Characterizing the Impact of Time Error on General Systems Victor S. Reinhardt Raytheon Space and...

Characterizing the Impact of Characterizing the Impact of Time Error on General SystemsTime Error on General Systems

Victor S. ReinhardtVictor S. ReinhardtRaytheon Space and Airborne SystemsRaytheon Space and Airborne Systems

El Segundo CA, USAEl Segundo CA, USA

2008 IEEE International Frequency Control Symposium

Honolulu, Hawaii, USA, May 18 - 21, 2008

FCS 2008 Time Error -- V. Reinhardt

Time Error x(t) Impacts Systems Mainly by Generating ME & MN

• ME =ME = Multiplicative Signal Error Multiplicative Signal ErrorMN = MN = Multiplicative NoiseMultiplicative Noise Short term MEShort term MECan be causal or randomCan be causal or random

x(t) induces ME & MN in generated or x(t) induces ME & MN in generated or processed signals through slope modulationprocessed signals through slope modulation

• MN Also called MN Also called Inter-symbol interferenceInter-symbol interference Noise powerNoise powerSignal processing noiseSignal processing noise Scaling noiseScaling noise

v(t) = v(t+x(t)) - v(t) v’(t)x(t)

t

v(t)

v(t) v(t+x)

t+x(t)

RFCarrier v(t)

(t) = ox(t)ot

Baseband


Paper will Discuss How to Characterize x(t) Induced ME & MN

• Especially in presence of random negative Especially in presence of random negative power law (neg-p) noisepower law (neg-p) noise

Noise with PSD Noise with PSD

Lx(f)

dB

c/H

z f -1

Log10(f)

f -2

f -3f -4

Typical Neg-p Values pFlicker of Phaseor Time Error x(t)

-1

Random Walk of Phase or White Frequency

-2

Flicker of Frequency -3Random Walk of Frequency

or Random Run -4

Lx(f) f p (p < 0 )


Paper Will Use Concept of a Timebase (TB)

• A TB = tA TB = tTBTB(t) is(t) is a a continuouscontinuous time source for time source for generating or processing a signal v(t)generating or processing a signal v(t)

Ideal v(t) is generated or processed as v(tIdeal v(t) is generated or processed as v(tTBTB(t))(t))t t ideal TB ideal TB Discrete epochs in a real TB ignoredDiscrete epochs in a real TB ignored

ttTBTB(t) = t + x(t) = t + xTBTB(t) (t)

Not through a phase errorNot through a phase errorImportant when signals are Important when signals are aperiodicaperiodic

~ tTB(t)

v(t) SignalGenerator

OutputSignal

v(tTB(t))~ tTB(t)

v(t) SignalProcessor

Inputv(t)

ProcessorSees

v(tTB(t))

Time error defined through Time error defined through impact on v(t + ximpact on v(t + xTBTB(t))(t))


Will Use This General System Model for ME/MN Discussion

• Models classic information transfer Models classic information transfer systems systems Communications, digital Communications, digital

• Also models systems that transfer info to Also models systems that transfer info to measure channel properties measure channel properties Navigation, Navigation, ranging, radarranging, radar

Tx BBTB

Gener- ate BB UC

Tx RFTB

DC

Rx RF TB

ProcessBB

Information InformationTx Subsystem Rx Subsystem

Rx BB TB ~

V-Channel

~

X-Channels

Delay v

Delay x

BaseBaseBandBandLoopLoop

RF LoopRF Loop

~ ~

PLL PLL


Tx BBTB

Gener-ate BB UC

Tx RFTB

DC

Rx RF TB

ProcessBB

Information InformationTx Subsystem Rx Subsystem

Rx BB TB ~

V-Channel

~

X-Channels

Delay v

Delay x

BaseBaseBandBandLoopLoop

RF LoopRF Loop

Loop Response Function Hp(f) Can Model More than Classic PLLs

Width 8”

Hp(f)

~ ~

PLL PLL

Free-running Null 0 1Synchronous Hardline 1 0

System Type Hp(f) Name Hp(f) 1-Hp(f)

1st order PLL 1st ord LP 1st ord HP

2nd order PLL 2nd ord LP 2nd ord HPAsynchronous


Statistical Properties of Signals in General Systems

• Autocorrelation functionAutocorrelation functionRv(tg,) = E{v(tg+/2)v*(tg-/2)}ttgg = Global (average) time = Global (average) time = Local (delta) time= Local (delta) time

