Modeling Negative Power Law Noise

Modeling Negative Power Law Noise Modeling Negative Power Law Noise

Victor S. ReinhardtVictor S. ReinhardtRaytheon Space and Airborne SystemsRaytheon Space and Airborne Systems

El Segundo, CA, USAEl Segundo, CA, USA

2008 IEEE International Frequency Control SymposiumHonolulu, Hawaii, USA, May 18 - 21, 2008

FCS 2008 Neg-p -- V. Reinhardt Page 2

Negative Power Law Noise Gets its Name from its Neg-p PSD

• But autocorrelation function must be wide-But autocorrelation function must be wide-sense stationary (WSS) to have a PSD sense stationary (WSS) to have a PSD

• Then can define PSD LThen can define PSD LXX(f) as Fourier (f) as Fourier Transform (FT)Transform (FT)over over of R of Rxx(())

ttgg = Global (average) time = Global (average) time

)(Redt)}({R(f)L xj-

-xf,x ττ ωττ

)}0.5-)x(t0.5E{x(t)(tR gggx τττ , = Local (delta) time= Local (delta) time

)(Rx τ

dBc/

Hz

f -1

Log10(f)

f -2

f -3f -4

For x(t) PSD Lx(f) f p for p < 0 pFlicker of Time Error x(t) or f(t) -1

Random Walk of x(t) or White y(t) -2Flicker of Frequency y(t) -3

Random Walk of y(t) -4

For x(t) PSD Lx(f) f p for p < 0


Neg-p Noise Also Called Non-Stationary (NS) • Must use dual-freq Loève spectrum LMust use dual-freq Loève spectrum Lxx(f(fgg,f) ,f)

not single-freq PSD Lnot single-freq PSD Lxx(f)(f)Loève Spectrum Loève Spectrum

• Paper will show neg-p noise can be Paper will show neg-p noise can be pictured as either WSS or NS processpictured as either WSS or NS processAnd these pictures are not in conflictAnd these pictures are not in conflictBecause different assumptions used for each Because different assumptions used for each

• Will also show how to generate practical Will also show how to generate practical freq & time domain models for neg-p noisefreq & time domain models for neg-p noiseAnd avoid pitfalls associated with divergencesAnd avoid pitfalls associated with divergences

)(R)(tR xgx ττ ,

)}}(t{R{f),(fL gxf,t,fgx ggττ ,


Classic Example of Neg-p Noise – Random Walk• Integral of a white noiseIntegral of a white noise

process is a random walkprocess is a random walk

• But starting in f-domainBut starting in f-domainCan write Can write So So BecauseBecause

• Will show different assumptions used for Will show different assumptions used for each picture so not in conflicteach picture so not in conflict

)(t'vdt'(t)v 0t

0 2-

||0.5t),(tR gg2- ττ Not WSS

ω(f)/Vj(f)V 02-

White Noise

Vo V-2

-1

1/j

)(Rv τ

202- /L(f)L ω Is WSS?

(f)H(f)V(f)V 0h (f)L|H(f)|(f)L 02

h

1f

-18

-12

-6

0

6

12

18

0 10 20 30t

v -2(t

)

Random Walk

f-domain Integrator


A Historical Aside — Random Walk

• 11stst discussed by Lucretius [~ 60 BC] discussed by Lucretius [~ 60 BC]Later Jan Ingenhousz [1785]Later Jan Ingenhousz [1785]

• Traditionally attributed toTraditionally attributed toRobert Brown [1827] Robert Brown [1827]

• Treated by Lord Rayleigh [1877] Treated by Lord Rayleigh [1877] • Full mathematical treatment by Thorvald Full mathematical treatment by Thorvald

Thiele [1880]Thiele [1880]• Made famous in physics by Albert Einstein Made famous in physics by Albert Einstein

[1905] and Marian Smoluchowski [1906] [1905] and Marian Smoluchowski [1906] • Continuous form named Wiener process Continuous form named Wiener process

in honor of Norbert Wienerin honor of Norbert Wiener

Random Walk

2(tg) tg


Generating Colored Noise from White Noise Using Wiener Filter • Can change spectrum of white noise vCan change spectrum of white noise v00(t) (t)

by filtering it with h(t),H(f)by filtering it with h(t),H(f)H(f) called a Wiener filterH(f) called a Wiener filter

• NS picture NS picture Starts at t=0Starts at t=0

• WSS picture WSS picture Must start at t=-Must start at t=- for t-translation invariance for t-translation invariance Necessary condition for WSS processNecessary condition for WSS process

• Wiener filters divergent for neg-p noise Wiener filters divergent for neg-p noise Need to write neg-p filter asNeed to write neg-p filter as

limit of bounded sister filterlimit of bounded sister filterto stay out of troubleto stay out of trouble

