1. The Properties of Time and Phase Variances in the Presence of Power Law Noise for Various Systems Victor S. Reinhardt Raytheon Space and Airborne Systems El Segundo California Copyright 2006 Victor S. Reinhardt--Rights to copy material is granted so long as a source reference is listed on each page, section, or graphic utilized.

2. Page 2 Characterizing Time and Phase Variances by Using a System Response Function -V Reinhardt- FCS 2006 What Paper will Discuss  Properties of time (x=  /  o ) & phase (  ) error variances In presence of negative power law noise [  < 0] * For various types of systems: Digital, communications, signal processing, radar, ranging, & timing  Variances defined using an explicit system phase response function H s (f) H s (f) allows N-sample (as well as Allan) variance of x (or  ) to be used in many systems when   -4 H s (f) is a means of categorizing variance properties  Application of approach presented in FCS 2004 & 2005 * Note exponent notation change from previous FCS papers to make it consistent with IEEE Std 1139

3. Page 3 Characterizing Time and Phase Variances by Using a System Response Function -V Reinhardt- FCS 2006 Previous Approaches for Defining Variances  Variances defined with a kernel & simple HF cut-off Kernel K x# (f) describes frequency response properties of variance used (i.e., x# = x1, x2, ….) System response only represented by HF cut-off f h Relies completely on K x# (f) for convergence of  2 x# when  < 0  Variances defined using bandpass approximation Heuristic low freq cut-off f l ensures convergence when  < 0 How to interpret and determine LF cut-off f l ?

4. Page 4 Characterizing Time and Phase Variances by Using a System Response Function -V Reinhardt- FCS 2006 Approach Used for Defining Variances in This Paper  H s (f) determined by examining phase filtering properties of system under consideration  More rigorous than introducing heuristic f l and f h f l & f h can be determined for BP approximation by examining H s (f)  For  < 0 H s (f) & K x# (f) together determine convergence Relieves some of burden on K x# (f)  H s (f) falls into general classes Can be used to categorize variance behavior in presence of f  noise

5. Page 5 Characterizing Time and Phase Variances by Using a System Response Function -V Reinhardt- FCS 2006  Digital sampling H s (f) Many samples with anti-aliasing LP filter Various digital & signal processing applications  Delay H s (f) Signal compared to delayed version of itself Clock misalignment, synchronous digital Coherent radar, delay line discriminators  Delay with averaging H s (f) Delay & averaging over time T d 2-way ranging & time transfer  Phase locked H s (f) One system tracks another through PLL Communications systems Asynchronous digital systems Disciplined osc and coordinated time systems PLL can be hidden in calibrations or corrections PLL can be after-the-fact paper correction General Classes of H s (f) Classes of H s (f) Delay Delay+Av PLL Sampling f 

6. Page 6 Characterizing Time and Phase Variances by Using a System Response Function -V Reinhardt- FCS 2006 Review of Variances to be Used  N-sample variance of x (or  )  N-sample zero dead time Allan variance of x (or  ) Note  x2 2 normalization change from  xa 2 in previous FCS papers System  Response Function H s (f) & h s (t) x(t) or  (t) x n or  n Continuous Error at Input Regular Samples at Output Model of System S x (f) (or S  (f)) Defined at Input N Samples at Interval  over Time T = N 

7. Page 7 Characterizing Time and Phase Variances by Using a System Response Function -V Reinhardt- FCS 2006 Properties of Variance Kernels  N-sample variance kernel K x1 (f) For fT << 1: K x1 (f)  f 2  2 x1 converges for S x (f)  f -1, f -2 If H s (f) contains highpass filter  2 x1 converges for S x (f)  f -3 & f -4  N-sample Allan variance kernel K x2 (f) For f  << 1: K x2 (f)  f 2 For fT << 1: K x2 (f)  f 4  2 x2 converges for S x (f)  f -1... f -4 without any HP filtering in H s (f) Log(f  ) dB K x1 (N=100) fT = 1 K x1 & K x2 f4f4 f2f2 f2f2 K x2 4 f  = 1

8. Page 8 Characterizing Time and Phase Variances by Using a System Response Function -V Reinhardt- FCS 2006 Properties of  2 x1 (or  2  1 )  When N >> 1 (T = N  finite)  When T   K x1  1 Can be used when H s (f) alone ensures convergence  Useful approximation for K x1 (N>>1) - K x1 - K’ x1

9. Page 9 Characterizing Time and Phase Variances by Using a System Response Function -V Reinhardt- FCS 2006  Digital sampling H s (f)  Delay H s (f)  Delay with averaging H s (f)  Phase locked H s (f) Properties of Variances for Various Systems Sorted by Class of H s (f) Classes of H s (f) Delay Delay+Av PLL Sampling f 

10. Page 10 Characterizing Time and Phase Variances by Using a System Response Function -V Reinhardt- FCS 2006 Digital Sampling H s (f)  H s (f) = Anti-aliasing LP filter of BW B B ~ 1/  (f l = 0 f h = B) - Nyquist sampled:  = 0.5/B - Over-sampled:  < 0.5/B  N = T/  >>1 f  |Hs(f)|  B

11. Page 11 Characterizing Time and Phase Variances by Using a System Response Function -V Reinhardt- FCS 2006 Delay H s (f)  H s (f) = Signal compared to itself delayed by  d   2 x1 (or  2  1 ) converge for   -4 If T is finite because |H s (f)| 2  f 2 for f  d << 1 K x1 (f)  f 2 for fT<<1  Justifies heuristic band-pass approx for  2  1 used in radar ~ Tx Clock Misalignment Signal Delay  c Rx Clock Delay  s  d =  c -  s ~ Coherent Radar x D dd  (t)  (t-  d )  ProcessA/D |H s (f)|  f   d -1 f2f2

12. Page 12 Characterizing Time and Phase Variances by Using a System Response Function -V Reinhardt- FCS 2006 Delay with Averaging H s (f)  H s (f) = Delay by  d  ave over T d Delay response  Average over T d response  For  =  d & T = T d  2 x1 for delay + ave same as  2 x2 for ave only because 2 equivalent pictures ~ 2-Way Ranging & Time Transfer D x  d +x tt Xponder ~ x’ tt Ranging R   d   1 Time Transfer x-x’   2 -  1 /2 x+  d /2 dd R Ave for T d 11 22 |H s (f)|  f   d -1 T d -1 f2f2

13. Page 13 Characterizing Time and Phase Variances by Using a System Response Function -V Reinhardt- FCS 2006 H p (f) = PLL response function H h (f) = Channel response for signal path  x (or  ) is error of difference between master and slave  When H s (f) ensures  convergence for all T  For 2 nd order PLL:  2 x1 converges for all T for   -4  For 1 st order PLL: Finite T need for  2 x1 convergence for   -2 Loop can cycle-slip for f -3 or f -4 noise at T given by  2  1  1 radian PLL H s (f) |H s (f)|  fhfh f  BpBp Master ~ PLL Slave Channel f h ~ Clock (or LO) H h (f) Signal Channel Phase Locked System H p (f) with Loop BW B p

14. Page 14 Characterizing Time and Phase Variances by Using a System Response Function -V Reinhardt- FCS 2006 Summary & Conclusions  Incorporating explicit H s (f) in variance definitions is useful for Categorizing variance behavior Showing that  2 x1 can be used in presence of power law noise for  < -2 Defining proper values of f h and f l to be used in bandpass approximation  4 general classes H s (f) cover a wide range of systems Digital sampling H s (f) Delay H s (f) Delay plus averaging H s (f) Phase locked H s (f)