• Wide-sense stationary (WSS) Wide-sense stationary (WSS) Rv(tg,) = Rv() PSDPSD Lv(f) = Fourier Transform (FT) of R = Fourier Transform (FT) of Rvv(())

• Non-stationary (NS) Non-stationary (NS) Rv(tg,) Rv() Loève SpectrumLoève Spectrum Lv(fg,f) = Double FT of R = Double FT of Rvv(t(tgg,,))Cyclo-stationaryCyclo-stationary (CS) (CS) Rv(tg+mT,) =Rv(tg,)

tg

vt

WSS NS CS


The MN Convolution for Lv(fg,f)

• From can write From can write

• For RF carrier For RF carrier Generating this MN Generating this MN convolution straightforward for neg-p Lconvolution straightforward for neg-p Lxx(f)(f)

& & v is WSS sov is WSS so

(t)x(t)v'v(t) δ

x

So pole inSo pole inneg-p Lneg-p Lxx(f) at 0(f) at 0

0 f

Lx(f)

)f'-(f)Lf',(fL/4]'[df'f),(fL xgv2g

2gv ωω AssumesAssumes

WSS x(t)WSS x(t)

(f))L(ff),(fL vggv δδ δ

fof

Lv(f)

v(t) is WSS v(t) is WSS & single freq& single freq

)f-(fL(f)L oxv δ

Translated to Translated to pole at fpole at foo

0 f

Lv(fg,f)

fo

fg


But for BB Generating Lv(fg,f) from Neg-p Lx(f) is Problematic

• BB signals broadband & centered on f = 0BB signals broadband & centered on f = 0• Now neg-p LNow neg-p Lxx(f) goes to infinity in middle (f) goes to infinity in middle

of convolutionof convolution

• So can’t define convolution for neg-p x(t) So can’t define convolution for neg-p x(t) noisenoise

• Unless …Unless …

x

)f',(fL/4]'[df'f),(fL gv2g

2gv ωω )f'-(fLx

0 f

Lv(fg,f)

fg 0 f

Lx(f)

0 f

Lv(fg,f)????


There is HP Filtering of Neg-p Noise in Lx(f)

• Will show there is such HP filtering in LWill show there is such HP filtering in Lxx(f) (f) due to two mechanismsdue to two mechanismsSystem topological structuresSystem topological structuresRemoval of causal behavior in defining MN Removal of causal behavior in defining MN

• This problem has been driver in search for This problem has been driver in search for neg-p HP filtering mechanisms neg-p HP filtering mechanisms

)f'-(fL|)f'-H(f|)f',(fL'df'f),(fL TB2

gv2

gv ω

0 f

Lx(f)

x0 f

Lv(fg,f)

fg 0 f

Lv(fg,f)

)f'-(fLx


HP Filtering of Time Error by System Topological Structures

• Well-known that PLL HP filters xWell-known that PLL HP filters xRxRx - x - xTxTx

• Delay mismatch Delay mismatch also alsoHP filters xHP filters xTxTx

Delay-line discriminator effectDelay-line discriminator effectIn f-domainIn f-domain

• HP filtering of x(t) modeled as HP filtering of x(t) modeled as System System Response Function HResponse Function Hss(f)(f) acting on x(t) acting on x(t)See Reinhardt FCS 2005 & FCS 2006 for detailsSee Reinhardt FCS 2005 & FCS 2006 for details

ΔτΔτΔτ (t)y(t)x - )(txx TxTxTx

(f)(f)XH(f)X TxΔτΔτ

1)(ff)f(4sin|(f)H| 222 ΔτΔτπΔτ

Tx Rx

PLL

~~xTx(t) xRx(t)Free Running

TB Errors

Hp(f)

Delay x

Delay v

= x - v PLL

(f)]X - (f)][XH[1(f)X TxRxpPLL


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

• Such HSuch Hvv(f) can only(f) can onlyLP filter LLP filter Lxx(f)(f)Even when HEven when Hvv(f) HP filter’s v(t)(f) HP filter’s v(t)Because hBecause hvv(t) t- translation(t) t- translation

invariant must conserve xinvariant must conserve xoo

Also for broadband v(t) Also for broadband v(t) H Hss(f) can only (f) can only approx effect of Happrox effect of Hvv(f) on x(t) (f) on x(t) Because HBecause Hvv(f) distorts the broadband signal(f) distorts the broadband signal

