Characterizing the Impact of Time Error on General Systems

1. Characterizing the Impact of Time Error on General Systems Victor S. ReinhardtRaytheon Space and Airborne SystemsEl Segundo CA, USA 2008 IEEE International Frequency Control Symposium Honolulu, Hawaii, USA, May 18 - 21, 2008

2. v(t) v(t) ot RFCarrier v(t+x) v(t) (t) = ox(t) t t+x(t) Time Error x(t) Impacts Systems Mainly by Generating ME & MN • 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 • Inter-symbol interference Noise power • Signal processing noise Scaling noise Baseband v(t) = v(t+x(t)) -v(t)  v’(t)x(t)

3. Lx(f) f -1 dBc/Hz f -2 f -4 f -3 Log10(f) Paper will Discuss How to Characterize x(t) Induced ME & MN • Especially in presence of random negative power law (neg-p) noise • Noise with PSD  Lx(f)  f p (p < 0)

4. OutputSignal v(tTB(t)) ProcessorSees v(tTB(t)) Input v(t) v(t) SignalGenerator v(t) SignalProcessor ~ ~ tTB(t) tTB(t) Paper Will Use Concept of a Timebase (TB) • A TB = tTB(t) is a continuous time source for generating or processing a signal v(t) • Ideal v(t) is generated or processed as v(tTB(t)) • t  ideal TB • Discrete epochs in a real TB ignored • tTB(t) = t + xTB(t) • Not through a phase error • Important when signals are aperiodic  Time error defined through impact on v(t + xTB(t))

5. Tx Subsystem Rx Subsystem Information Information V-Channel Gener- ate BB UC DC ProcessBB Delay v Tx BBTB Rx BB TB ~ ~ Tx RFTB Rx RF TB ~ ~ RF Loop X-Channels  BaseBandLoop PLL PLL Delay x Will Use This General System Model for ME/MN Discussion • Models classic information transfer systems  Communications, digital • Also models systems that transfer info to measure channel properties  Navigation, ranging, radar

6. Hp(f) Loop Response Function Hp(f) Can Model More than Classic PLLs Tx Subsystem Rx Subsystem Information Information V-Channel Gener-ate BB UC DC ProcessBB Delay v Tx BBTB Rx BB TB ~ ~ Tx RFTB Rx RF TB ~ ~ RF Loop X-Channels  BaseBandLoop PLL PLL Delay x Width 8”

7. v tg t  WSS NS CS Statistical Properties of Signals in General Systems • Autocorrelation function Rv(tg,) = E{v(tg+/2)v*(tg-/2)} • tg = Global (average) time •  = Local (delta) time • Wide-sense stationary (WSS) Rv(tg,) = Rv() • PSD Lv(f) = Fourier Transform (FT) of Rv() • Non-stationary (NS) Rv(tg,)  Rv() • Loève SpectrumLv(fg,f) = Double FT of Rv(tg,) • Cyclo-stationary (CS) Rv(tg+mT,) =Rv(tg,)

8. Lx(f)  Lv(f) Lv(fg,f)  f x 0 f f fo fg 0 fo v(t) is WSS & single freq Translated to pole at fo So pole inneg-p Lx(f) at 0 The MN Convolution for Lv(fg,f) • From can write • For RF carrier  Generating this MN convolution straightforward for neg-p Lx(f) • & v is WSS so AssumesWSS x(t) 

9. Lv(fg,f)  ???? Lv(fg,f) f 0 fg f 0 Lx(f) f 0 But for BB  Generating Lv(fg,f) from Neg-p Lx(f) is Problematic • BB signals broadband & centered on f = 0 • Now neg-p Lx(f) goes to infinity in middle of convolution • So can’t define convolution for neg-p x(t) noise • Unless … x

10. x Lx(f) Lv(fg,f) f 0 Lv(fg,f) f 0 fg f 0 There is HP Filtering of Neg-p Noise in Lx(f) • Will show there is such HP filtering in Lx(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

