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On Difference Variances as Residual Error Measures in Geolocation

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### On Difference Variances as Residual Error Measures in Geolocation

Victor S. ReinhardtRaytheon Space and Airborne SystemsEl Segundo, CA, USA

ION National Technical MeetingJanuary 28-30, 2008San Diego, California

Trajectory

ResError

t

x(tn)

x(tn+)

()x(tn)

Two Types of Random Error Variances Used in Navigation- Residual error (R) variancesare used in measuringgeolocation error

= Mean Sq (MS) ofdifference between position or time data x(tn) and a trajectory est from data

- Mth order difference() variances used inmeasuring T&F error

MS of Mth order difference of data x(tn) over

- 1st order difference ()x(tn) = x(tn+) - x(tn)

Trajectory

ResError

t

()2x(tn)

()x(tn)

()x(tn+)

Two Types of Random Error Variances Used in Navigation- Residual error (R) variancesare used in measuringgeolocation error

= Mean Sq (MS) ofdifference between position or time data x(tn) and a trajectory est from data

- Mth order difference() variances used inmeasuring T&F error

MS of Mth order difference of data x(tn) over

- 1st order difference ()x(tn) = x(tn+) - x(tn)
- MS of ()2x(tn) Allan variance (of x)
- MS of ()3x(tn) Hadamard variance (of x)

R-variances not known for good convergence properties

- When negative power law (neg-p) noise is present
- Neg-p noise PSD Lx(f) f p with p < 0
- p generally -1, -2, -3, -4 for T&F sources
- R-variances are the proper residual error measures in geolocation
- Despite any such convergence problems
- Address statistical questions being posed
- -variances known for good convergence properties when neg-p noise present
- But -variances don’t seem to relate to residual geolocation error as R-variances do

Paper Will Show

- -variances do measure residual geolocation error under certain conditions
- Mainly when an (M-1)th order polynomial is used to estimate the trajectory
- & R variances equivalent for these conditions
- R-variances do have good convergence properties for neg-p noise
- Because trajectory estimation process highpass (HP) filters the noise in the data
- True under more general conditions than for equivalence between & R variances

x(tn)

●

TrueTrajectoryxc(t)

●

●

Model FnEst xw,M(t,A)

True Noisexp(t)

●

●

●

●

●

t

●

N Data Samples over T = N∙o

Residual Errors in Geolocation Problems- Have N data samples x(tn) over interval T
- Data contains a (true) causal trajectory xc(t)that we want to estimate from the data
- And data also contains neg-p noise xp(t)
- Assume we estimate xc(t) by fitting
- A model function xw,M(t,A) to the data
- Through adjustment of M parameters A = (ao,a1,…aM-1)

x(tn)

●

●

●

ObservableRes (R) Error xj(tn)

●

●

●

●

●

t

●

N Data Samples over T = N∙o

Observable Residual (R) Error (of Data from Fit)xj(tn) = x(tn) - xw,M(tn,A)

- Define point R variance at x(tn) E{xj(tn)2}
- E{…} = Ensemble average over random noise
- Average R variance x-j2 Average of E{xj(tn)2} over N samples
- Average can be uniformly or non-uniformly weighted (depending on weighting used in fit)

xc(t)

xw,M(t,A)

x(tn)

●

●

●

True Function(W) Error xw(tn)

ObservableRes (R) Error xj(tn)

●

●

●

●

●

t

●

N Data Samples over T = N∙o

The True (W) Error (Between Fit Function & Actual Trajectory)xw(tn) = xw,M(tn,A)- xc(tn)

- True measure of fit accuracy but not observable from the data
- Point W variance E{xw(tn)2}
- Average W variance x-w2

xc(t)

xw,M(t,A)

