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ECE145C /218C notes, M. Rodwell, copyrighted 2007
Mixed Signal IC DesignNotes set 6:Mathematics of Electrical Noise
Mark RodwellUniversity of California, Santa Barbara
[email protected] 805-893-3244, 805-893-3262 fax
ECE145C /218C notes, M. Rodwell, copyrighted 2007
Strategy
ive.comprehens isbook sZeil'der Van .literature theinor web), the(on notes class noisemy in found be can detail More
rates.error ns,correlatio signal functions, ncorrelatio densities, spectral SNR,
of ncalculatiocorrect for sufficient backround give :Strategy
detail. insubject thisdevelop to145c/218c in not time is There
ECE145C /218C notes, M. Rodwell, copyrighted 2007
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.
ECE145C /218C notes, M. Rodwell, copyrighted 2007
random variables
ECE145C /218C notes, M. Rodwell, copyrighted 2007
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
xxxxX
X
x
xX∫=<<
fX(x)
xx1 x2
ECE145C /218C notes, M. Rodwell, copyrighted 2007
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
21)(
:ondistributi Gaussian The
2
2
2
2
x
xxX
x
xxxf
σ
σσ
−−
π=
fX(x)
xx
~2σx
ECE145C /218C notes, M. Rodwell, copyrighted 2007
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
ECE145C /218C notes, M. Rodwell, copyrighted 2007
Variance
( ) ( )[ ] ( )
variance theofroot square thesimply is X of deviation standard The
)(
valueaverage its from deviation square-mean-root its is X of varianceThe
2222
2
x
XX
x
dxxfxxxXExX
σ
σ
σ
∫∞+
∞−
−=−=−=
ECE145C /218C notes, M. Rodwell, copyrighted 2007
Returning to the Gaussian Distribution
fX(x)
xx
~2σx
( )
clear. benow should
2exp
21)(
:ondistributi Gaussian thedescribing notation The
2
2
2
−−
π=
xxX
xxxfσσ
ECE145C /218C notes, M. Rodwell, copyrighted 2007
Variance vs Expectation of the Square
( ) ( )( )
( )
( )
( )( )
n.expectatio theof square theminussquare theof nexpectatio theis varianceThe
2
2
2
222
22
22
22
22
xX
xxxX
xXxX
xxXX
xXxXxX
X
X
−=
+⋅⋅−=
+⋅−=
+⋅−=
−−=−=
σ
σ
ECE145C /218C notes, M. Rodwell, copyrighted 2007
Pairs of Random Variables
∫ ∫=<<<<D
C
B
AXY
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
ECE145C /218C notes, M. Rodwell, copyrighted 2007
Pairs of Random Variables
∫
∫ ∫
∫
∫ ∫
=
=<<
=
=<<
∞+
∞−
∞+
∞−
D
CY
D
CXY
B
AX
B
AXY
dyyf
dxdyyxfDyCP
dxxf
dxdyyxfBxAP
)(
),(} {
:for Ysimilarly and
)(
),(} {
defined be alsomust onsdistributi Marginal
ECE145C /218C notes, M. Rodwell, copyrighted 2007
Statistical Independence
expected.generally not is This
t.independenlly statistica be tosaid are variablesthe , )()(),(
wherecase theInyfxfyxf YXXY =
ECE145C /218C notes, M. Rodwell, copyrighted 2007
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
∫∫ ∫
∫∫ ∫
∫ ∫
∞+
∞−
∞+
∞−
∞+
∞−
∞+
∞−
∞+
∞−
∞+
∞−
∞+
∞−
∞+
∞−
===
===
=
ECE145C /218C notes, M. Rodwell, copyrighted 2007
Correlation between random variables
[ ]
( )( )[ ] [ ]
values.mean zero have or Y Xeither if same theare covariance and ncorrelatio that Note
is Yand X of covariance The
),(•
is Yand X of ncorrelatio The
yxRyxyXYxXYEyYxXEC
dxdyyxfxyXYER
XY
XY
XYXY
⋅−=+−−=−−=
== ∫ ∫∞+
∞−
∞+
∞−
ECE145C /218C notes, M. Rodwell, copyrighted 2007
Correlation versus Covariance
careful. Be
ons.distributi lconditionacalculate 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
ECE145C /218C notes, M. Rodwell, copyrighted 2007
Correlation Coefficient
.covariance and ncorrelatio betweengy terminoloin confusion (standard) theNote
/is Yand X oft coefficien ncorrelatio The
YXXYXY C σσρ =
ECE145C /218C notes, M. Rodwell, copyrighted 2007
Sum of TWO Random Variables
[ ] [ ] [ ][ ] [ ]
[ ] [ ] [ ]
n.correlatio of role theemphasizes This
2means zero have both and If
2 2)(
: variablesrandom twoof Sum
222
22
2222
XY
XY
CYEXEZEYX
RYEXEYXYXEYXEZE
YXZ
++=
++=
++=+=
+=
ECE145C /218C notes, M. Rodwell, copyrighted 2007
Pairs of Jointly Gaussian Random Variables
[ ] [ ]jiiii
n
YYXXXY
XYYXXY
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
121),(
:GaussianJointly are Yand X If
21
2
2
2
2
2
2
−+
−−+
−⋅
−−×
−=
σσσσρ
ρσπσ
ECE145C /218C notes, M. Rodwell, copyrighted 2007