So can use a simple HF cut-off fSo can use a simple HF cut-off fhh to to approximate the effect of an Happroximate the effect of an Hvv(f) on x(t)(f) on x(t)

hv(t)vout(t+xo)vin(t+xo)

Slow x(t) xo

Tx Rx

PLL

~~

SignalFilterHv(f)


Summary of HP Filtering of Lx(f) by Topological Structures

W 9”

(f)L|(f)H|(f)L Tx2

Δττ DelayMismatch

(f)]L(f)[L|(f)H-1|(f)L TxRx2

pp PLL

(f)L (f)L(f)L px τTotal filtered x-PSD

System HP Filtering HP Order TB Error Systems

Mismatch |H(f)|2 f2 LTx(f) Sync & Async

1st ord |1-Hp(f)|2 f2 LTx(f)+LRx(f) Async

2nd ord |1-Hp(f)|2 f4 LTx(f)+LRx(f) Async

Hardline |1-Hp(f)|2 SyncZero Error

None ---- LTx(f)+LRx(f) Free Running

Tx Rx

PLL

~~LTx(f) LRx(f)Free Running

TB Errors

Hp(f)

Delay x

Delay v

= x - v


Hs(f) HP Order Not Always Sufficient to Ensure Convergence of Lx(f) • Example: Delay mismatch for fExample: Delay mismatch for f -3 -3 TB noise TB noise

• To deal with this problem note thatTo deal with this problem note thatCausal behavior should be removed from x(t) Causal behavior should be removed from x(t)

for Lfor Lxx(f) in MN convolution (short term noise)(f) in MN convolution (short term noise)Causal behavior either part of ME (ex: drift) or Causal behavior either part of ME (ex: drift) or corrected for & not part of either ME or MNcorrected for & not part of either ME or MN

Without a priori knowledge must estimate Without a priori knowledge must estimate causal behavior from measured datacausal behavior from measured data

This estimation process causes further HP This estimation process causes further HP filtering filtering [Reinhardt PTTI 2007 & ION NTM 2008][Reinhardt PTTI 2007 & ION NTM 2008]

Lx(f) = |H(f)|2 LTB(f) f -3 f

-1 f 2

Tx Rx

~ = x-v

TB LTB(f) f -3


Effect of Removing Fixed Causal Freq Offset in Previous Example

• diverges for fdiverges for f -3 -3 noisenoiseLet’s remove estimateLet’s remove estimate

of freq offset of freq offset Residual x(t) for LResidual x(t) for Lxx(f)(f)

in MN conv is nowin MN conv is now

Proportional to error measure for non-zero Proportional to error measure for non-zero dead-time Allan variancedead-time Allan variance

Well known fWell known f 4 4 HP behavior suppresses f HP behavior suppresses f -3 -3 L LTBTB(f) (f) divergencedivergence

• Now LNow Lxx(f) for MN converges for f(f) for MN converges for f -3 -3 noise noise (even without H(even without H(f) HP filtering)(f) HP filtering)

Δτ][ T)(ty-(t)yx TxTxMN

ΔτΔτ (t)yx

T)-(ty(t)y

-

T)(ty-(t)y

Est freqoffset

T)-(ty


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 Can be represented using a Green’s function

solution gsolution gww(t,t’) (t,t’) [Reinhardt PTTI 2007 & ION NTM 2008]

xxss(t) = H(t) = Hss(f) filtered TB error(f) filtered TB errorGGww(t,-f) = FT of g(t,-f) = FT of gww(t,t’) over t’ (t,t’) over t’

• Residual x-error for MN Residual x-error for MN

)(t'x)t'(t,gdt'(t)x sw

- est

(f)(f)XHf)(t,Gdf(t)x TBsj

f

f- MN

h

h

(t)x-(t)x(t)x estsMN

f)(t,G-ef)(t,G wtj

jω

(f)(f)XHf)(t,Gdf TBsw

f

f-

h

h


Loève Spectrum of xMN(t) Now Becomes

LLjj(f,f’) = Double FT of g(f,f’) = Double FT of gjj(t,t’) over t & t'(t,t’) over t & t'

