Characterizing the Impact of Time Error on General Systems

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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
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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(tx(t)) - v(t) ? v(t)x(t)
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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 lt 0 )
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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))
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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

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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
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Statistical Properties of Signals in General
Systems

• Autocorrelation function
• Rv(tg,?) Ev(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 Spectrum Lv(fg,f) Double FT of
  Rv(tg,?)
• Cyclo-stationary (CS) Rv(tgmT,?) Rv(tg,?)

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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)
?
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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 cant define convolution for neg-p x(t) noise
• Unless

x
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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

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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)
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What About Effect of Signal Filters Hv(f) on
Lx(f)?

• Such Hv(f) can onlyLP filter Lx(f)
• Even when Hv(f) HP filters 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
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Summary of HP Filtering of Lx(f) by Topological
Structures
Hp(f)
W 9
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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 -3
? f -1
? f 2
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Effect of Removing Fixed Causal Freq Offset in
Previous Example

• diverges for f -3 noise
• Lets 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)

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Can Generalize to Any Causal Estimate Linear in
x(t)

• A causal estimation process linear in x(t)
• Can be represented using a Greens 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 ?

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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

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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

?
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(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
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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

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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/