki/File:Si_sinc.svg
)2/(
)2/sin( reponsefrequency hasFilter
pulses. square into impulses convertsFilter
![Page 11: ECE 2C, notes set 12: Sampling and Analog-Digital ConversionECE 2C, notes set 12: Sampling and Analog-Digital Conversion Mark Rodwell University of California, Santa Barbara rodwell@ece.ucsb.edu](https://reader033.vdocuments.net/reader033/viewer/2022060912/60a6ff34459e7b6d080aba8e/html5/thumbnails/11.jpg)
class notes, M. Rodwell, copyrighted 2013
Real vs Ideal Sampling (4)
)()()( :signal sampled filter the then We
/)()()( :impulses of sequenceagainst signalmultiply first We
,,
0,
jHjVjV
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SRS
impCS
![Page 12: ECE 2C, notes set 12: Sampling and Analog-Digital ConversionECE 2C, notes set 12: Sampling and Analog-Digital Conversion Mark Rodwell University of California, Santa Barbara rodwell@ece.ucsb.edu](https://reader033.vdocuments.net/reader033/viewer/2022060912/60a6ff34459e7b6d080aba8e/html5/thumbnails/12.jpg)
class notes, M. Rodwell, copyrighted 2013
Spectrum of the impulse train (1)
...... )(
:So
)(1
)(1
where
/2 where )(
:seriesFourier a as thiscan write we, period with periodic is )( Because
)( ...)2()()()(
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nimp
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![Page 13: ECE 2C, notes set 12: Sampling and Analog-Digital ConversionECE 2C, notes set 12: Sampling and Analog-Digital Conversion Mark Rodwell University of California, Santa Barbara rodwell@ece.ucsb.edu](https://reader033.vdocuments.net/reader033/viewer/2022060912/60a6ff34459e7b6d080aba8e/html5/thumbnails/13.jpg)
class notes, M. Rodwell, copyrighted 2013
Spectrum of the impulse train (2)
...,3 ,2 , ,, ,2 ,3, ...at s with tonespectrum a is This
......
...)2()()()(
000000
2
0000
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000
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eVeVVeVeV
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![Page 14: ECE 2C, notes set 12: Sampling and Analog-Digital ConversionECE 2C, notes set 12: Sampling and Analog-Digital Conversion Mark Rodwell University of California, Santa Barbara rodwell@ece.ucsb.edu](https://reader033.vdocuments.net/reader033/viewer/2022060912/60a6ff34459e7b6d080aba8e/html5/thumbnails/14.jpg)
class notes, M. Rodwell, copyrighted 2013
Spectrum of the Sampled Signal (1)
trainimpulse theof spectrum and
input theof spectrum theofn convolutio the
thereforeis signal sampled theof ansformFourier tr The
)'()'()'(2
1
2
)( and )( ofn convolutio
2
)(*)()(
/)()()(
0
00
,
0,
jdjjVjVV
V
jVjV
V
jVjVjV
Vtvtvtv
impc
impcimpc
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class notes, M. Rodwell, copyrighted 2013
Spectrum of the Sampled Signal (2)
0
,2
)( and )( ofn convolutio)(
V
jVjVjV
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overlap not to are spectra theif 2/ than less bandwidth havemust )( signal The 0H tVC
criterion samplingNyquist theagain, is, This
overlap. not to are spectra theif 2or 2 0 highsamplehighsample ff
![Page 16: ECE 2C, notes set 12: Sampling and Analog-Digital ConversionECE 2C, notes set 12: Sampling and Analog-Digital Conversion Mark Rodwell University of California, Santa Barbara rodwell@ece.ucsb.edu](https://reader033.vdocuments.net/reader033/viewer/2022060912/60a6ff34459e7b6d080aba8e/html5/thumbnails/16.jpg)
class notes, M. Rodwell, copyrighted 2013
Spectral Overlap (Aliasing) if sampling is too slow
criterion samplingNyquist theagain, is, This
overlap. not to are spectra theif 2 haveMust highsample ff
overlap Spectral
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class notes, M. Rodwell, copyrighted 2013
Analog-Digital Conversion
precisiongreater :bits More
sample.each oftion representabinary bit -Nan generates This
coversion. digital-analogby followed then is Sampling
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class notes, M. Rodwell, copyrighted 2013