ECE 2C, notes set 12: Sampling and Analog-Digital ConversionECE 2C, notes set 12: Sampling and...
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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
Transcript of ECE 2C, notes set 12: Sampling and Analog-Digital ConversionECE 2C, notes set 12: Sampling and...
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
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
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
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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2sin2)(
1
2cos2)(
1
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2sin22
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1cos2)(
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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
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highsample
high
ff
f
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s.25every samples 40KHz;at samplemust wesignal, kHz 20-DC aFor
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
class notes, M. Rodwell, copyrighted 2013
This is a Real Sample-Hold Gate
sequencedesign IC level-senior :about thislearn To
Daneshgar Saeid
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
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/
2/
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tjj
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tth
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
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
Vtvtvtv
SRS
impCS
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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000
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2/0
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impn
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imp
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Kimp
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VeV
etVetvc
ectv
tv
KtVtVtVtVtv
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
2
0
000
0000
DC
eVeVVeVeV
tVtVtVtv
tjtjtjtj
imp
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
S
impCS
class notes, M. Rodwell, copyrighted 2013
Spectrum of the Sampled Signal (2)
0
,2
)( and )( ofn convolutio)(
V
jVjVjV
impc
S
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
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
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
class notes, M. Rodwell, copyrighted 2013