• Note HP filtered x-spectrum not WSSNote HP filtered x-spectrum not WSSBecause xBecause xestest(t) not modeled as being time (t) not modeled as being time

translation invarianttranslation invariantggww(t,t’) not g(t,t’) not gww(t-t’)(t-t’)

• LLvv(f(fgg,f) now given by double convolution,f) now given by double convolution)f'-f,f'-(f)Lf'-f,(fL /4]'-'[df'df'f),(fL ggj-xgv

2g

2ggv ωω

*gjgj

f

fgjx )f',0.5f(f)Lf',0.5f(fLdf'f),(fL

h

h

)(f'L|)(f'H| TB2

s


When Causal Model xest(t) is Time Translation Invariant

• And filtered x(t) is WSSAnd filtered x(t) is WSSNow MN convNow MN conv

reduces toreduces to

WhereWhere

• Note t-translation invariant gNote t-translation invariant gww(t-t’) means(t-t’) meansxxestest(t) has new fit solution at each x(t) has new fit solution at each xMNMN(t)(t)Ex: moves with t in xEx: moves with t in xMNMN(t)(t)

• Non t-invariant gNon t-invariant gww(t,t’) means solution (t,t’) means solution fixed as t in xfixed as t in xMNMN(t) changes(t) changesEx: Single xEx: Single xestest(t) solution for all t in T(t) solution for all t in T

(f)L|(f)H||(f)H|(f)L TB2

s2

jx

)(f')Lf',(fL'df'f),(fL xgv2

gv ω

)t'-(tg)t'(t,g jj (f)f)H(f'f),(f'L jj δ

(f))L(ff),(fL xggx δ

T)-(ty


(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

Kx-j(f) in dB forUnweighted LSQF over T

-2 -1 0 1 2-150

-100

-50

0

1Log10(fT)

P 5(t

) f

10

P 4(t)

f

8

P 3(t)

f

6a 0+a 1

t f

4a0 est f 2

f = 1/T

1M-

0m

mmM ta(t)P

(Reinhardt PTTI 2007)

-2 -1 0 1 2-150

-100

-50

0

Kx-j(f) in dB forWeighted LSQF over T

Log10(fT)

Weighting

Teff

T

f = 1/Teff

(Reinhardt ION NTM 2008)

P 5(t

)

P 4(t

) P 3(t)

a 0+a 1

t est

a 0 est


Final Summary & Conclusions

• To properly characterize x(t) induced MNTo properly characterize x(t) induced MNMust include HP filtering effects ofMust include HP filtering effects ofSystem topological structures System topological structures H Hss(f) (f) Removal of causal estimate Removal of causal estimate G Gjj(t,f)(t,f)

Otherwise cannot properly define LOtherwise cannot properly define Lvv(f(fgg,f) ,f) convolution in presence of neg-p noise for convolution in presence of neg-p noise for broadband signalsbroadband signals

• Can guarantee convergence of LCan guarantee convergence of Lvv(f(fgg,f) in ,f) in presence of neg-p noise for any neg-ppresence of neg-p noise for any neg-pBy using (M-1)By using (M-1)thth order polynomial model for order polynomial model for

removing causal x(t) behaviorremoving causal x(t) behaviorWith HP filtering from HWith HP filtering from Hss(f) can use lower(f) can use lower

M-order model M-order model


Final Summary & Conclusions

• Note that ME or MN due to delay mismatch Note that ME or MN due to delay mismatch determined by determined by Means that absolute accelerations of a TB are Means that absolute accelerations of a TB are

objectively observable a closed systemobjectively observable a closed systemWithout a 2Without a 2ndnd TB as a reference TB as a referenceSimply by observing changes in ME or MNSimply by observing changes in ME or MN

• Ex: Observing MN induced BER changesEx: Observing MN induced BER changesIs relativity principle for TBsIs relativity principle for TBsFrequency changes have objective Frequency changes have objective observabilty while time and freq offsets do notobservabilty while time and freq offsets do not

• For preprint & presentation seeFor preprint & presentation seewww.ttcla.org/vsreinhardtwww.ttcla.org/vsreinhardt//

(t)x - )(tx TxTx Δτ

Characterizing the Impact of Time Error on General Systems Victor S. Reinhardt Raytheon Space and...

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