ECE 145B / 218B, notes 2: Mathematics of Electrical Noise · Thermal Noise as a Random Variable R...
Transcript of ECE 145B / 218B, notes 2: Mathematics of Electrical Noise · Thermal Noise as a Random Variable R...
class notes, M. Rodwell, copyrighted 2012
ECE 145B / 218B, notes 2: Mathematics of Electrical Noise
Mark Rodwell
University of California, Santa Barbara
[email protected] 805-893-3244, 805-893-3262 fax
class notes, M. Rodwell, copyrighted 2012
Strategy
ive.comprehens isbook sder Zeil'Van .literature in theor
web),(on the notes class noisemy in found becan detail More
rates.error ns,correlatio signal
functions,n correlatio densities, spectral SNR,
ofn calculatiocorrect for sufficient backround give
:Strategy
detail.in subject thisdevelop toclass in this not time is There
class notes, M. Rodwell, copyrighted 2012
Topics
Math:
distributions, random variables, expectations,
pairs of RV, joint distributions, covariance and correlations.
Random processes, stationarity, ergodicity, correlation functions,
autocorrelation function, power spectral density.
Noise models of devices:
thermal and shot noise.
Models of resistors, diodes, transitors, antennas.
Circuit noise analysis:
network representation. Solution.
Total output noise. Total input noise. 2 generator model.
En/In model. Noise figure, noise temperature.
Signal / noise ratio.
class notes, M. Rodwell, copyrighted 2012
random variables
class notes, M. Rodwell, copyrighted 2012
The first step: random Variables
function. ondistributiy probabilit theis )(
)(}{
is and between lies y that probabilit The
. valueparticulara on takes variablerandoma ,experiment an During
2
1
21
21
xf
dxxfxxxP
xxx
xX
X
x
x
X
fX(x)
xx1
x2
class notes, M. Rodwell, copyrighted 2012
Example: The Gaussian Distribution
Gaussian. theofthat
toclose onsdistributiy probabilit have effects smallmany of sum thefrom
arising processes random physical rem*,limit theo central* theof Because
).( deviation standard theand )( mean eshortly th define willWe
2exp
2
1)(
:ondistributi Gaussian The
2
2
2
2
x
xx
X
x
xxxf
fX(x)
xx
~2x
class notes, M. Rodwell, copyrighted 2012
Mean values and expectations
dxxfxXEX
X
dxxxfXEXX
dxxfxgxgE
X
X
X
)(
of valueExpected
)(
X of Value Mean
)()()(
X variablerandom theof g(X) functiona of nExpectatio
222
2
class notes, M. Rodwell, copyrighted 2012
Variance
variance theofroot square the
simply is X of deviation standard The
)(
valueaverage its from
deviation square-mean-root its is X of varianceThe
2222
2
x
XX
x
dxxfxxxXExX
class notes, M. Rodwell, copyrighted 2012
Returning to the Gaussian Distribution
fX(x)
xx
~2x
clear. benow should
2exp
2
1)(
:ondistributi Gaussian thedescribing notation The
2
2
2
xx
X
xxxf
class notes, M. Rodwell, copyrighted 2012
Variance vs Expectation of the Square
n.expectatio theof square theminus
square theof nexpectatio theis varianceThe
2
2
2
222
22
22
22
22
xX
xxxX
xXxX
xxXX
xXxXxX
X
X
class notes, M. Rodwell, copyrighted 2012
Pairs of Random Variables
D
C
B
A
XY
XY
dxdyyxfDyCBxAP
yxf
),(} and {
),( ondistributijoint by the described isbehavior joint Their
y.andx valuesparticular specific on takes
Yand X variablesrandom ofpair a ,experiment an In
. variablesrandom of pairs understandfirst must we
processes, random understand To
class notes, M. Rodwell, copyrighted 2012
Pairs of Random Variables
D
C
Y
D
C
XY
B
A
X
B
A
XY
dyyf
dxdyyxfDyCP
dxxf
dxdyyxfBxAP
)(
),(} {
:for Ysimilarly and
)(
),(} {
defined be alsomust onsdistributi Marginal