)(t')vt'h(tdt'(t)v 0t

-h

)(t')vt'h(tdt'(t)v 0t

0h

(f)HLim(f)H p,pp

εεε

Lo H(f)|H(f)|2Lo(f)

Wiener Filter


Random Walk as the Limit of a Sister Process• Sister process is single poleSister process is single pole

LP filtered white noiseLP filtered white noise

• In WSS picture vIn WSS picture v-2-2(t) = (t) = for any tfor any t

Need sister processes to keep vNeed sister processes to keep v-2-2(t) finite(t) finiteTrue for any neg-p valueTrue for any neg-p value

t 0 )(t'vdt'(t)v 2

VV0

-1

1/j1/

Sister Filter

1)-(j(f)H εωε

12202, )(L(f)L

εωε

t'

h(t-t

')

t

h(t-t’)

0

t’ t

-2 -1 0 1 2

dB(|H

(f)|)

|H(f)|-dB

0

Log(f)

-1

(f)LLim(f)L 2,-2- εε 0


Even When Final Variable Bounded (Due to HP Filtering of Neg-p Noise)• Intermediate variables areIntermediate variables are

unbounded (in WSS picture)unbounded (in WSS picture)Can cause subtle problemsCan cause subtle problemsSister process helps diagnoseSister process helps diagnose

& fix such problems& fix such problems• In NS picture vIn NS picture v-2-2(t) is(t) is

bounded for finite tbounded for finite t• But vBut v-1-1(t) (f(t) (f -1 -1 noise) is not noise) is notSister process needed forSister process needed for

ff -1 -1 noise even in NS picture noise even in NS pictureto keep t-domain process boundedto keep t-domain process bounded

v-2(t)-

WSS Pictureof

Random Walk

-18

-12

-6

0

6

12

18

0 10 20 30

t

v -2(t

)

0

NS Picture ofRandom Walk


Models For f -1 Noise

• The diffusive line modelThe diffusive line modelWhite current noise into a diffusive line White current noise into a diffusive line

generates flicker voltage noisegenerates flicker voltage noiseDiffusive line modeled as R-C ladder networkDiffusive line modeled as R-C ladder networkIn limit of generates fIn limit of generates f -1 -1 voltage noise voltage noise

with white current noise input with white current noise input

101- ||L(f)L ω

0δ1/2)(jZ(f) ω

(f)I)(j(f)V 01/2

1- ω

0δ

δ1)(j ωδ

δ

0I1-V

1)(j ωδ


Sister Model for Diffusive Line

• Adds shunt resistor to bound DC voltageAdds shunt resistor to bound DC voltage

• Not well-suited for t-domain modelingNot well-suited for t-domain modelingBecause Wiener filter not rational polynomialBecause Wiener filter not rational polynomial

Z 1)( εδ

1/2220 )(L(f)L εωε

-10 |L(0)L εε |

h

h

f

f 1/2

tj0

1, ]j[ dfeL(t)v

εω

ω

ε

1/2)(jZ(f) εω


A Historical Aside — The Diffusive Line• Studied by Lord Kelvin [1855]Studied by Lord Kelvin [1855]For pulse broadening problem in submarine For pulse broadening problem in submarine

telegraph cablestelegraph cables• Refined by Oliver Heaviside [1885] Refined by Oliver Heaviside [1885] Developed modern telegrapher’s equationDeveloped modern telegrapher’s equationAdded inductances & patented impedance Added inductances & patented impedance

matched transmission linematched transmission line• Adolf Fick developed Fick’s Law & Adolf Fick developed Fick’s Law &

diffusion equation [1855]diffusion equation [1855]1-dimensional diffusion equation following 1-dimensional diffusion equation following

Fick’s (Ohm’s) Law is diffusive lineFick’s (Ohm’s) Law is diffusive lineUsed in heat & molecular transportUsed in heat & molecular transport


The Trap f -1 Model is More Suited for f & t Domain Modeling• Each “trap” independentEach “trap” independent

white noise source filteredwhite noise source filteredby single-poleby single-poleWiener filterWiener filterSum over Sum over mm from from 00 to to MMSister model (M Sister model (M ))00 > 0 > 0 MM < <

Well-behaved inWell-behaved inf & t domains f & t domains

• For For 00 0 0 MM becomes fbecomes f -1 -1 noise noise

mj1(f)H

m γωγ

M

022

0TR

dL(f)Lγ

γ γωγ

|f|4LdL 0

0 220

γωγ

0

0

M

V0,m(f)H

mγ V0,0

V0,M

(f)H0γ

(f)HMγ

V-1●●

●●

(L0 same for all m)


