A Physical Interpretation of Difference Variances Victor S. Reinhardt 2007 Joint Meeting of the...

A Physical Interpretation of A Physical Interpretation of Difference VariancesDifference Variances

Victor S. ReinhardtVictor S. Reinhardt

2007 Joint Meeting of the European Time and 2007 Joint Meeting of the European Time and Frequency Forum (EFTF) and the IEEE International Frequency Forum (EFTF) and the IEEE International

Frequency Control Symposium (IEEE-FCS)Frequency Control Symposium (IEEE-FCS)

Geneva, SwitzerlandGeneva, Switzerland

May 29 - June 1, 2007May 29 - June 1, 2007

Copyright 2007 Victor S. Reinhardt--Rights to copy material is granted so long as a source reference is listed on each page, section, or graphic utilized.

A Physical Interpretation of Difference Variances V.S. Reinhardt

A Physical Interpretation of Difference Variances of the Time Error x(t)

• Most variances of x(t) used in the industry can be written asMost variances of x(t) used in the industry can be written asMMthth order difference variances order difference variances x,Mx,M

22 or their sample statistics or their sample statistics

E{..} more appropriate for this paper than <..>E{..} more appropriate for this paper than <..>MM defined so all defined so all x,Mx,M

22 equal for uncorrelated white noise equal for uncorrelated white noise

• But not all But not all x,Mx,M22 equal for neg- equal for neg- (neg power law) noise (neg power law) noise

Then which order M of Then which order M of x,Mx,M22 most appropriate to measure random most appropriate to measure random

residual error in a given problem?residual error in a given problem? Paper will answer question by providing a rationale for choosing M Paper will answer question by providing a rationale for choosing M FSCS2007 figs.doc 9.05” width

x,M2 = M

-1E{[()Mx(t)]2} x,M2 = M

-1E{[()Mx(t)]2} ()x(t)

x(t+) - x(t)

M

0m

2M ]

m)!(Mm!

M![λ

M ()Mx(t) M Associated Names x,M2

0 x(t) - ( = E{x(t)}) [1] 1 Standard, Sample E{[x(t)-]2}

1 x(t+)-x(t) 2 TIErms (ITU-T G.810) TIErms2/2

2 x(t+2)-2x(t+)+x(t) 6 x2(), y

2(), TVAR 2y2()/3

3 x(t+3 )-3x(t+2)+3x(t+)-x(t) 20 Hy2() = Hadamard 32

Hy2()/10

[1] M=0 is degenerate case. ()0x(t) x(t) - . ()M = 0 for M>0


To Address Question Must Investigate Behavior of x(t) Over Long Interval T• x(t) generally separated into causal environmental & aging x(t) generally separated into causal environmental & aging

factors plus a random residual errorfactors plus a random residual error Ignoring spurs, etc., but won’t effect over-all conclusionsIgnoring spurs, etc., but won’t effect over-all conclusions

• Environmental sensitivity to temp, power supply, etc.Environmental sensitivity to temp, power supply, etc. Environmental coefficients can be determined from data over time Environmental coefficients can be determined from data over time

<<T where other factors small<<T where other factors small Will assume x(t) has environment already removed & removalWill assume x(t) has environment already removed & removal

has little impact on long term behavior of x(t)has little impact on long term behavior of x(t)

• Aging – Model as (M-1)Aging – Model as (M-1)thth order polynomial order polynomial

Will assume A & env coeffs not functions of time over TWill assume A & env coeffs not functions of time over T Must determine A over ~T for accurate determinationMust determine A over ~T for accurate determination Thus long term random errors will impact aging determinationThus long term random errors will impact aging determination

• Residual Residual xr,M(t) = x(t) – xa,M(t,A) & variance & variance r,M(t)2 = E{xr,M(t)2}

CCoefficients in xa,M(t,A) are determined by fit to data over T Behavior of aging fit will lead us to our rationale for choosing Behavior of aging fit will lead us to our rationale for choosing x,Mx,M

22

1M

0m

momMa, t-(taA)(t,x ) )aa,(aA 1M-1,...0

T

x(t) w/oEnv t

Tot

Aging xa,M(t,A)

