1. Characterizing the Impact of Time Error on General Systems 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 Time Error -- V. Reinhardt Time Error x(t) Impacts Systems Mainly by Generating ME & MN ME = Multiplicative Signal Error ME = Multiplicative Signal Error  MN = Multiplicative Noise  Short term ME  Can be causal or random  x(t) induces ME & MN in generated or processed signals through slope modulation MN Also called MN Also called  Inter-symbol interference  Noise power  Signal processing noise  Scaling noise  v(t) = v(t+x(t)) - v(t)  v’(t)x(t) t  v(t ) v(t) v(t+x) t+x(t) RF Carrier  v(t)  (t) =  o x(t) otot Base band

3. Page 3 FCS 2008 Time Error -- V. Reinhardt Paper will Discuss How to Characterize x(t) Induced ME & MN Especially in presence of random negative power law (neg-p) noise Especially in presence of random negative power law (neg-p) noise  Noise with PSD  L x (f) dBc/Hz f -1 Log 10 (f) f -2 f -3 f -4 L x (f)  f p (p < 0 )

4. Page 4 FCS 2008 Time Error -- V. Reinhardt Paper Will Use Concept of a Timebase (TB) A TB = t TB (t) is a continuous time source for generating or processing a signal v(t) A TB = t TB (t) is a continuous time source for generating or processing a signal v(t)  Ideal v(t) is generated or processed as v(t TB (t))  t  ideal TB  Discrete epochs in a real TB ignored  t TB (t) = t + x TB (t)  Not through a phase error  Important when signals are aperiodic ~ t TB (t) v(t) Signal Generator Output Signal v(t TB (t)) ~ t TB (t) v(t) Signal Processor Input v(t) Processor Sees v(t TB (t))  Time error defined through impact on v(t + x TB (t))  Time error defined through impact on v(t + x TB (t))

5. Page 5 FCS 2008 Time Error -- V. Reinhardt Will Use This General System Model for ME/MN Discussion Models classic information transfer systems  Communications, digital Models classic information transfer systems  Communications, digital Also models systems that transfer info to measure channel properties  Navigation, ranging, radar Also models systems that transfer info to measure channel properties  Navigation, ranging, radar Tx BB TB Gener- ate BB UC Tx RF TB DC Rx RF TB Process BB Information Tx SubsystemRx Subsystem Rx BB TB ~ V-Channel ~ X-Channels Delay  v Delay  x Base Band Loop RF Loop  RF Loop  ~ ~ PLL

6. Page 6 FCS 2008 Time Error -- V. Reinhardt Tx BB TB Gener- ate BB UC Tx RF TB DC Rx RF TB Process BB Information Tx SubsystemRx Subsystem Rx BB TB ~ V-Channel ~ X-Channels Delay  v Delay  x Base Band Loop RF Loop  RF Loop  Loop Response Function H p (f) Can Model More than Classic PLLs Width 8” H p (f) ~ ~ PLL

7. Page 7 FCS 2008 Time Error -- V. Reinhardt Statistical Properties of Signals in General Systems Autocorrelation function Autocorrelation function R v (t g,  ) = E{v(t g +  /2)v*(t g -  /2)}  t g = Global (average) time   = Local (delta) time Wide-sense stationary (WSS) Wide-sense stationary (WSS) R v (t g,  ) = R v (  )  PSD = Fourier Transform (FT) of R v (  )  PSD L v (f) = Fourier Transform (FT) of R v (  ) Non-stationary (NS) Non-stationary (NS) R v (t g,  )  R v (  )  Loève Spectrum = Double FT of R v (t g,  )  Loève Spectrum L v (f g,f) = Double FT of R v (t g,  )  Cyclo-stationary (CS)  Cyclo-stationary (CS) R v (t g +mT,  ) =R v (t g,  ) tgtg  v t WSSNSCS

