Modeling Negative Power Law Noise

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Modeling Negative Power Law Noise

• Victor S. ReinhardtRaytheon Space and Airborne
  SystemsEl Segundo, CA, USA

2008 IEEE International Frequency Control
Symposium Honolulu, Hawaii, USA, May 18 - 21, 2008
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Negative Power Law Noise Gets its Name from its
Neg-p PSD

• But autocorrelation function must be wide-sense
  stationary (WSS) to have a PSD
• Then can define PSD LX(f) as Fourier Transform
  (FT)over ? of Rx(?)

tg Global (average) time
? Local (delta) time
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Neg-p Noise Also Called Non-Stationary (NS)

• Must use dual-freq Loève spectrum Lx(fg,f) not
  single-freq PSD Lx(f)
• Loève Spectrum ?
• Paper will show neg-p noise can be pictured as
  either WSS or NS process
• And these pictures are not in conflict
• Because different assumptions used for each
• Will also show how to generate practical freq
  time domain models for neg-p noise
• And avoid pitfalls associated with divergences

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Classic Example of Neg-p Noise Random Walk

• Integral of a white noiseprocess is a random
  walk
• But starting in f-domain
• Can write ?
• So ?
• Because
• Will show different assumptions used for each
  picture so not in conflict

Not WSS
f-domain Integrator
Is WSS?
?
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A Historical Aside Random Walk

• 1st discussed by Lucretius 60 BC
• Later Jan Ingenhousz 1785
• Traditionally attributed toRobert Brown 1827
• Treated by Lord Rayleigh 1877
• Full mathematical treatment by Thorvald Thiele
  1880
• Made famous in physics by Albert Einstein 1905
  and Marian Smoluchowski 1906
• Continuous form named Wiener process in honor of
  Norbert Wiener

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Generating Colored Noise from White Noise Using
Wiener Filter

• Can change spectrum of white noise v0(t) by
  filtering it with h(t),H(f)
• H(f) called a Wiener filter
• NS picture ?
• Starts at t0
• WSS picture ?
• Must start at t-? for t-translation invariance
• Necessary condition for WSS process
• Wiener filters divergent for neg-p noise
• Need to write neg-p filter aslimit of bounded
  sister filterto stay out of trouble

Wiener Filter
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Random Walk as the Limit of a Sister Process

• Sister process is single poleLP filtered white
  noise
• In WSS picture v-2(t) ?for any t
• Need sister processes to keep v-2(t) finite
• True for any neg-p value

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Even When Final Variable Bounded (Due to HP
Filtering of Neg-p Noise)

• Intermediate variables areunbounded (in WSS
  picture)
• Can cause subtle problems
• Sister process helps diagnose fix such problems
• In NS picture v-2(t) isbounded for finite t
• But v-1(t) (f -1 noise) is not
• Sister process needed forf -1 noise even in NS
  pictureto keep t-domain process bounded

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Models For f -1 Noise

• The diffusive line model
• White current noise into a diffusive line
  generates flicker voltage noise
• Diffusive line modeled as R-C ladder network
• In limit of generates f -1 voltage
  noise with white current noise input

?
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Sister Model for Diffusive Line

• Adds shunt resistor to bound DC voltage
• Not well-suited for t-domain modeling
• Because Wiener filter not rational polynomial

?
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A Historical Aside The Diffusive Line

• Studied by Lord Kelvin 1855
• For pulse broadening problem in submarine
  telegraph cables
• Refined by Oliver Heaviside 1885
• Developed modern telegraphers equation
• Added inductances patented impedance matched
  transmission line
• Adolf Fick developed Ficks Law diffusion
  equation 1855
• 1-dimensional diffusion equation following Ficks
  (Ohms) Law is diffusive line
• Used in heat molecular transport

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The Trap f -1 Model is More Suited for f t
Domain Modeling

• Each trap independentwhite noise source
  filteredby single-poleWiener filter
• Sum over ?m from ?0 to ?M
• Sister model (M ? ?)?0 gt 0 ?M lt ?
• Well-behaved inf t domains
• For ?0 ? 0 ?M ? ?becomes f -1 noise

(L0 same for all m)
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A Historical Aside The Trap Model

• Developed by McWorter 1955 to explain flicker
  noise in semiconductors
• Traps ? loosely coupled storage cells for
  electrons/holes that decay with TCs 1/?m
• Surface cells for Si bulk for GaAs/HEMT
• GaAs/HEMT semi-insulating (why much higher
  flicker noise)
• Simplified theory by van der Ziel 1959
• Flicker of v-noise from traps converted to
  flicker of ?-noise in amps through AM/PM

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A Practical Trap Simulation Model Using Discrete
Number of Filters

• Trap filter every decade
• 1/4 dB error over 6 decades with 8 filters
• Can reduce error by narrowing filter spacing

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Other f -1 Noise Models

• Barnes Jarvis 1967, 1970
• Diffusion-like sister model with finite
  asymmetrical ladder network
• Finite rational polynomial with one input white
  noise source
• 4 filter stages generate f -1 spectrum over
  nearly 4 decades of f with lt 1/2 dB error
• Barnes Allan 1971
• f -1 model using fractional integration

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Discrete t-Domain Simulators for Neg-p Noise

• For f -2 noise can use NS integrator model in
  discrete t-domain
• NS model bounded in t-domain for finite t
• Discrete integrator (1st order autoregressive
  (AR) process)
• wn uncorrelated random shocks or
  innovations
• wn need not be Gaussian (i.e. random 1) to
  generate appropriate spectral behavior
• Central limit theorem ? Output becomes Gaussian
  for large number of shocks

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Trap f -1 Discrete t-Domain Simulator

• Must use sister model
• Full f -1 model unboundedin NS picture
• Wiener filter for each trap
• t-domain AR model
• Sum overtraps forf -1 noise

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From f -1 and f -2 models Can Generate any
Integer Neg-p Model
Right Crop 66x72
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Summary and Conclusions

• Either WSS or NS pictures can be used for neg-p
  noise as convenient
• Not in conflict ? Different assumptions used
• Need sister models to resolve problems
• Can generate practical models for any integer
  neg-p noise
• By concatenating integrator trap models
• Are simple to implement in f t domains
• For preprint presentation see
• www.ttcla.org/vsreinhardt/