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On Difference Variances as Residual Error Measures in Geolocation Victor S. Reinhardt Raytheon Space and Airborne Systems El Segundo, CA, USA ION National Technical Meeting January 28-30, 2008 San Diego, California x(t n ) Trajectory Res Error t x(t n ) x(t n +) ()x(t n ) 

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on difference variances as residual error measures in geolocation

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

two types of random error variances used in navigation

x(tn)

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)
two types of random error variances used in navigation3

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
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
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
residual errors in geolocation problems

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)
observable residual r error of data from fit

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)

the true w error between fit function actual trajectory

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 m th order variance for paper
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

variances as measures of residual error in geolocation
-Variancesas Measures of Residual Error in Geolocation
  • Can prove for N = M + 1 data points that
    • R-variance = Mth order -Variance with  = T/Mx-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
can extend equivalence to any n as follows

Errors vs N(M=2)

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
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
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

graphical explanation of hp filtering of l x f in r variances

Fit Solutions for Various p

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

this tracking causes hp filtering of l x f in r variances

Fit Solutions for Various p

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 representing r variances
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 representing r variances17
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) = ejt - Gw(t,f)
spectrally representing r variances18
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 representing r variances19
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
k x j f calculated for polynomial x w m t a lsqf

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) & LSQF
spectral equations for true function model errors

xc(t)

xw,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)?
h s f system response function see reinhardt fcs 2006

Topological Hs(f) for 2-Way Ranging

d

~

D

Xponder

x(t)

x(t-d)

|Hs(f)|2 = 4sin2(fd)

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 (fd <<1)

summary of spectral properties of r w errors with respect to l x f

− Obs Residual

− 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
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
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/