1. Modeling Negative Power Law Noise Victor S. Reinhardt Raytheon Space and Airborne Systems El Segundo, CA, USA 2008 IEEE International Frequency Control Symposium Honolulu, Hawaii, USA, May 18 - 21, 2008

2. Page 2 FCS 2008 Neg-p -- V. Reinhardt 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 But autocorrelation function must be wide- sense stationary (WSS) to have a PSD Then can define PSD L X (f) as Fourier Transform (FT) over  of R x (  ) Then can define PSD L X (f) as Fourier Transform (FT) over  of R x (  ) t g = Global (average) time  = Local (delta) time dBc/Hz f -1 Log 10 (f) f -2 f -3 f -4

3. Page 3 FCS 2008 Neg-p -- V. Reinhardt Neg-p Noise Also Called Non- Stationary (NS) Must use dual-freq Loève spectrum L x (f g,f) not single-freq PSD L x (f) Must use dual-freq Loève spectrum L x (f g,f) not single-freq PSD L x (f)  Loève Spectrum  Paper will show neg-p noise can be pictured as either WSS or NS process 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 Will also show how to generate practical freq & time domain models for neg-p noise  And avoid pitfalls associated with divergences

4. Page 4 FCS 2008 Neg-p -- V. Reinhardt Classic Example of Neg-p Noise – Random Walk Integral of a white noise process is a random walk Integral of a white noise process is a random walk But starting in f-domain But starting in f-domain  Can write   So   Because Will show different assumptions used for each picture so not in conflict Will show different assumptions used for each picture so not in conflict Not WSS White Noise V o V -2 -- 1 1/j  Is WSS?  t v -2 (t) Random Walk f-domain Integrator

5. Page 5 FCS 2008 Neg-p -- V. Reinhardt A Historical Aside — Random Walk 1 st discussed by Lucretius [~ 60 BC] 1 st discussed by Lucretius [~ 60 BC]  Later Jan Ingenhousz [1785] Traditionally attributed to Robert Brown [1827] Traditionally attributed to Robert Brown [1827] Treated by Lord Rayleigh [1877] Treated by Lord Rayleigh [1877] Full mathematical treatment by Thorvald Thiele [1880] Full mathematical treatment by Thorvald Thiele [1880] Made famous in physics by Albert Einstein [1905] and Marian Smoluchowski [1906] Made famous in physics by Albert Einstein [1905] and Marian Smoluchowski [1906] Continuous form named Wiener process in honor of Norbert Wiener Continuous form named Wiener process in honor of Norbert Wiener Random Walk  2 (t g )  t g

6. Page 6 FCS 2008 Neg-p -- V. Reinhardt Generating Colored Noise from White Noise Using Wiener Filter Can change spectrum of white noise v 0 (t) by filtering it with h(t),H(f) Can change spectrum of white noise v 0 (t) by filtering it with h(t),H(f)  H(f) called a Wiener filter NS picture  NS picture   Starts at t=0 WSS picture  WSS picture   Must start at t=-  for t-translation invariance  Necessary condition for WSS process Wiener filters divergent for neg-p noise Wiener filters divergent for neg-p noise  Need to write neg-p filter as limit of bounded sister filter to stay out of trouble LoLo H(f) |H(f)| 2 L o (f) Wiener Filter

7. Page 7 FCS 2008 Neg-p -- V. Reinhardt Random Walk as the Limit of a Sister Process Sister process is single pole LP filtered white noise Sister process is single pole LP filtered white noise In WSS picture v -2 (t) =  for any t In WSS picture v -2 (t) =  for any t  Need sister processes to keep v -2 (t) finite  True for any neg-p value VV V0V0 -- 1 1/j  1/1/ Sister Filter h  (t-t’)   0 t’ t |H  (f)|-dB   0 Log(f)  -1

8. Page 8 FCS 2008 Neg-p -- V. Reinhardt Even When Final Variable Bounded (Due to HP Filtering of Neg-p Noise) Intermediate variables are unbounded (in WSS picture) Intermediate variables are unbounded (in WSS picture)  Can cause subtle problems  Sister process helps diagnose & fix such problems In NS picture v -2 (t) is bounded for finite t In NS picture v -2 (t) is bounded for finite t But v -1 (t) (f -1 noise) is not But v -1 (t) (f -1 noise) is not  Sister process needed for f -1 noise even in NS picture to keep t-domain process bounded v -2 (t) -- WSS Picture of Random Walk t  v -2 (t)  0 NS Picture of Random Walk