8. Page 8 FCS 2008 Time Error -- V. Reinhardt The MN Convolution for L  v (f g,f) From can write From can write For RF carrier  Generating this MN convolution straightforward for neg-p L x (f) For RF carrier  Generating this MN convolution straightforward for neg-p L x (f)  &  v is WSS so x So pole in neg-p L x (f) at 0 0 f L x (f)  Assumes WSS x(t)  fofo f L v (f) v(t) is WSS & single freq Translated to pole at f o 0 f L  v (f g,f) fofo  fgfg

9. Page 9 FCS 2008 Time Error -- V. Reinhardt But for BB  Generating L  v (f g,f) from Neg-p L x (f) is Problematic BB signals broadband & centered on f = 0 BB signals broadband & centered on f = 0 Now neg-p L x (f) goes to infinity in middle of convolution Now neg-p L x (f) goes to infinity in middle of convolution So can’t define convolution for neg-p x(t) noise So can’t define convolution for neg-p x(t) noise Unless … Unless … x 0 f L v (f g,f) fgfg 0 f L x (f)  0 f L  v (f g,f) ????

10. Page 10 FCS 2008 Time Error -- V. Reinhardt There is HP Filtering of Neg-p Noise in L x (f) Will show there is such HP filtering in L x (f) due to two mechanisms Will show there is such HP filtering in L x (f) due to two mechanisms  System topological structures  Removal of causal behavior in defining MN This problem has been driver in search for neg-p HP filtering mechanisms This problem has been driver in search for neg-p HP filtering mechanisms 0 f L x (f) x 0 f L v (f g,f) fgfg 0 f L  v (f g,f)

11. Page 11 FCS 2008 Time Error -- V. Reinhardt HP Filtering of Time Error by System Topological Structures Well-known that PLL HP filters x Rx - x Tx Well-known that PLL HP filters x Rx - x Tx Delay mismatch  also HP filters x Tx Delay mismatch  also HP filters x Tx  Delay-line discriminator effect  In f-domain HP filtering of x(t) modeled as System Response Function H s (f) acting on x(t) HP filtering of x(t) modeled as System Response Function H s (f) acting on x(t)  See Reinhardt FCS 2005 & FCS 2006 for details Tx Rx PLL ~~ x Tx (t)x Rx (t) Free Running TB Errors H p (f) Delay  x Delay  v  =  x -  v PLL

12. Page 12 FCS 2008 Time Error -- V. Reinhardt What About Effect of Signal Filters H v (f) on L x (f)? Such H v (f) can only LP filter L x (f) Such H v (f) can only LP filter L x (f)  Even when H v (f) HP filter’s v(t)  Because h v (t) t- translation invariant must conserve x o  Also for broadband v(t)  H s (f) can only approx effect of H v (f) on x(t)  Because H v (f) distorts the broadband signal  So can use a simple HF cut-off f h to approximate the effect of an H v (f) on x(t) h v (t) v out (t+x o )v in (t+x o ) Slow x(t)  x o Tx Rx PLL ~~ Signal Filter H v (f)

13. Page 13 FCS 2008 Time Error -- V. Reinhardt Summary of HP Filtering of L x (f) by Topological Structures W 9” Delay Mismatch PLL Total filtered x-PSD Tx Rx PLL ~~ L Tx (f)L Rx (f) Free Running TB Errors H p (f) Delay  x Delay  v  =  x -  v

14. Page 14 FCS 2008 Time Error -- V. Reinhardt H s (f) HP Order Not Always Sufficient to Ensure Convergence of L x (f) Example: Delay mismatch for f -3 TB noise Example: Delay mismatch for f -3 TB noise To deal with this problem note that To deal with this problem note that  Causal behavior should be removed from x(t) for L x (f) in MN convolution (short term noise)  Causal behavior either part of ME (ex: drift) or corrected for & not part of either ME or MN  Without a priori knowledge must estimate causal behavior from measured data  This estimation process causes further HP filtering [Reinhardt PTTI 2007 & ION NTM 2008] L x (f) = |H  (f)| 2 L TB (f)  f -3  f -1  f 2 Tx Rx ~  =  x -  v TB  L TB (f)  f -3

