Modeling Negative Power Law Noise Victor S. Reinhardt Raytheon Space and Airborne Systems El...
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Transcript of Modeling Negative Power Law Noise Victor S. Reinhardt Raytheon Space and Airborne Systems El...
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 ωτ
τ
)}0.5-)x(t0.5E{x(t)(tR gggx τττ ,
= Local (delta) time= Local (delta) time
)(Rx τ
dB
c/H
z
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
FCS 2008 Neg-p -- V. Reinhardt Page 3
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 ,
FCS 2008 Neg-p -- V. Reinhardt Page 4
Classic Example of Neg-p Noise – Random Walk
• Integral of a white noiseIntegral of a white noiseprocess 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 0
t
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
FCS 2008 Neg-p -- V. Reinhardt Page 5
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
FCS 2008 Neg-p -- V. Reinhardt Page 6
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 0
t
-h
)(t')vt'h(tdt'(t)v 0
t
0h
(f)HLim(f)H p,pp
εεε
Lo H(f)|H(f)|2Lo(f)
Wiener Filter
FCS 2008 Neg-p -- V. Reinhardt Page 7
Random Walk as the Limit of a Sister Process
• Sister process is single poleSister process is single poleLP 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/j
1/
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
FCS 2008 Neg-p -- V. Reinhardt Page 8
Even When Final Variable Bounded (Due to HP Filtering of Neg-p Noise)
• Intermediate variables areIntermediate variables areunbounded (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) isbounded 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
FCS 2008 Neg-p -- V. Reinhardt Page 9
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
0I
1-V1)(j ωδ
FCS 2008 Neg-p -- V. Reinhardt Page 10
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
Z1)( 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
FCS 2008 Neg-p -- V. Reinhardt Page 11
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
FCS 2008 Neg-p -- V. Reinhardt Page 12
The Trap f -1 Model is More Suited for f & t Domain Modeling
• Each “trap” independentEach “trap” independentwhite noise source filteredwhite noise source filteredby single-poleby single-poleWiener filterWiener filter
Sum over Sum over mm from from 00 to to MM
Sister 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
mj
1(f)H
m γωγ
M
022
0TR
dL(f)L
γ
γ γω
γ
|f|4
LdL 0
0 220
γω
γ
0
0
M
V0,m
(f)Hmγ
V0,0
V0,M
(f)H0γ
(f)HMγ
V-1
●●
●●
(L0 same for all m)
FCS 2008 Neg-p -- V. Reinhardt Page 13
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
FCS 2008 Neg-p -- V. Reinhardt Page 14
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
-1/4d
B
M
0m2m
2m0
trL
(f)Lγω
γ
L(f)
0
-20
-40
-600 2 4 6
Log(f)
dB
L-1(f)
FCS 2008 Neg-p -- V. Reinhardt Page 15
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
FCS 2008 Neg-p -- V. Reinhardt Page 16
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
FCS 2008 Neg-p -- V. Reinhardt Page 17
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
Lo
g1
0(L
(f))
fro
m x
nLog10(f)
FittedSlopef -1.03
Spectrum Recovered from t-Domain Simulation
FCS 2008 Neg-p -- V. Reinhardt Page 18
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
FCS 2008 Neg-p -- V. Reinhardt Page 19
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 see
www.ttcla.org/vsreinhardtwww.ttcla.org/vsreinhardt//