Residual xr,M(t)

to


Behavior of Aging Fit When Neg- Noise is Present

• Model x(t) over T asModel x(t) over T as TrueTrue aging x aging xa,Ma,M(A(A(0)(0),t) plus ,t) plus

random power law noise Lrandom power law noise Lxx(f) (f) f f

• Can use least squares fit (LSQF) toCan use least squares fit (LSQF) todetermine A from N samples xdetermine A from N samples xnn over T over T r,Mr,M = RMS residual from = RMS residual from estimatedestimated x xa,Ma,M(A,t)(A,t) xa,M = xa,M(t,A) – xa,M(t,A(o)) = Est – true aging = Est – true aging

• For white-x (For white-x (ff 0 0) noise ) noise r,Mr,M

2 2 & & xxa,Ma,M 0 as N 0 as N (BW large) (BW large)

• However for neg-However for neg- noise ( noise ( -2) -2) ff -1 -1 noise more like white-x noise noise more like white-x noise

r,Mr,M22 & & xxa,Ma,M do notdo not 0 as N 0 as N

True aging can’t be recovered below levelTrue aging can’t be recovered below levelof long term noise (in single meas over T)of long term noise (in single meas over T)

Must ensemble average over many dataMust ensemble average over many datasets to reduce error further sets to reduce error further

Problem for real frequency sources unless aging model valid overProblem for real frequency sources unless aging model valid overt >> T t >> T Each device has unique aging coefficients Each device has unique aging coefficientslong term error-1.xls

f 0 Noise

r,M approx the same for all

f -2 Noise

EstAging

xa,M

True Aging xn

t

f -4 Noise

xr,M


Further Properties of Aging Solutions for Neg- Noise• Numerical LSQF solutions for M=2Numerical LSQF solutions for M=2

T = 4K points generated ofT = 4K points generated ofx(t) = xx(t) = xa,2a,2(A(A(0)(0),t) + f,t) + f

noise noise Estimated A determined by LSQF usingEstimated A determined by LSQF using

N = 2 to 2K samples spaced over TN = 2 to 2K samples spaced over T RMS RMS r,2r,2 & & xxa,2a,2 averaged over all 4K averaged over all 4K

points & LSQF for all possible start points & LSQF for all possible start offsets offsets

r,2r,2 relatively insensitive to # of relatively insensitive to # of samples in LSQF (esp for neg-samples in LSQF (esp for neg-))

For neg-For neg- noise: noise: RMS RMS xxa,2a,2 > RMS > RMS r,2r,2 Expected for correlated noiseExpected for correlated noise

• Theoretical LSQF SolutionsTheoretical LSQF Solutions Can write formal solution in terms of RCan write formal solution in terms of Rxx(() )

& Green’s functions& Green’s functions Doesn’t lead us to our goal of obtaining a Doesn’t lead us to our goal of obtaining a

physical interpretation for physical interpretation for x,Mx,M22

Solutions increase in complexity with MSolutions increase in complexity with M A simpler approach that will lead us to A simpler approach that will lead us to

our goal our goal is next topicis next topic

Errors in LSQF forxa,2(A,t) = a0 + a1t

0 deg of freedom(2 Samples in LSQF)

Samples in LSQF 1 10 100 1K

RMS r,2 f

0 Noise2

1

0

2

1

0

f -2 Noise

RMS xa,2

2

1

0

f -4 Noise

RMS xa,2

RMS xa,2

RMS r,2

RMS r,2


A Simpler Approach – The Zero Degree of Freedom (N=M) Approximation for r,M

2

• Can use fact that Can use fact that r,Mr,M22 rel insensitive to rel insensitive to

# points to simplify calculation of # points to simplify calculation of r,Mr,M22

• Zero degree of freedom solution (N=M)Zero degree of freedom solution (N=M) M+1 xM+1 xnn (n = 0 to M) spaced by (n = 0 to M) spaced by over T over T Calc A from only M points (Calc A from only M points (excludeexclude n=p) n=p) Can solve for Can solve for r,Mr,M

22 at n=p at n=p withoutwithout actually calculating x actually calculating xa,Ma,M(A,t)(A,t)Generalization of Newton’s Forward Difference Formula Generalization of Newton’s Forward Difference Formula