class notes, M. Rodwell, copyrighted 2012
Statistical Independence
expected.generally not is This
t.independenlly statistica be tosaid are variablesthe
, )()(),(
wherecase theIn
yfxfyxf YXXY
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Expectations of a pair of random variables
. and for similarly ...and
)( ),(
ofn Expectatio
)(),(
: ofn Expectatio
),(),(),(
is Y and Y variablesrandom theof Y)g(X,function a ofn expectatio The
2
2222
2
YY
dxxfxdxdyyxfxXXE
X
dxxxfdxdyyxxfxXE
X
dxdyyxfyxgyxgE
XXY
XXY
XY
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Correlation between random variables
values.mean zero have or Y X
either if same theare covariance and ncorrelatio that Note
is Yand X of covariance The
),(•
is Yand X of ncorrelatio The
yxR
yxyXYxXYEyYxXEC
dxdyyxfxyXYER
XY
XY
XYXY
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Correlation versus Covariance
careful. Be
ons.distributi lconditiona
calculate e.g. we whenreturn can valuesmean nonzero But,
ably.interchang terms two theuse to
analysis noisecircuit in common thereforeisIt
.covariance toequal thenis nCorrelatio
mean. zero have then variablesrandom The
component. varying- time thefrom bias) (DC valuemean the
separateusually wecurrents, and voltages with workingare weWhen
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Correlation Coefficient
.covariance and ncorrelatio between
gy terminoloin confusion (standard) theNote
/
is Yand X oft coefficien ncorrelatio The
YXXYXY C
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Sum of TWO Random Variables
n.correlatio of role theemphasizes This
2
means zero have both and If
2
2)(
: variablesrandom twoof Sum
222
22
2222
XY
XY
CYEXEZE
YX
RYEXE
YXYXEYXEZE
YXZ
class notes, M. Rodwell, copyrighted 2012
Pairs of Jointly Gaussian Random Variables
jiiii
n
YYXXXY
XYYX
XY
xxExxEx
XXX
yyyyxxxx
yxf
scovariance and , variances, means
ofset a by specified is which
),,,(
vector random GaussianJointly a have can wegeneral, In
. variablesof #larger a toextended be can definition This
)())(()(
)1(2
1exp
12
1),(
:GaussianJointly are Yand X If
21
2
2
2
2
2
2
class notes, M. Rodwell, copyrighted 2012
Linear Operations on JGRV's
number.any of JGRVsfor holdsresult The
function. Gaussiana produces
functions Gaussian 2 of nconvolutio
because arisesresult theproof; without stated is This
Gaussian.Jointly also are and Then
and
define weif and Gaussian,Jointly are Yand X If
WV
dYcXWbYaXV
class notes, M. Rodwell, copyrighted 2012
Probability distribution after a Linear Operation on JGRV's
2
2
2
2
2
2
22
22
22222222
22222222
)())(()(
)1(2
1exp
12
1),(
. and of ondistributijoint thecalculatenow can We
))(()(
)(
))((
)(2
)(2
and
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wwwwvvvv
wvf
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YdXcYbXaYEbdXYEbcadXacE
WVbdYXYbcadacXE
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YdXcXYEcdYEdXEcWWE
YbXaXYEabYEbXEaVVE
YdXcWYbXabYaXEVEV
tedious details
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Why are JGRV's Important ?
circuits).(linear
systemslinear in npropagatio noise of nscalculatio simplifies vastly This
found.simply be always can functions ondistributi n,informatio thisWith
. variancesand ns,correlatio means, of
track keep tosufficient isit ,operationslinear tosubjected sJGRV' With
:conclusionclear a is but there tedious wasslidelast theon math The
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Uncorrelated Variables.