11. Delay v Tx Rx xTx(t) xRx(t) ~ ~ Free RunningTB Errors Delay x PLL  = x - v HP Filtering of Time Error by System Topological Structures • Well-known that PLL HP filters xRx - xTx • Delay mismatch  alsoHP filters xTx • Delay-line discriminator effect • In f-domain • HP filtering of x(t) modeled as System Response Function Hs(f)acting on x(t) • See Reinhardt FCS 2005 & FCS 2006 for details PLL Hp(f)

12. SignalFilter Tx Rx Hv(f) ~ ~ PLL vin(t+xo) vout(t+xo) hv(t) What About Effect of Signal Filters Hv(f) on Lx(f)? • Such Hv(f) can onlyLP filter Lx(f) • Even when Hv(f) HP filter’s v(t) • Because hv(t) t- translationinvariant must conserve xo • Also for broadband v(t)  Hs(f) can only approx effect of Hv(f) on x(t) • Because Hv(f) distorts the broadband signal • So can use a simple HF cut-off fh to approximate the effect of an Hv(f) on x(t) Slow x(t)  xo

13. Delay v Total filtered x-PSD Tx Rx LTx(f) LRx(f) ~ ~ DelayMismatch Free RunningTB Errors Delay x PLL PLL  = x - v Summary of HP Filtering of Lx(f) by Topological Structures Hp(f) W 9”

14. Tx Rx Lx(f) = |H(f)|2 LTB(f)  = x-v ~ TB  LTB(f)  f -3 Hs(f) HP Order Not Always Sufficient to Ensure Convergence of Lx(f) • Example: Delay mismatch for f -3 TB noise • To deal with this problem note that • Causal behavior should be removed from x(t) for Lx(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]  f-1  f 2  f -3

15. Est freqoffset - Effect of Removing Fixed Causal Freq Offset in PreviousExample • diverges for f -3 noise • Let’s remove estimateof freq offset  • Residual x(t) for Lx(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 LTB(f) divergence • Now Lx(f) for MN converges for f -3 noise (even without H(f) HP filtering)

16. Can Generalize to Any Causal Estimate Linear in x(t) • A causal estimation process linear in x(t) • Can be represented using a Green’s function solution gw(t,t’) [Reinhardt PTTI 2007 & ION NTM 2008] • xs(t) = Hs(f) filtered TB error • Gw(t,-f) = FT of gw(t,t’) over t’ • Residual x-error for MN 

17. Loève Spectrum of xMN(t) Now Becomes • Lj(f,f’) = Double FT of gj(t,t’) over t & t' • Note HP filtered x-spectrum not WSS • Because xest(t) not modeled as being time translation invariant • gw(t,t’) not gw(t-t’) • Lv(fg,f) now given by double convolution

18. When Causal Model xest(t) is Time Translation Invariant • And filtered x(t) is WSS • Now MN convreduces to • Where • Note t-translation invariant gw(t-t’) means • xest(t) has new fit solution at each xMN(t) • Ex: moves with t in xMN(t) • Non t-invariant gw(t,t’) means solution fixed as t in xMN(t) changes • Ex: Single xest(t) solution for all t in T 

19. Kx-j(f) in dB forUnweighted LSQF over T Kx-j(f) in dB forWeighted LSQF over T (Reinhardt PTTI 2007) (Reinhardt ION NTM 2008) a0 est f 2 a0 est a0+a1t f 4 f = 1/Teff f = 1/T a0+a1t est P3(t)f 6 Weighting Teff P3(t) P4(t)  f 8 P5(t) f 10 P4(t) P5(t) T 1 Log10(fT) Log10(fT) (M-1)th Order Polynomial Estimation Will Lead to f 2M HP Filtering in MN Kx-j(f) = Average of |Gj(t,f)|2 over T

20. Final Summary & Conclusions • To properly characterize x(t) induced MN • Must include HP filtering effects of • System topological structures  Hs(f) • Removal of causal estimate  Gj(t,f) • Otherwise cannot properly define Lv(fg,f) convolution in presence of neg-p noise for broadband signals • Can guarantee convergence of Lv(fg,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 Hs(f) can use lowerM-order model

21. Final Summary & Conclusions • 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 2nd TB as a reference • Simply by observing changes in ME or MN • 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 www.ttcla.org/vsreinhardt/