• Physical interpretation of x,M()2 Approximate measure of r,M

2 for any # of points in LSQF Rationale for choosing M in x,M()2 Match to M in xa,M(t,A) when

poly aging removal specified in problem When poly aging not specified Match x,M()2 to lowest order

xa,M(t,A) that fits actual aging over T (Occam’s razor)

xr,M(to+p) = ()Mx(to)/C(M, p)xr,M(to+p) = ()Mx(to)/C(M, p) r,M2(to+p) = M x,M

2()/C(M,p)2r,M2(to+p) = M x,M

2()/C(M,p)2

xM

xa,M

x0

T = M

xpxr,M

xn x(to + n) n = 0 to M

M

0nn

M n)xC(M,x(t))Δ(τp)!(Mp!

M!1)(p)C(M,

pM


Interpreting x,M2 as a Measure of Residual

Error After Aging Removal

M Associated Names x,M2 Aging Removed

0 Standard, Sample E{[x(t)-]2} None

1 TIErms (ITU-T G.810) TIErms2/2 a0

2 x2(), y

2(), TVAR 2y2()/3 a0+ a1t

3 Hy2() = Hadamard 32

Hy2()/10 a0+ a1t+a2t

2

FSCS2007 figs.doc 9” width

xr,1 = ()1x0

x0

x1

xa

xr

xr,0 = ()0x0 x0 -

0th order aging(time offset) removal

1st order aging (time &freq offset) removal

r,12 = TIErms2 r,0

2 = Standard

No removal

r,22 = 0.52y

2()

xr,2 = 0.5()2x0

x0

x1

x2

xa

p 0.5M for min r,M2

x0

xr


Can Extend the Definition to Finite Sample Statistics

• Overlapping statistic (NOverlapping statistic (N, finite data length T, finite data length Tdd = Ndt > M = Ndt > M))

• Modified statistic Modified statistic

• Can write similar equationCan write similar equationfor total statisticfor total statistic

• These are statistics of r,M2 when xa,M(A,t) fit to data

over T (not over Td) Implicit solution for A changes as t (tot) is slid over Td - M Aging model needs only be valid representation over T (not Td)

2MMT

0

1d

1M

2modM,x, ](t)x)[dt)M(T

d

τ

τΔ(ττλσ

x(t)dt(t)xt

t

1τ

τ τ

Finite time average of M

-1[()M x(t)]2 2 MMT

0

1d

1M

2ovM,x, x(t))(dt)M(T

d

τΔτλστ

Finite time average of M

-1[()M x(t)]2

x(t) = time average over

Solution for A changes with t

xM

xa,M

x0

T = M

xpxr,M

t t+T


Spectral Properties of x,M2

• Can write spectral integral for Can write spectral integral for x,Mx,M22 as as

LLxx(f) = SSB PSD of x(t)(f) = SSB PSD of x(t) KKx,Mx,M(f) = x-kernel for (f) = x-kernel for x,Mx,M

22

• |H|Hss(f)|(f)|22 = System response function (See FCS 2006 paper) = System response function (See FCS 2006 paper) Spectral properties Spectral properties specificspecific to system under consideration to system under consideration Have shown HHave shown Hss(f) often has k(f) often has kthth order zero at f=0 order zero at f=0 Aids in the convergence of low order Aids in the convergence of low order x,Mx,M

22 for neg- for neg- noise noise

• Kx,M(f) f 2M for f<<1 for f<<1 Well-known property of Well-known property of x,Mx,M22 kernels kernels

x-kernels of statistics have same f<<1 behavior for M>1x-kernels of statistics have same f<<1 behavior for M>1 Statistic of Statistic of x,0x,0

22 (sample variance) is special case because of (sample variance) is special case because of

• Removing xRemoving xa,Ma,M(t,A) from x(t) causes residual noise to be HP filtered!!(t,A) from x(t) causes residual noise to be HP filtered!! Consequence of interpretation of Consequence of interpretation of x,Mx,M

22 as approx measure of as approx measure of r,Mr,M22

Can interpret xCan interpret xa,Ma,M(t,A) as information removed from x(t) as well as aging(t,A) as information removed from x(t) as well as aging