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
WVdYcXWbYaXV +=+=
ECE145C /218C notes, M. Rodwell, copyrighted 2007
Probability distribution after a Linear Operation on JGRV's
[ ] [ ][ ] [ ] [ ] [ ][ ] [ ] [ ] [ ][ ] [ ][ ]
[ ] [ ] [ ]
−+
−−+
−⋅
−−×
−=
++−⋅+++=
−+++=
−++=−=
+−⋅++=−=
+−⋅++=−=
+=+=+==
2
2
2
2
2
2
22
22
22222222
22222222
)())(()()1(2
1exp
121),(
. and of ondistributijoint thecalculatenow can We
))(()( )(
))(()(2
)(2 and
WWVVVW
VWWVVW
VW
W
V
wwwwvvvv
wvf
WV
YdXcYbXaYEbdXYEbcadXacEWVbdYXYbcadacXE
WVdYcXbYaXEWVVWECYdXcXYEcdYEdXEcWWE
YbXaXYEabYEbXEaVVEYdXcWYbXabYaXEVEV
σσσσρ
ρσπσ
σ
σ
tedious details
ECE145C /218C notes, M. Rodwell, copyrighted 2007
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
ECE145C /218C notes, M. Rodwell, copyrighted 2007
Uncorrelated Variables.
ceindependenimply does eduncorrelat s,JGRV'For
ce.independenimply not does ncorrelatio Zeron.correlatio zero implies ceIndependen
)()(),(:tindependenlly Statistica
0:edUncorrelat
yfxfyxf
C
YXXY
XY
=
=
ECE145C /218C notes, M. Rodwell, copyrighted 2007
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
1211
2111
1211
1211
21)(1 P variablerandoma isresistor thein dissipatedpower The
resistor R the toapplied are voltagesTwo
2221
21
22
21
22
21
22
21
2221
21
221
2121
21
21
VR
VVR
VR
RV
RV
R
VRR
VR
VR
CR
VR
VVVVR
VVR
PPE
VVVV
VV
VV
++=
++=
++=
++=
++=+==
σσρ
σ
ECE145C /218C notes, M. Rodwell, copyrighted 2007
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 statisticais 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
2221
221
21
22111
1
1
1
NMppM
ppMMMpNEMNE
NM
ppNNEpNEpNNE
Np
N
NN
N
=→
>><<>>
−=⋅==⋅=
−=−====
σ
σσ
σ
ECE145C /218C notes, M. Rodwell, copyrighted 2007
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
=
=
=
ECE145C /218C notes, M. Rodwell, copyrighted 2007
random processes
ECE145C /218C notes, M. Rodwell, copyrighted 2007
Random Processes
v(t)is outcome particular The).( is process random The
time.of function randomour is Thisrandom.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
ECE145C /218C notes, M. Rodwell, copyrighted 2007
[ ]
[ ]
[ ] ∫
∫
∫
∞+
∞−
∞+
∞−
∞+
∞−
=
=
==
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
ECE145C /218C notes, M. Rodwell, copyrighted 2007
[ ] [ ]
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
ECE145C /218C notes, M. Rodwell, copyrighted 2007
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
tVtVtVtt
Time Samples of Random Processes
v(t)
t
v(t1 )v(t2 )
t1 t2
ECE145C /218C notes, M. Rodwell, copyrighted 2007
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
ECE145C /218C notes, M. Rodwell, copyrighted 2007
( )[ ] ( )[ ]
( )[ ] ( )[ ]( )[ ] ( )[ ])()(orderslower
)(),()(),(:tystationariorder 2
orderslower ..and )(),...,(),()(),...,(),(
:tystationariorder N
time.ithnot vary w do process stationarya of statistics The
11
2121
nd
2121
th
τττ
τττ
+=→++=
−
+++=−
tVfEtVfEtVtVfEtVtVfE
tVtVtVfEtVtVtVfE nn
Stationary Random Processes
v(t)
t
v(t1 )v(t2 )
t1 t2
ECE145C /218C notes, M. Rodwell, copyrighted 2007
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.
ECE145C /218C notes, M. Rodwell, copyrighted 2007
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 τ( ) = E X t( )X t + τ( )[ ] 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
ECE145C /218C notes, M. Rodwell, copyrighted 2007
Variation of a random process with time
Note that RXX 0( )= E X t( )X t( )[ ]= σX2 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.
)(τxxR
τ
)(τxxR
τ
Narrow autocorrelation:Fast variation
Broad autocorrelationSlow variation
ECE145C /218C notes, M. Rodwell, copyrighted 2007
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
σY2 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 τ+t A narrow autocorrelation is indicative of a quickly-varying random process
ECE145C /218C notes, M. Rodwell, copyrighted 2007
Power spectral densities
The autocorrellation function describes how a random process evolves with time. Find its Fourier transform:
( ) ( ) τωττω 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.
