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On Distinction Fluctuations as Remaining Mistake Measures in Geolocation Victor S. Reinhardt Raytheon Space and Airborne Frameworks El Segundo, CA, USA Particle National Specialized Meeting January 28-30, 2008 San Diego, California x(t n ) Direction Res Blunder t x(t n ) x(t n +) ()x(t n )

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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 +ï\') ï(ï\')x(t n ) ï\' Two Types of Random Error Variances Used in Navigation Residual slip (R) fluctuations are utilized as a part of measuring geolocation lapse = Mean Sq (MS) of distinction between position or time information x(t n ) and a direction est from information M th request contrast (ï) changes utilized as a part of measuring T&F blunder ïµ MS of M th request distinction of information x(t n ) over ï\' 1 st request contrast ï(ï\')x(t n ) = x(t n +ï\') - x(t n )

x(t n ) Trajectory Res Error t ï(ï\') 2 x(t n ) ï(ï\')x(t n ) ï(ï\')x(t n +ï\') ï\' ï\' Two Types of Random Error Variances Used in Navigation Residual mistake (R) differences are utilized as a part of measuring geolocation blunder = Mean Sq (MS) of contrast between position or time information x(t n ) and a direction est from information M th request distinction (ï) changes utilized as a part of measuring T&F blunder ïµ MS of M th request contrast of information x(t n ) over ï\' 1 st request contrast ï(ï\')x(t n ) = x(t n +ï\') - x(t n ) MS of ï(ï\') 2 x(t n ) ïµ Allan change (of x) MS of ï(ï\') 3 x(t n ) ïµ Hadamard change (of x)

R-changes not known for good merging properties When negative force law (neg-p) commotion is available Neg-p clamor ï PSD L x (f) ïµ f p with p < 0 p for the most part - 1, - 2, - 3, - 4 for T&F sources R-differences are the best possible lingering lapse measures in geolocation Despite any such union issues Address factual inquiries being postured ï-changes known for good joining properties when neg-p clamor introduce But ï-changes donât appear to identify with leftover geolocation slip as R-differences do

Paper Will Show ï-changes do gauge leftover geolocation mistake under specific conditions Mainly when a (M-1) th request polynomial is utilized to evaluate the direction ï & R changes proportionate for these conditions R-fluctuations benefit have union properties for neg-p clamor Because direction estimation process highpass (HP) channels the commotion in the information True under more broad conditions than for proportionality between ï & R fluctuations

â x(t n ) â True Trajectory x c (t) â Model Fn Est x w,M (t,A) True Noise x p (t) â t â N Data Samples over T = Nâï\' o Residual Errors in Geolocation Problems Have N information tests x(t n ) over interim T Data contains a (genuine) causal direction x c (t) that we need to appraise from the information And information additionally contains neg-p clamor x p (t) Assume we assess x c (t) by fitting A model capacity x w,M (t,A) to the information Through conformity of M parameters A = (an o ,a 1 ,â¦a M-1 )

â x(t n ) â Observable Res (R) Error x j (t n ) â t â N Data Samples over T = Nâï\' o Observable Residual (R) Error (of Data from Fit) x j (t n ) = x (t n ) - x w,M (t n ,A) Define point R difference at x(t n ) ï E{x j (t n ) 2 } E{â¦} = Ensemble normal over irregular commotion Average R difference ï³ x-j 2 ï Average of E{x j (t n ) 2 } over N tests Average can be consistently or non-consistently weighted (contingent upon weighting utilized as a part of fit) x c (t) x w,M (t,A)

â x(t n ) â True Function (W) Error x w (t n ) Observable Res (R) Error x j (t n ) â t â N Data Samples over T = Nâï\' o The True (W) Error (Between Fit Function & Actual Trajectory) x w (t n ) = x w,M (t n ,A) - x c (t n ) True measure of fit exactness yet not perceptible from the information Point W fluctuation ï E{x w (t n ) 2 } Average W fluctuation ï ï³ x-w 2 x c (t) x w,M (t,A)

Precise Definition of M th Order ï-Variance for Paper Overlapping number juggling normal of square of ï(ï\') M x(t n ) over information in addition to E{â¦} Not talking about aggregate or altered midpoints ï¬ M ï All requests break even with for white (p=0) commotion Can demonstrate M th request ï-Variance HP channels L x (f) with 2M th request zero (at f = 0) ï³ x,1 (ï\') 2 ïµ MS Time Interval Error ï 2 nd Order zero ï³ x,2 (ï\') 2 ïµ Allan fluctuation of x ï 4 th Order zero ï³ x,3 (ï\') 3 ïµ Hadamard var of x ï 6 th Order zero

