96 IEEE TRANSACTIONS ON AEROSPACE AND NAVIGATIONAL ELECTRONICS

the time rate of change of the signal voltage at the out-put of the subtraction network was NA /r. This assump-tion, however, is valid only for a narrow region aboutthe origin (i.e., about the actual time of arrival). Forlarger error (noise) voltages, the time rate of change ofthe signal voltage becomes less than 2A/T, and the dif-ferential error in the time of arrival estimate becomeslarger as the error voltage increases. Therefore, theexpected result is that the range accuracies obtainedfrom the computer simulation should be larger than thetheoretically predicted values for lower signal-to-noiseratios. This result is shown in Fig. 3.

In an operational system, the range accuracy would

approach asymptotically a constant value at large SINratios. This asymptotic value is determined by the quan-tum error due to a finite sampling time of the zero-cross-ing detector. For the simulation discussed here, how-ever, the effect of the quantum error is not significant inthe region of S/N ratios plotted.

ACKNOWLEDGMENT

The authors are deeply grateful to S. Rittenburg andH. Ward for their helpful discussions of the theoreticalconsiderations and for suggestions on the digital simu-lation, and to B. DiLorenzo and A. Tebbetts for theirefforts in conducting the digital simulation.

Error Distributions of Best Estimate of Position

from Multiple Time Difference

Hyperbolic Networks

NATHAN MARCHAND, FELLOW, IEEE

Summary-A mathematical model for the error distribution ofthe best estimate of position of a vehicle as determined from a setof simultaneous measurements of times of arrival of electromagneticwaves from an arbitrary number of ground stations is obtained. Themajor and minor axes of the error ellipse, as well as the angle thattheymake with the assumed axes, are determined. The results are insuch form as to be easily adaptable for use with a computer.

INTRODUCTION

Tp HE PURPOSE of this paper is to obtain a mathe-matical model of the best estimate of position, andthe theoretical two-dimensional error distribution

of this best estimate of position, of a vehicle as deter-mined from a set of simultaneous time measurements ofdistances determined by electromagnetic wave times ofarrival to (or from) an arbitrary number of ground sta-tions. Since the problem here is one of time-differencehyperbolic lines of position, the minimum number offixed ground stations is three, provided that they arenot located on a straight line. There is no maximumnumber for the number of stations in the network.

All times are arbitrarily referenced to the same timebase but since this reference is normally not known, only

Manuscript received January 3, 1964. The work reported in thispaper was supported by Federal Aviation Agency, Systems Researchand Development Service, Atlantic City, N. J., under Contract No.FAA/BRD-26.

The author is with the Marchand Electronic Laboratories,Greenwich, Conn.

time differences are used to determine a set of hyper-bolic lines as in loran and other hyperbolic position de-termination systems. It will be assumed that the timemeasurements are uncorrelated (the errors are uncorre-lated) and that the statistical distribution of the timemeasurements for each time measurement is known. Itmust be pointed out that the statistical distributions ofreadings at any station is dependent upon many things:atmospheric conditions along the path, the type ofradiation, maintenance and many other factors, so thatthe statistical distribution for the particular set of ob-servations being studied should be employed in anyerror distribution calculations. Care should be taken tounderstand all the variables before trying to correlatethe results of one calculation with that of a new prob-lem.

NETWORK CONFIGURATION

It will be assumed that there are k ground-based sta-tions involved and that the time of arrival of the kthelectromagnetic radiation (referenced to a commonarbitrary time base) is 7k. It will be assumed that in thecomplete population of readings the mean error at anystation (or from any station) will be zero and that thedeviation 0r-g and the variance o-,,2 of the distribution oferrors in the time readings (for each of the stations) isknown, the subscript g varying from 1 to k and desig-

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Marchand: Error Distributions at Best Position Estimate

nating the deviiation and variance of each particularpopulation of readings.To obtain the statistical distributions of the time dif-

ferences the following conversions are used: (In thosecases where the statistical distributions of the time dif-ferences are known, they can be used directly and it isnot necessary to know the statistical distributions atany station). If rj is the jth time difference then

Ti = Ta - Tb (a # b) (1)where 7Ta and mb are individual times of arrivals of elec-tromagnetic waves from stations a and b. Thus a and bwill assume all values from 1 to k (for k stations) exceptthose values where a is equal to b. Since rj is the resultof combinations of k things taken two at a time, thenthe

k!maximum value of j =M. (2)

The subscript j will be used to designate the particu-lar hyperbola determined by any pair of times, anytime difference, determined by the receptions (or trans-missions) at any two ground stations. The deviationand variance of rj will be determined by the deviationsand variances of the two time measurements whichenter into the calculation of Tj. If the two measurementsof time differences, Ta and Tb, have variances O-ra2 and¢T1b2, respectively, then the variances of the resultantTjs, nameIy,-Tj2 will be given by

2Tj2= ra2 + , b2 (a 7 b). (3)

Thus, in the general case, there will be m hyperbolaswhose determining time differences will have variancesas defined above, the subscript j varying from 1 to mand designating the variance of the particular jth hy-perbola.