15. Page 15 FCS 2008 Time Error -- V. Reinhardt Effect of Removing Fixed Causal Freq Offset in Previous Example diverges for f -3 noise diverges for f -3 noise  Let’s remove estimate of freq offset   Residual x(t) for L x (f) in MN conv is now  Proportional to error measure for non-zero dead-time Allan variance  Well known f 4 HP behavior suppresses f -3 L TB (f) divergence Now L x (f) for MN converges for f -3 noise (even without H  (f) HP filtering) Now L x (f) for MN converges for f -3 noise (even without H  (f) HP filtering) -- Est freq offset

16. Page 16 FCS 2008 Time Error -- V. Reinhardt Can Generalize to Any Causal Estimate Linear in x(t) A causal estimation process linear in x(t) A causal estimation process linear in x(t)  Can be represented using a Green’s function solution g w (t,t’)  Can be represented using a Green’s function solution g w (t,t’) [Reinhardt PTTI 2007 & ION NTM 2008]  x s (t) = H s (f) filtered TB error  G w (t,-f) = FT of g w (t,t’) over t’ Residual x-error for MN  Residual x-error for MN 

17. Page 17 FCS 2008 Time Error -- V. Reinhardt Loève Spectrum of x MN (t) Now Becomes  L j (f,f’) = Double FT of g j (t,t’) over t & t' Note HP filtered x-spectrum not WSS Note HP filtered x-spectrum not WSS  Because x est (t) not modeled as being time translation invariant  g w (t,t’) not g w (t-t’) L  v (f g,f) now given by double convolution L  v (f g,f) now given by double convolution

18. Page 18 FCS 2008 Time Error -- V. Reinhardt When Causal Model x est (t) is Time Translation Invariant And filtered x(t) is WSS And filtered x(t) is WSS  Now MN conv reduces to  Where Note t-translation invariant g w (t-t’) means Note t-translation invariant g w (t-t’) means  x est (t) has new fit solution at each x MN (t)  Ex: moves with t in x MN (t) Non t-invariant g w (t,t’) means solution fixed as t in x MN (t) changes Non t-invariant g w (t,t’) means solution fixed as t in x MN (t) changes  Ex: Single x est (t) solution for all t in T 

19. Page 19 FCS 2008 Time Error -- V. Reinhardt (M-1) th Order Polynomial Estimation Will Lead to f 2M HP Filtering in MN K x-j (f) = Average of |G j (t,f)| 2 over T K x-j (f) in dB for Unweighted LSQF over T 1 Log 10 (fT) P 5 (t)  f 10 P 4 (t)  f 8 P 3 (t)  f 6 a 0 +a 1 t  f 4 a 0 est  f 2 f = 1/T (Reinhardt PTTI 2007) K x-j (f) in dB for Weighted LSQF over T Log 10 (fT) Weighting T eff T f = 1/T eff (Reinhardt ION NTM 2008) P 5 (t) P 4 (t) P 3 (t) a 0 +a 1 t est a 0 est

20. Page 20 FCS 2008 Time Error -- V. Reinhardt Final Summary & Conclusions To properly characterize x(t) induced MN To properly characterize x(t) induced MN  Must include HP filtering effects of  System topological structures  H s (f)  Removal of causal estimate  G j (t,f)  Otherwise cannot properly define L  v (f g,f) convolution in presence of neg-p noise for broadband signals Can guarantee convergence of L  v (f g,f) in presence of neg-p noise for any neg-p Can guarantee convergence of L  v (f g,f) in presence of neg-p noise for any neg-p  By using (M-1) th order polynomial model for removing causal x(t) behavior  With HP filtering from H s (f) can use lower M-order model

21. Page 21 FCS 2008 Time Error -- V. Reinhardt Final Summary & Conclusions Note that ME or MN due to delay mismatch determined by Note that ME or MN due to delay mismatch determined by  Means that absolute accelerations of a TB are objectively observable a closed system  Without a 2 nd TB as a reference  Simply by observing changes in ME or MN Ex: Observing MN induced BER changes Ex: Observing MN induced BER changes  Is relativity principle for TBs  Frequency changes have objective observabilty while time and freq offsets do not For preprint & presentation see For preprint & presentation see www.ttcla.org/vsreinhardt/