ceindependenimply does eduncorrelat s,JGRV'For
ce.independenimply not does ncorrelatio Zero
n.correlatio zero implies ceIndependen
)()(),(
:tindependenlly Statistica
0
:edUncorrelat
yfxfyxf
C
YXXY
XY
class notes, M. Rodwell, copyrighted 2012
Summing of Noise (Random) Voltages
V1
V2
R
add. do voltagesnoise random theof values timeousinstantane The
included. bemust termoncorrellatia -
-addnot do generators random two theof powers noise The
12
11
2111
12
11
12
11
21
)(1
P variablerandoma isresistor thein dissipatedpower The
resistor R the toapplied are voltagesTwo
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221
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1
2
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1
2
2
2
1
2
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21
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VR
VVR
VR
RV
RV
R
VRR
VR
VR
CR
VR
VVVVR
VVR
PPE
VVVV
VV
VV
class notes, M. Rodwell, copyrighted 2012
Shot Noise as a Random Variable
count. theof valuemean theapproachescount theof varianceThe
,1 and ,1 ,1 supposeNow
)( and
, photons received of # thecalling --- so t,independenlly statistica
is each of sion transmis, them)of ( photonsmany sendnow weIf
so and
. photons received of # thecall and photon, one Send
.y probabilit ion transmisshasfiber The
2
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111
1
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1
N
MppM
ppMMMpNEMNE
N
M
ppNNEpNEpNNE
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p
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NN
N
class notes, M. Rodwell, copyrighted 2012
Thermal Noise as a Random Variable
R
kT/C. variancehas voltagenoise The
/
2/2/
2/
hence kT/2,energy mean has T tureat tempera
systema of freedom of degreet independenany amics, thermodynFrom
resistor. via theroom the
withmequilibriu thermalsestablisheit :power no dissipate can C
heat. of form thein room thehenergy wit exchange can R
T tureat tempera room) (a warm reservoir""a withmequilibriu in isresistor The
. resistor Ra toconnected is Ccapacitor A
2
2
CkTV
kTCV
kTE
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random processes
class notes, M. Rodwell, copyrighted 2012
Random Processes
v(t)is outcome particular The
).( is process random The
time.of function randomour is This
random.at sheet oneout Pick
space. sampley probabilit the
called is can garbage This
can. garbagea into Put them
time. vs. voltageof functions of
paper, of sheets separate on graphs, ofset a Draw
tV
class notes, M. Rodwell, copyrighted 2012
dttvgtvgA
tvdtvftvgtvgE
t
gg
gdxxfxgxgE
V
X
))(())((
:over time function
theof average an define also can We
))(())(())(())((
: timeparticular someat space sample
over the average an define can we
,definition process randomour With
space. sample over the is average the where, of * valueaverage* theis
)()()(
X variablerandoma of g(X) functiona of nexpectatio theof definion theRecall
1111
1
Time Averages vs. Sample Space Averages
class notes, M. Rodwell, copyrighted 2012
space" sample over the variationrandom"
toequivalent
time" with variationrandom"
made have wesense, some In
))(())((
space sample lstatistica over the averages
toequal over time averages
has process random ErgodicAn
1 tvgAtvgE
Ergodic Random Processes
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Gaussian.jointly be not)might (or might They
on.distributiy probabilitjoint some have )( and )(
).( and )( valueshas )( process random the
and at times samples timeWith
21
21
21
tVtV
tVtVtV
tt
Time Samples of Random Processes
v(t)
t
v(t1 )
v(t2 )
t1
t2
class notes, M. Rodwell, copyrighted 2012
Random Waveforms are Random Vectors
v(t)
tt1
t2
tn
class. thisof scope thebeyondmostly is This
geometry. and analysisvector
using signals random analyze thuscan We
vector.randoma into process randomour convert we
,... samples timespaced-regularlypick weif and
d,bandlimite is signal randoma if
theorem,sampling sNyquist' Using
1 ntt
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)()(orderslower
)(),()(),(
:tystationariorder 2
orderslower ..and
)(),...,(),()(),...,(),(
:tystationariorder N
time.ithnot vary w do process stationarya of statistics The
11
2121
nd
2121
th
tVfEtVfE
tVtVfEtVtVfE
tVtVtVfEtVtVtVfE nn
Stationary Random Processes
v(t)
t
v(t1 )
v(t2 )
t1
t2
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Restrictions on the random processes we consider
We will make following restrictions to make analysis tractable:
The process will be Ergodic.