Precise Definition of Mth Order-Variance for Paper

- Overlapping arithmetic average of square of ()Mx(tn) over data plus E{…}
- Not discussing total or modified averages
- M All orders equal for white (p=0) noise
- Can show Mth order -Variance HP filters Lx(f) with 2Mth order zero (at f = 0)

x,1()2 MS Time Interval Error 2nd Order zero

x,2()2 Allan variance of x 4th Order zero

x,3()3 Hadamard var of x 6th Order zero

-Variancesas Measures of Residual Error in Geolocation

- Can prove for N = M + 1 data points that
- R-variance = Mth order -Variance with = T/Mx-j2 = x,M(T/M)2when
- xa,M(t,A) is (M-1)th order polynomial
- Uniform weighted Least SQ Fit (LSQF) is used
- x-j2 “unbiased” MS ( sum sq by N – M)
- Well-known for Allan variance of x
- Equivalent to 3-sample x-j2when time & freq offset (1st order polynomial in x) removed
- Hadamard variance of x equivalent to
- 4-sample x-j2 when time & freq offset & freq drift (2nd order polynomial in x) removed

f0 Noise

2

RMS{xj}

1

x-w

0

1

10

100

1K

Samples N

N

2

x-w

f -2 Noise

N-M

x-j2(N) x,M(T/M)2

1

RMS{xj}

0

2

f -4 Noise

x-w

1

RMS{xj}

0

N = M+1

Can Extend Equivalence to Any N as Follows- “Biased” x-j RMS{xj}
- “Biased” sum sq by N
- Can show RMS{xj} Constant as N varies (while T remains fixed)
- Thus for “unbiased” x-j2
- Exact relationship exists for each p & N
- Similar Allan-Barnes “bias” functions for Allan variance

Consequences of Equivalence Between & R Variances

- -variances measure res geolocation error
- When xw,M(t,A) is poly & uniform LSQF used
- For non-uniform weighting (Kalman?) x,M(Teff/M) should also be estimate of x-j
- Teff Correlation time for non-uniform fit
- Don’t have to remove xw,M(t,A) from data if use x,M(Teff/M) to estimate x-j
- Because ()Mxw,M(t,A) = 0when xw,M(t,A) = (M-1)th (or lower) order poly
- Explains sensitivity of Allan variance to causal frequency drift & insensitivity of Hadamard to such drift

HP Filtering of Noise in R-Variances

- Paper proves fitting process
- HP filters Lx(f) in R-variances
- HP filtering order depends on complexity of model function xw,M(tn,A) used
- R-variances guaranteed to converge if free to choose model function
- True under very general conditions
- Fit solution is linear in data x(tn)
- Fit is exact solution when no noise is present & xw,M(tn,A) is correct model for xc(tn)
- True even when x,M() not measure of x-j
- Applies to any weighting, LSQF, Kalman, …

long term error-1.xls

f0 Noise

f0 Noise

x(tn)

x(tn)

f -2 Noise

f -2 Noise

xj

xj

xw

xw

f -4 Noise

f -4 Noise

xc

xc

xa,M

xa,M

T

Graphical Explanation of HP Filtering of Lx(f) in R-Variances- For white (p=0) noise the fit behaves in classical manner
- As N xw,M(tn,A) xc(tn)& x-w 0
- Again T is fixed as N is varied
- But for neg-pnoise
- As N xw,M(tn,A) not xc(tn)
- Because fit solution necessarily tracks highly correlated low freq (LF) noise components in data

long term error-1.xls

f0 Noise

x(tn)

f -2 Noise

f -2 Noise

xj

vj

xw

vw

f -4 Noise

f -4 Noise

xc

vc

xa,M

va,M

T

This tracking causes HP filtering of Lx(f) in R-Variances- With HP knee fT
- fT 1/T (uniform weighted fit)
- fT 1/Teff (non-uniform)
- True for all noise
- Implicit in fitting theory for correlated noise
- Can’t distinguish correlated noise from causal behavior
- Only apparent for neg-p noise because most power in f< fT
- While for white noise power uniformly distributed over f

long term error-1.xls

Spectrally RepresentingR-Variances

- Gj(t,f) & Kx-j(f) HP filtering due to fit
- To understand what Gj(t,f) & Kx-j(f) areconsider the following
- Can write fit solution in terms of Green’s function gw(t,t’) because assumed fit is linear in x(tn)