Thus removing information from x(t) by fitting to data causes residual noise to be HP filtered

(f)dfK|(f)H|(f)L2σ Mx,

2sx

2x.M

0

|Hs(f)|2 Replaces fh

)f(sin2(f)K 2M2M1MMx, τπλ


Summary of Consequences of Interpretation of x,M

2 as measure of r,M2

• Removing aging or information from data turns a seeming Removing aging or information from data turns a seeming standard variance into a higher order variancestandard variance into a higher order variance Removing xRemoving xa,Ma,M(t,A) from data acts as a highpass filter on the noise (t,A) from data acts as a highpass filter on the noise

HP filtering order depends on complexity of info extracted (as HP filtering order depends on complexity of info extracted (as measured by poly order)measured by poly order)

Based on assumption that x-kernel of general Based on assumption that x-kernel of general r,Mr,M22 will have same will have same

ff2M2M behavior for f<<1 behavior for f<<1Need to investigate behavior of exact x-kernels of Need to investigate behavior of exact x-kernels of r,Mr,M

22 further further• Order of Order of x,Mx,M

22 should match order of x should match order of xa,Ma,M(t,A) that is specified (t,A) that is specified by problem or most appropriately fits the data over Tby problem or most appropriately fits the data over T Aging will contaminate Aging will contaminate x,Mx,M

22 if x if xa,Ma,M(t,A) doesn’t properly represent (t,A) doesn’t properly represent the actual aging over T the actual aging over T Explains sensitivity of Allan based variances to freq drift and Explains sensitivity of Allan based variances to freq drift and

insensitivity of Hadamard based variancesinsensitivity of Hadamard based variances• Should not Should not arbitrarilyarbitrarily change the order of change the order of x,Mx,M

22 to avoid a to avoid a divergence problem due to neg-divergence problem due to neg- noise noise Order defined by spec or by best match to actual aging behavior Order defined by spec or by best match to actual aging behavior

over Tover T Not free to change without changing problem addressedNot free to change without changing problem addressed Must find other ways of dealing with such divergencesMust find other ways of dealing with such divergences


Dealing with Divergences when Neg- Noise is Present

• Well-known that Well-known that x,0x,022 & & x,1x,1

22 can diverge for neg- can diverge for neg- noise noise It has been argued that these variances should not be usedIt has been argued that these variances should not be used But But x,0x,0

22 & & x,1x,122 measure measure r,Mr,M

22 for no aging & time-offset removal for no aging & time-offset removal Cannot change these variances (if want Cannot change these variances (if want x,Mx,M

22 to be a measure of to be a measure of r,Mr,M

22) without changing system spec or problem statement) without changing system spec or problem statementExample: Coherent synthesizer Example: Coherent synthesizer Freq offset error is a problem Freq offset error is a problemx,2x,2

22 not appropriate because takes out ave freq error over T not appropriate because takes out ave freq error over T

• Will show such divergences indicate real design, spec, or Will show such divergences indicate real design, spec, or analysis problems in presence of neg-analysis problems in presence of neg- noise noise Appropriate response is to fix these problemsAppropriate response is to fix these problems––not to not to arbitrarilyarbitrarily

change the variancechange the variance

• Will discuss examples of how to deal with divergencesWill discuss examples of how to deal with divergences Non-essential divergence—Requires only analysis changeNon-essential divergence—Requires only analysis change

Didn’t use correct HDidn’t use correct Hss(f) (or only f(f) (or only fhh) or wrong variance for problem ) or wrong variance for problem Essential divergence—Requires design or spec changeEssential divergence—Requires design or spec change

Design change: Change hardware Design change: Change hardware change H change HSS(f)(f)Spec change: Change problem to be addressed and/or add side Spec change: Change problem to be addressed and/or add side

conditions conditions


An Essential Divergence1st order PLL with f -3 Noise• Standard variance Standard variance x,0x,0