The process will be stationary to any order: all statistical properties are independent of
time. Many common processes are not stationary, including integrated white noise and 1/f
noise.
The process will be Jointly Gaussian. This means that if the values of a random process
X(t) are sampled at times t1, t2, etc, to form random variables X1=X(t1), etc, then X1,X2,
etc. are a jointly Gaussian random variable.
In nature, many random processes result from the sum of a vast number of small
underlying random processses. From the central limit theorem, such processes can
frequently be expected to be Jointly Gaussian.
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Variation of a random process with time
For the random process X(t), look at X1=X(t1) and X2=X(t2).
21212121 ),(•2121
dxdxxxfxxXXER XXXX
To compute this we need to know the joint probability distribution. We have assumed a
Gaussian process. The above is called the Autocorrellation function. IF the process is
stationary, it is a function only of (t1-t2)=tau, and hence
RXX E X t X t
this is the autocorrellation function. It describes how rapidly a random voltage varies
with time….
PLEASE recall we are assuming zero-mean random processes (DC bias subtracted). Thus
the autocorrellation and the auto-covariance are the same
class notes, M. Rodwell, copyrighted 2012
Variation of a random process with time
Note that RXX 0 E X t X t X2
gives the variance of the random process.
The autocorrelation function gives us variance of the random process and the correlation
between its values for two moments in time. If the process is Gaussian, this is enough to
completely describe the process.
)(xxR
)(xxR
Narrow autocorrelation:
Fast variation
Broad autocorrelation
Slow variation
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Autocorrelation is an Estimate of the Variation with Time
If random variables X and Y are Jointly Gaussian, and have zero mean,
then knowledge of the value y of the outome of Y results in a best estimate of X as
follows:
E X Y y X Y y RXY
Y2 y
"The expected value of the random variable X, given that the random variable Y has value
y is ..."
Hence, the autocorrellation function tells us the degree to which the signal at time t is
related to the signal at time t
A narrow autocorrelation is indicative of a quickly-varying random process
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Power spectral densities
The autocorrellation function describes how a random process evolves with time.
Find its Fourier transform:
djRS XXXX )exp(
This is called the power spectral density of the signal.
Remembering the usual Fourier transform relationships, if the power spectrum is broad,
the autocorrellation function is narrow, and the signal varies rapidly--it has content at high
frequencies, and the voltages of any two points are strongly related only if the two points
are close together in time.
If the power spectrum is narrow, the autocorrellation function is broad, and the signal
varies slowly--it has content only at low frequencies and the voltages of any two points are
strongly related unless if the two points are broadly separated in time.
class notes, M. Rodwell, copyrighted 2012
Power spectral densities
Rxx(tau)
tau
Rxx(tau)
tau
time
time
x(t)
x(t)
Sxx(omega)
omega
Sxx(omega)
omega
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Power Spectral Densities
density" spectralpower " term,for the ionjustificat theis This
power. theus give lldensity wi spectralpower the
gintegratin thenprocess, theinpower thecalled is if So,
2
1)0(
ly,specifical
)exp(2
1
thatso holds, transforminverse The
)exp(
function ationautocorrel theof ransform Fourier tthe
isdensity spectralpower that theRecall
2
2
X
XXXXX
XXXX
XXXX
dSR
djSR
djRS
class notes, M. Rodwell, copyrighted 2012
Correlated Random Processes
djSR
djRS
tYtXER
XYXY
XYXY
XY
)exp(2
1 thereforeand
)exp(
:follows asdensity spectral-crossa have They will
processes theof function oncorrellati-cross theDefine
Y(t).and X(t) processes random woConsider t
related.lly statistica be can processes Two
class notes, M. Rodwell, copyrighted 2012
Single-Sided Hz-based Spectral Densities
dfjRjfS
dffjjfStXtXER
djRjS
djjStXtXER
XXXX
XXXX
XXXX
XXXX
)2exp(2~
)2exp(~
2
1)()(
DensitiesSpectral based- HzSided-Single
)exp(
)exp(2
1)()(
DensitiesSpectral Sided-Double
class notes, M. Rodwell, copyrighted 2012
Single-Sided Hz-based Spectral Densities- Why ?
