Mixed Signal IC Design Notes set 6: Mathematics of ... · Random Processes The particular outcome...

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


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


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.


random variables


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


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


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


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

∫∞+

∞−

−=−=−=


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


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


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


Pairs of Random Variables

∫

∫ ∫

∫

∫ ∫

=

=<<

=

=<<

∞+

∞−

∞+

∞−

D

CY

D

CXY

B

AX

B

AXY

dyyf

dxdyyxfDyCP

dxxf

dxdyyxfBxAP

)(

),(} {

:for Ysimilarly and

)(

),(} {

defined be alsomust onsdistributi Marginal


Statistical Independence

expected.generally not is This

t.independenlly statistica be tosaid are variablesthe , )()(),(

wherecase theInyfxfyxf YXXY =


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

∫∫ ∫

∫∫ ∫

∫ ∫

∞+

∞−

∞+

∞−

∞+

∞−

∞+

∞−

∞+

∞−

∞+

∞−

∞+

∞−

∞+

∞−

===

===

=


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

⋅−=+−−=−−=

== ∫ ∫∞+

∞−

∞+

∞−


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


Correlation Coefficient

.covariance and ncorrelatio betweengy terminoloin confusion (standard) theNote

/is Yand X oft coefficien ncorrelatio The

YXXYXY C σσρ =


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

++=

++=

++=+=

+=


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


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 +=+=


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


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


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

=

=


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


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


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

=

=

=


random processes


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


[ ]

[ ]

[ ] ∫

∫

∫

∞+

∞−

∞+

∞−

∞+

∞−

=

=

==

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


[ ] [ ]

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


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


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


( )[ ] ( )[ ]

( )[ ] ( )[ ]( )[ ] ( )[ ])()(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


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.


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


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


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


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.


Power spectral densities

Rxx(tau)

tau

Rxx(tau)

tau

time

time

x(t)

x(t)

Sxx(omega)

omega

Sxx(omega)

omega


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=

−=


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=

−=

+=


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=+=

∫

∫

∫

∫

∞+

∞−

∞+

∞−

∞+

∞−

∞+

∞−


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

→

=+= ∫∫∫−

−


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=+=

∫

∫

∫

∫

∞+

∞−

∞+

∞−

∞+

∞−

∞+

∞−


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


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


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

=

=

=

=→

Mixed Signal IC Design Notes set 6: Mathematics of ... · Random Processes The particular outcome...

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Transcript of Mixed Signal IC Design Notes set 6: Mathematics of ... · Random Processes The particular outcome...