22 diverges in linear PLL model diverges in linear PLL model 11stst order PLL: |H order PLL: |Hss(f)|(f)|22 f f 2 2 (f<<1) (f<<1) KKx,0x,0(f) = 1 (f) = 1 |H |Hss(f)|(f)|22KKx,0x,0(f)L(f)Lxx(f) (f) f f -1 -1 (f<<1) (f<<1)

• Divergence is indicator of a Divergence is indicator of a physical problemphysical problem Divergence in linear model indicator of Divergence in linear model indicator of

cycle slipscycle slips in actual PLL in actual PLL Arbitrarily changing order of varianceArbitrarily changing order of variance

doesn’t fix cycle slip problemdoesn’t fix cycle slip problem

• Change design to 2Change design to 2ndnd order PLL order PLL Eliminates cycle slips Eliminates cycle slips |H |Hss(f)|(f)|22 f f 4 4 (f<<1) (f<<1) System changed not variance System changed not variance

• Allow slips (don’t change design) butAllow slips (don’t change design) butchange specchange spec Specify mean time to cycle slip Specify mean time to cycle slip Specify that variance be evaluated using sample varianceSpecify that variance be evaluated using sample variance

over finite T over finite T excludingexcluding data containing cycle slips data containing cycle slips Sample variance has x-kernel Sample variance has x-kernel f f 2 2 (f<<1) for finite T (f<<1) for finite T

Variance change based on spec changeVariance change based on spec change

Sample Variance

1N

)μ(x

σ

1N

0n

2Nn

2Nx,0,

1N

0nn

1N xNμ

-d

etec

tor

x or

Cycle Slips


A Non-Essential DivergenceRadar or Ranging System with f -3 Noise

• Characterized by “delay” systemCharacterized by “delay” systemresponse response |H|Hss(f)|(f)|22 f f22 (f<<1) (f<<1) ,o,o

22 often used to specify often used to specify -error residual -error residual But But ,o,o

22 diverges for f diverges for f -3 -3 noise noise

• Heuristic radar solution: Use BP varianceHeuristic radar solution: Use BP variance TTcc = Coherent data processing interval = Coherent data processing interval

for extracting radar informationfor extracting radar information No rigorous justification for using No rigorous justification for using ,bp,bp

22

• Rigorous solution: Proper error measure is sample variance Rigorous solution: Proper error measure is sample variance over Tover Tdd (not standard variance) (not standard variance) Higher order variance if more than just one parameter (aHigher order variance if more than just one parameter (a00) is ) is

extracted from data extracted from data x-kernel x-kernel f f 2 2 or greater so no divergence problem for correct or greater so no divergence problem for correct

variancevariance Problem was that incorrect variance was usedProblem was that incorrect variance was used

df(f)L2σh

C

f

1/T φ

2bpφ,

~Coherent Radar

x

D d

(t) (t- d)

Process


Summary and Conclusions

x,Mx,M22 can be interpreted as an approximate measure of can be interpreted as an approximate measure of r,Mr,M

2 2

when a poly is removed from datawhen a poly is removed from data x,Mx,M

22 should be matched to x should be matched to xa,Ma,M(t,A) removed from the data(t,A) removed from the data

• Extracting information from data HP filters the residual noiseExtracting information from data HP filters the residual noise Based on assumption that f<<1 behavior of Based on assumption that f<<1 behavior of r,Mr,M

22 same as same as x,Mx,M22

Need to investigate exact properties of Need to investigate exact properties of r,Mr,M22 kernels further kernels further

• One type of variance is not appropriate for all problemsOne type of variance is not appropriate for all problems• One should not One should not arbitrarilyarbitrarily change a variance order simply to change a variance order simply to

avoid a divergence due to neg-avoid a divergence due to neg- noise noise There must be a rationale behind any variance changeThere must be a rationale behind any variance change

• A divergence is a signal to fix a real design, spec, & analysis A divergence is a signal to fix a real design, spec, & analysis problemproblem May be essential or non-essential divergence May be essential or non-essential divergence

• Paper results can also be applied to variances of Paper results can also be applied to variances of (t) & y(t)(t) & y(t)

• Would like to acknowledge JPL time & frequency section for Would like to acknowledge JPL time & frequency section for serving as sounding board and for making several serving as sounding board and for making several suggestions that improved this papersuggestions that improved this paper

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