.frequency the toclose lying sfrequencieat
bandwidth signal of per Hz
power signal of Watts hedirectly t is ~
~~
2
1~
2
1Power
, bandwidth theinpower signal The
? notation Why this
high
low
high
low
low
high
highlow
f
jfS
dfjfSdfjfSdfjfS
ff
XX
f
f
XX
f
f
XX
f
f
XX
class notes, M. Rodwell, copyrighted 2012
Single-Sided Hz-based Cross Spectral Densities
XYdf
djfS
dfjRjfS
dffjjfStYtXER
djRjS
djjStYtXER
XY
XYXY
XYXY
XYXY
XYXY
as writtenoften also is ~
)2exp(2~
)2exp(~
2
1)()(
DensitiesSpectral Cross based- HzSided-Single
)exp(
)exp(2
1)()(
DensitiesSpectral Cross Sided-Double
class notes, M. Rodwell, copyrighted 2012
Example: Cross Spectral Densities
jfSjfSjfSjfS
jSjSjS
jSjSjSjSjS
RRRR
tYtXtYtXER
tYtXtV
XYYYXXVV
XYYYXX
XYXYYYXXVV
YXXYYYXX
VV
~Re2
~~~densities spectral sided-single in Or,
Re2
)()()()(
)()()(
*
X
RY
V
P=V2/R
class notes, M. Rodwell, copyrighted 2012
Example: Cross Spectral Densities
relevant. isdensity spectral-cross that theNote
R.in dissipatedpower (expected) total thegives
relevant) is bandwidthever (over whatfrequency respect to withgIntegratin
...~
, and between bandwidth thein And
/0/)()(
valueexpected has /)( Power The 2
high
low
f
f
VV
highlow
VV
dfjfSP
ff
RRRtVtVE
RtVP X
RY
V
P=V2/R
class notes, M. Rodwell, copyrighted 2012
Our Notation for Spectral Densities and Correlations
)2/(2)(~
)()()(
)()(
)2/(2)(~
)()(density spectral cross
)()()()()()(function lationcrosscorre
)2/(2)(~
)()()(
)()(
)2/(2)(~
)()(density spectralpower
)()()()()()(function ationautocorrel
)(),()(),(frequency of function
)()( timeof function
Outcome ProcessRandom
*
*
jSjfS
jyjxjS
RjS
jSjfS
RjS
tytvARtYtXER
jSjfS
jvjvjS
RjS
jSjfS
RjS
tvtvARtVtVER
jvjfvjVjfV
tvtV
xyxy
xy
xyxy
XYXY
XYXY
xyXY
vvvv
vv
vvvv
VVVV
VVVV
vvVV
FF
FF
)()()()( and )()()()(
processes ergodic stationaryFor
** jyjxjSjSjvjvjSjS xyXYvvVV
. tesimply wri can we),(or )(her clear whetit makescontext When vjvvtvv
class notes, M. Rodwell, copyrighted 2012
Example: Noise passing through filters & linear electrical networks
Vin(t) h(t) Vout(t))()()(
)()()(
)()()()()(
)()()(
)()()(
)()()()()()(
So
)()()( ),()(any for then
),( function transfer and )( response impulse hasfilter theIf
**
2
2
***
jSjhjS
jSjhjS
jvjvjhjvjv
jSjhjS
jSjhjS
jvjhjvjhjvjv
jvjhjvtvtv
jhth
inininout
inininout
ininoutout
ininoutout
VVVV
vvvv
inininout
VVVV
vvvv
ininoutout
inoutoutin
densities. spectral based- Hzsided-single tochange to trivialisIt