9. Page 9 FCS 2008 Neg-p -- V. Reinhardt Models For f -1 Noise The diffusive line model 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 

10. Page 10 FCS 2008 Neg-p -- V. Reinhardt Sister Model for Diffusive Line Adds shunt resistor to bound DC voltage Adds shunt resistor to bound DC voltage Not well-suited for t-domain modeling Not well-suited for t-domain modeling  Because Wiener filter not rational polynomial 

11. Page 11 FCS 2008 Neg-p -- V. Reinhardt A Historical Aside — The Diffusive Line Studied by Lord Kelvin [1855] Studied by Lord Kelvin [1855]  For pulse broadening problem in submarine telegraph cables Refined by Oliver Heaviside [1885] Refined by Oliver Heaviside [1885]  Developed modern telegrapher’s equation  Added inductances & patented impedance matched transmission line Adolf Fick developed Fick’s Law & diffusion equation [1855] Adolf Fick developed Fick’s Law & diffusion equation [1855]  1-dimensional diffusion equation following Fick’s (Ohm’s) Law is diffusive line  Used in heat & molecular transport

12. Page 12 FCS 2008 Neg-p -- V. Reinhardt The Trap f -1 Model is More Suited for f & t Domain Modeling Each “trap” independent white noise source filtered by single-pole Wiener filter Each “trap” independent white noise source filtered by single-pole Wiener filter  Sum over  m from  0 to  M  Sister model (M   )  0 > 0  M 0  M <   Well-behaved in f & t domains For  0  0  M   becomes f -1 noise For  0  0  M   becomes f -1 noise 00  00 MM V 0,m  V 0,0 V 0,M V -1 ● ● (L 0 same for all m)

13. Page 13 FCS 2008 Neg-p -- V. Reinhardt A Historical Aside — The Trap Model Developed by McWorter [1955] to explain flicker noise in semiconductors 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] Simplified theory by van der Ziel [1959] Flicker of v-noise from traps converted to flicker of  -noise in amps through AM/PM Flicker of v-noise from traps converted to flicker of  -noise in amps through AM/PM

14. Page 14 FCS 2008 Neg-p -- V. Reinhardt A Practical Trap Simulation Model Using Discrete Number of Filters Trap filter every decade Trap filter every decade  ±1/4 dB error over 6 decades with 8 filters Can reduce error by narrowing filter spacing Can reduce error by narrowing filter spacing Error from f -1 = ±1/4 dB +1/4 -1/4 dB L  (f) 0 -20 -40 -60 024 6 Log(f) dB L -1 (f)

15. Page 15 FCS 2008 Neg-p -- V. Reinhardt Other f -1 Noise Models Barnes & Jarvis [1967, 1970] 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 < ±1/2 dB error Barnes & Allan [1971] Barnes & Allan [1971]  f -1 model using fractional integration

16. Page 16 FCS 2008 Neg-p -- V. Reinhardt Discrete t-Domain Simulators for Neg-p Noise For f -2 noise can use NS integrator model in discrete t-domain For f -2 noise can use NS integrator model in discrete t-domain  NS model bounded in t-domain for finite t  Discrete integrator (1 st order autoregressive (AR) process) w n = uncorrelated random “shocks” or “innovations” w n = uncorrelated random “shocks” or “innovations”  w n need not be Gaussian (i.e. random ±1) to generate appropriate spectral behavior  Central limit theorem  Output becomes Gaussian for large number of shocks

17. Page 17 FCS 2008 Neg-p -- V. Reinhardt Trap f -1 Discrete t-Domain Simulator Must use sister model Must use sister model  Full f -1 model unbounded in NS picture Wiener filter for each trap Wiener filter for each trap t-domain AR model t-domain AR model Sum over traps for f -1 noise Sum over traps for f -1 noise Log 10 (L(f)) from x n Log 10 (f) Fitted Slope f -1.03 Spectrum Recovered from t- Domain Simulation

18. Page 18 FCS 2008 Neg-p -- V. Reinhardt From f -1 and f -2 models Can Generate any Integer Neg-p Model Right Crop 66%x72% f 0f 0f 0f 0 White Input f -2 Integrate f -2 White Input f -4 White Input Integrate f -2 f -1 Trap f -1 White Inputs f -3 Trap f -1 White Inputs Integrate f -2

19. Page 19 FCS 2008 Neg-p -- V. Reinhardt Summary and Conclusions Either WSS or NS pictures can be used for neg-p noise as convenient 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 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 seeFor preprint & presentation see www.ttcla.org/vsreinhardt/