ECE145C /218C notes, M. Rodwell, copyrighted 2007
Power spectral densities
Rxx(tau)
tau
Rxx(tau)
tau
time
time
x(t)
x(t)
Sxx(omega)
omega
Sxx(omega)
omega
ECE145C /218C notes, M. Rodwell, copyrighted 2007
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,
21)0(
ly,specifical
)exp(21
thatso holds, transforminverse The
)exp(
function ationautocorrel theof ransform Fourier tthe isdensity spectralpower that theRecall
2
2
X
XXXXX
XXXX
XXXX
dSR
djSR
djRS
σ
ωωσ
ωωτωτ
τωττω
∫
∫
∫
∞+
∞−
∞+
∞−
∞+
∞−
π==
π=
−=
ECE145C /218C notes, M. Rodwell, copyrighted 2007
Correlated Random Processes
( ) ( ) ( )[ ]
( ) ( )
( ) ( ) ωωτωτ
τωττω
ττ
djSR
djRS
tYtXER
XYXY
XYXY
XY
)exp(21 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
∫
∫∞+
∞−
∞+
∞−
π=
−=
+=
ECE145C /218C notes, M. Rodwell, copyrighted 2007
Single-Sided Hz-based Spectral Densities
( ) [ ] ( )
( ) ( )
( ) [ ] ( )
( ) ( ) ττπτ
τπττ
τωττω
ωωτωττ
dfjRjfS
dffjjfStXtXER
djRjS
djjStXtXER
XXXX
XXXX
XXXX
XXXX
)2exp(2~
)2exp(~21)()(
DensitiesSpectral based- HzSided-Single
)exp(
)exp(21)()(
DensitiesSpectral Sided-Double
−=
=+=
−=
π=+=
∫
∫
∫
∫
∞+
∞−
∞+
∞−
∞+
∞−
∞+
∞−
ECE145C /218C notes, M. Rodwell, copyrighted 2007
Single-Sided Hz-based Spectral Densities- Why ?
{ }
( ) ( ) ( )
( )
.frequency the toclose lying sfrequencieat bandwidth signal of per Hz
power signal of Watts hedirectly t is ~
~~21~
21Power
, bandwidth theinpower signal The ? notation Why this
high
low
high
low
low
high
highlow
f
jfS
dfjfSdfjfSdfjfS
ff
XX
f
fXX
f
fXX
f
fXX
→
=+= ∫∫∫−
−
ECE145C /218C notes, M. Rodwell, copyrighted 2007
Single-Sided Hz-based Cross Spectral Densities
( ) [ ] ( )
( ) ( )
( ) [ ] ( )
( ) ( )
( ) XYdfdjfS
dfjRjfS
dffjjfStYtXER
djRjS
djjStYtXER
XY
XYXY
XYXY
XYXY
XYXY
as writtenoften also is ~
)2exp(2~
)2exp(~21)()(
DensitiesSpectral Cross based- HzSided-Single
)exp(
)exp(21)()(
DensitiesSpectral Cross Sided-Double
ττπτ
τπττ
τωττω
ωωτωττ
−=
=+=
−=
π=+=
∫
∫
∫
∫
∞+
∞−
∞+
∞−
∞+
∞−
∞+
∞−
ECE145C /218C notes, M. Rodwell, copyrighted 2007
Example: Cross Spectral Densities
( ) ( )( )[ ]( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( )( ) ( ) ( ){ }
( ) ( ) ( ) ( ){ }jfSjfSjfSjfS
jSjSjSjSjSjSjSjS
RRRRtYtXtYtXER
tYtXtV
XYYYXXVV
XYYYXX
XYXYYYXXVV
YXXYYYXX
VV
~Re2~~~densities spectral sided-single in Or,
Re2
)()()()(
)()()(
*
⋅++=
⋅++=+++=
+++=++++=
+=
ωωωωωωωω
τττττττ
X
RYV
P=V2/R
ECE145C /218C notes, M. Rodwell, copyrighted 2007
Example: Cross Spectral Densities
[ ] ( )
( )
relevant. isdensity spectral-cross that theNote
R.in dissipatedpower (expected) total thegivesrelevant) is bandwidthever (over whatfrequency respect to withgIntegratin
...~
, and between bandwidth thein And
/0/)()( valueexpected has /)( Power The 2
∫=
==
high
low
f
fVV
highlow
VV
dfjfSP
ff
RRRtVtVERtVP X
RYV
P=V2/R
ECE145C /218C notes, M. Rodwell, copyrighted 2007
Noise passing through filters & linear electrical networks
Vin(t) h(t) Vout(t)
)()()(
and
)()()(
)()()(
show can We
)()()( ),()(any for then),( function transfer and )( response impulse hasfilter theIf
2
*
ωωω
ωωω
ωωω
ωωωω
jSjHjS
jHjSjS
jSjHjS
jVjHjVtvtvjHth
ininoutout
ininoutin
inininout
VVVV
VVVV
VVVV
inoutoutin
=
=
=
=→