A Historical Aside — The Trap Model• Developed by McWorter [1955] to explain Developed by McWorter [1955] to explain

flicker noise in semiconductorsflicker noise in semiconductorsTraps Traps loosely coupled storage cells for loosely coupled storage cells for

electrons/holes that decay with TCs 1/electrons/holes that decay with TCs 1/m m Surface cells for Si & bulk for GaAs/HEMTSurface cells for Si & bulk for GaAs/HEMTGaAs/HEMT semi-insulating (why much higher GaAs/HEMT semi-insulating (why much higher flicker noise)flicker noise)

• Simplified theory by van der Ziel [1959]Simplified theory by van der Ziel [1959]• Flicker of v-noise from traps converted to Flicker of v-noise from traps converted to

flicker of flicker of ff-noise in amps through AM/PM -noise in amps through AM/PM


A Practical Trap Simulation Model Using Discrete Number of Filters

• Trap filter every decade Trap filter every decade ±1/4 dB error over 6 ±1/4 dB error over 6

decades with 8 filtersdecades with 8 filters• Can reduce error by Can reduce error by

narrowing filter spacing narrowing filter spacing Error from f -1 = ±1/4 dB+1/4

-1/4dB

M

0m2m

2m0

trL(f)L

γωγ

L(f)

0

-20

-40

-600 2 4 6

Log(f)

dB

L-1(f)


Other f -1 Noise Models

• Barnes & Jarvis [1967, 1970]Barnes & Jarvis [1967, 1970]Diffusion-like sister model with finite Diffusion-like sister model with finite

asymmetrical ladder network asymmetrical ladder network Finite rational polynomial with one input white Finite rational polynomial with one input white noise sourcenoise source

4 filter stages generate f4 filter stages generate f -1 -1 spectrum over spectrum over nearly 4 decades of f with < nearly 4 decades of f with < ±1/2 dB error±1/2 dB error

• Barnes & Allan [1971]Barnes & Allan [1971] ff -1 -1 model using fractional integration model using fractional integration


Discrete t-Domain Simulators for Neg-p Noise• For fFor f -2 -2 noise can use NS integrator model noise can use NS integrator model

in discrete t-domainin discrete t-domainNS model bounded in t-domain for finite tNS model bounded in t-domain for finite t

Discrete integrator (1Discrete integrator (1stst order autoregressive order autoregressive (AR) process)(AR) process)

• wwnn = uncorrelated random “shocks” or = uncorrelated random “shocks” or “innovations”“innovations”wwnn need not be Gaussian (i.e. random need not be Gaussian (i.e. random ±±1) to 1) to

generate appropriate spectral behaviorgenerate appropriate spectral behaviorCentral limit theorem Central limit theorem Output becomes Output becomes

Gaussian for large number of shocksGaussian for large number of shocks

n1n-nn wx)x(tx


Trap f -1 Discrete t-Domain Simulator

• Must use sister modelMust use sister modelFull fFull f -1 -1 model unbounded model unbounded

in NS picturein NS picture• Wiener filter for each trapWiener filter for each trap

• t-domain AR modelt-domain AR model

• Sum overSum overtraps fortraps forff -1 -1 noise noise

)/(s(s)H m1/2

mm γγ

mn,1/2

mm1,n-mmn, w)x(1x γγ

mn,

M

0mn xx

Mm-m 100.5γ

1 2 3

-5

-4

-3

-2

Log 1

0(L(f)

) fro

m x

nLog10(f)

FittedSlopef -1.03

Spectrum Recovered from t-Domain Simulation


From f -1 and f -2 models Can Generate any Integer Neg-p Model

Right Crop 66%x72%

ff 0 0WhiteInput

ff -2 -2Integratef -2

WhiteInput

ff -4 -4WhiteInput

Integratef -2

Integratef -2

ff -1 -1Trapf -1

WhiteInputs

ff -3 -3Trapf -1

WhiteInputs

Integratef -2


Summary and Conclusions

• Either WSS or NS pictures can be used for Either WSS or NS pictures can be used for neg-p noise as convenientneg-p noise as convenientNot in conflict Not in conflict Different assumptions used Different assumptions usedNeed sister models to resolve problemsNeed sister models to resolve problems

• Can generate practical models for any Can generate practical models for any integer neg-p noiseinteger neg-p noiseBy concatenating integrator & trap modelsBy concatenating integrator & trap modelsAre simple to implement in f & t domainsAre simple to implement in f & t domains

• For preprint & presentation seeFor preprint & presentation seewww.ttcla.org/vsreinhardtwww.ttcla.org/vsreinhardt//

http://www.ttcla.org/vsreinhardt/

http://www.ttcla.org/vsreinhardt/

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Modeling Negative Power Law Noise