ï-Variances as Measures of Residual Error in Geolocation Can demonstrate for N = M + 1 information focuses that R-difference = M th request ï-Variance with ï\' = T/M ï³ x-j 2 = ï³ x,M (T/M) 2 when x a,M (t,A) is (M-1) th request polynomial Uniform weighted Least SQ Fit (LSQF) is utilized ï³ x-j 2 ï âunbiasedâ MS (ï¸ entirety sq by N â M) Well-known for Allan change of x Equivalent to 3-test ï³ x-j 2 when time & freq counterbalance (1 st request polynomial in x) evacuated Hadamard fluctuation of x comparable to 4-test ï³ x-j 2 when time & freq balance & freq float (2 nd request polynomial in x) uprooted

Errors versus N (M=2) f 0 Noise 2 RMS{x j } 1 ï³ x-w 0 1 10 100 1K Samples N ï N 2 ï³ x-w f - 2 Noise N-M ï³ x-j 2 (N) ï ï³ x,M (T/M) 2 1 RMS{x j } 0 2 f - 4 Noise ï³ x-w 1 RMS{x j } 0 ï N = M+1 Can Extend Equivalence to Any N as Follows âBiasedâ ï³ x-j ï RMS{x j } âBiasedâ ï ï¸ whole sq by N Can indicate RMS{x j } ï Constant as N shifts (while T stays settled) Thus for âunbiasedâ ï³ x-j 2 Exact relationship exists for every p & N Similar Allan-Barnes âbiasâ capacities for Allan difference

Consequences of Equivalence Between ï & R Variances ï-differences measure res geolocation mistake When x w,M (t,A) is poly & uniform LSQF utilized For non-uniform weighting (Kalman?) ï³ x,M (T eff/M) ought to likewise be gauge of ï³ x-j T eff ï Correlation time for non-uniform fit Donât need to uproot x w,M (t,A) from information if use ï³ x,M (T eff/M) to assess ï³ x-j Because ï(ï\') M x w,M (t,A) = 0 when x w,M (t,A) = (M-1) th (or lower) arrange poly Explains affectability of Allan change to causal recurrence float & harshness of Hadamard to such float

HP Filtering of Noise in R-Variances Paper demonstrates fitting procedure HP channels L x (f) in R-fluctuations HP sifting request relies on upon multifaceted nature of model capacity x w,M (t n ,An) utilized R-changes ensured to unite if allowed to pick model capacity True under exceptionally broad conditions Fit arrangement is straight in information x(t n ) Fit is accurate arrangement when no clamor is available & x w,M (t n ,An) is right model for x c (t n ) True notwithstanding when ï³ x,M (ï\') not quantify of ï³ x-j Applies to any weighting, LSQF, Kalman, â¦ long haul blunder 1.xls

Fit Solutions for Various p f 0 Noise f 0 Noise x(t n ) x(t n ) f - 2 Noise f - 2 Noise x j x j x w x w f - 4 Noise f - 4 Noise x c x c x a,M x a,M T Graphical Explanation of HP Filtering of L x (f) in R-Variances For white (p=0) clamor the fit carries on in traditional way As N ï ï¥ x w,M (t n ,A) ï x c (t n ) & ï³ x-w ï 0 Again T is altered as N is changed But for neg-p clamor As N ï ï¥ x w,M (t n ,A) not ï x c (t n ) Because fit arrangement essentially tracks profoundly connected low freq (LF) commotion segments in information long haul lapse 1.xls

Fit Solutions for Various p f 0 Noise x(t n ) f - 2 Noise f - 2 Noise x j v j x w v w f - 4 Noise f - 4 Noise x c v c x a,M v a,M T This following reasons HP separating of L x (f) in R-Variances With HP knee f T f T ï 1/T (uniform weighted fit) f T ï 1/T eff (non-uniform) True for all clamor Implicit in fitting hypothesis for associated clamor Canât recognize corresponded clamor from causal conduct Only obvious for neg-p clamor on the grounds that most power in f< f T While for repetitive sound consistently appropriated over f long haul lapse 1.xls

Spectrally Representing R-Variances G j (t,f) & K x-j (f) ï HP separating because of fit To comprehend what G j (t,f) & K x-j (f) are consider the accompanying Can compose fit arrangement regarding Greenâs capacity g w (t,tâ) on the grounds that expected fit is straight in x(t n )

Spectrally Representing R-Variances G w (t,f) = Fourier change of g w (t,tâ) over tâ H s (f)X p (f) = Fourier change of x p (t) Greenâs fn for x j (t) ï g j (t,tâ) = ï¤(t-t n ) - g w (t,tâ) Fourier change ï G j (t,f) = e j ï·t - G w (t,f)

Spectrally Representing R-Variances K x-j (f) ï Average of | G j (t,f)| 2 over t (information) ï c 2 & ï³ x-c 2 ï Modeling slip terms Generated when x w,M (t n ,A) canât take after x c (t n ) over T

Spectrally Representing R-Variances The paper demonstrates the accompanying | G j (t,f)| 2 & K x-j (f) ïµ f 2M (f<<1) When x a,M (t,A) is (M-1) th request polynomial | G j (t,f)| 2 & K x-j (f) at any rate ïµ f 2 (f<<1) For any x a,