Spectrally RepresentingR-Variances

- Gw(t,f) = Fourier transform of gw(t,t’) over t’
- Hs(f)Xp(f) = Fourier transform of xp(t)
- Green’s fn for xj(t) gj(t,t’) = (t-tn) - gw(t,t’)
- Fourier transform Gj(t,f) = ejt - Gw(t,f)

Spectrally RepresentingR-Variances

- Kx-j(f) Average of |Gj(t,f)|2 over t (data)
- c2 & x-c2 Modeling error terms
- Generated when xw,M(tn,A) can’t follow xc(tn) over T

Spectrally RepresentingR-Variances

- The paper proves the following
- |Gj(t,f)|2 & Kx-j(f) f 2M(f<<1)
- When xa,M(t,A) is (M-1)th order polynomial
- |Gj(t,f)|2 & Kx-j(f) at least f 2(f<<1)
- For any xa,M(t,A) with DC component
- So R-variances guaranteed to converge
- If free to choose model function for estimating the trajectory

Kx-j(f) in dB forUniform Weighted Fit

Kx-j(f) in dB forNon-Uniform Weighted Fit

M = 1 f 2

M = 1

f = 1/Teff

M = 2 f 4

f = 1/T

M = 3 f 6

M = 2

Weighting

M = 5 f 10

M = 4 f 8

Teff

M = 3

M = 4

M = 5

T

1

Log10(fT)

Log10(fT)

(N =1000)

(N =1000)

Kx-j(f) Calculated for Polynomial xw,M(t,A) & LSQFxw,M(tn,A)

x(tn)

xw(t)

xj(tn)

xp(t)

Spectral Equations for True Function & Model Errors- We note that xj(tn) + xw(tn) = xp(tn)
- So noise must be LP filtered in xw(tn) because noise in xj(tn) is HP filtered
- In paper derive spectral equations for
- W-variances E{xw(tn)2} & x-w2 in terms of Lx(f)
- Model error variances c2 & x-c2 in terms of dual freq Loève Spectrum Lc(fg,f) of xc(t)
- Note Hs(f) appears in all spectral equations
- So what is this Hs(f)?

Topological Hs(f) for 2-Way Ranging

d

~

D

Xponder

x(t)

x(t-d)

|Hs(f)|2 = 4sin2(fd)

x

Hs(f) = System Response Function(See Reinhardt, FCS, 2006)- Models filtering action of system on x(t)
- Generated by actual filters in system & topological structures (such as PLLs)
- Acts on all variables the same way xp(t), xc(t), xj(t), xw(t)
- Hs(f) can HP filter as well as LP filter Lx(f)
- 2-way ranging Hs(f) generates 2nd order 0 at f=0
- So Hs(f) helps both R & W variances converge

f 2 (fd <<1)

− True Fn Error

f

fT

f

fT

fl

fh

fl

fh

Summary of Spectral Properties of R & W Errors with Respect to Lx(f)- At the knee freq fT the fitting process
- HP filters the obs residual error (R-variances)
- LP filters the true fn error (W-variances)
- Hs(f) filters both the same fl = HP fh = LP
- As Teff (fT << fl)true fn error 0
- If Hs(f) alone can overcome pole in Lx(f)
- Then W-variances also converge for neg-p noise
- Transition to stationary but correlated statistics

Consequences for GPS Navigation

- Confirms that R-variances measure consistency not accuracy for small Teff
- Can view control segment operations as PLL-like Hs(f) with HP cutoff fl = 1/TGPS
- TGPS determined by time constant of satellite parameter correction loops
- Can assume true function errors (W-variances) converge for this Hs(f)
- System tied to known ground sites
- Drift of timescale doesn’t effect nav accuracy
- R-variances measure true accuracy whenTeff >> TGPS

Final Summary & Conclusions

- -variances can be used for R-variancesin some geolocation problems
- Mainly when model function is polynomial
- R-variances HP filter noise in data due to trajectory estimation process
- True under very general conditions
- R-variances guaranteed to converge if free to choose model function
- R-variances represent true errors for large T when Hs(f) makes W-variances converge
- Preprints: www.ttcla.org/vsreinhardt/

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