In Fig. 1 are shown a few of the hyperbolas in thearea of intersection around a position location. It isassumed here that the area is small enough so that thehyperbolas may be represented as straight lines whichare tangent to the hyperbola at the point of intersectionor in the area of intersection. The angle which each ofthe hyperbolic tangents makes with an arbitrary refer-ence, in this case the x axis, is indicated as Oj (j varyingfrom 1 to m).

BEST ESTIMATE OF POSITION

Since the true position of the vehicle is unknown, it isassumed to be at the point 0 as shown in the figure. Thepoint 0 is at a perpendicular distance from the jthhyperbola, noted as the distance dj in the figure. Usingthe expression for joint probability, the likelihood of theset of position lines' is given by

For the derivation of this type of equation and its application toerror distribution in DF position determination, see R. G. Stansfield,"Statistical theory of D.F. fixing," J. IEE (London) pt. IIIA, pp.762 770; 1947.

x

Fig. 1-The intersection of hyperbolas determined by time differencesof arrival of signals at a few of the ground-based stations in the set.The assumed position of the transmitter-receiver is 0 and thetrue position is T. Di and d, are perpendicular distances to thehyperbola number j and 0, is the angle the tangent to the hyper-bola makes with the x axis.

P(d, . . dm)add j/dm27r)j1odl ..0d/ 1 m

exp - I adi2) * adm (4)

where 0dj iS the deviation and Odj2 is the variance of thedistance dj. By the method of maximum likelihood,2 thebest estimate of position is the one which maximizes theexponential term, or in other words minimizes the expo-nent

1 m dj2

2 j=l Tdj2(5)

However, dj is related to the change in r; necessary tomove the hyperbola over a perpendicular distance equalto dj. They are related by a constant Kj where

(6)KjA/jT= dj

where

0.081Kj =si nautical miles/Asec.

sin aj(7)

The angle aj is one half the included angle between linesdrawn from the assumed position to the two stationswhose reception times were used to determine the hy-perbola.

Similarly, the deviation Gdj iS given by

OJdj = Kjj. (8)

2 A. Hald, "Statistical Theory with Engineering Applications,"John Wiley and Sons, Inc., New York, N. Y., par. 8.8, the methodof maximum likelihood, pp. 204-208; 1952.

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98 IEEE TRANSACTIONS ON AEROSPACE AND NAVIGATIONAL ELECTRONICS

Thus the best estimate of position can be obtained byminimizing (5) after substituting (6) and (8) into it.Hence, to obtain the best estimate of position, (5)becomes

1 m(A,TJ)minimize (9)

2 j=1 7j2

where A'Tj is the change in time difference that has tobe made for the jth hyperbola to go through the chosenposition and o-j is the deviation for the jth hyperbolaas determined from (3). In other words, it means aminimization of the time variations squared weightedby their respective variances necessary to make all thehyperbolas intersect at a common point. The one-halfterm, since it is a constant, may be neglected.

THE ERROR DISTRIBUTION

The next step is to find the error distribution of thesebest estimates of positions. The error for this positiondetermination is the distance between the true positionof the vehicle, shown as the point T in the figure, andthe optimum calculated position. As shown in the fig-ure, the point T is at a distance Dj from the jth hyper-bola.

It is assumed that the points T and 0 are close enoughtogether that the same tangents to the hyperbolas maybe employed so that the constants K1 are the same.Thus

Dj = KjAT(10)TDi = Kjo7,

where Kj has the value as given in (7). The error in thetime difference which determines the jth hyperbola isATj. (Note the absence of the prime on the A which dif-ferentiates this time difference error from the time dif-ference error used in determining the optimum positiondetermination.) To make the calculations simpler, theorigin of the reference x, y coordinates is assumed to beat the true position T.From analytic geometry the normalized equation for

the tangent to the jth hyperbola is

x sin Oj - y cos 0j + KjArj = 0 (11)

where Oj is the angle which the tangent to the jth hyper-bola makes with the x axis. It is known that the tangentto a hyperbola at any point bisects the angle betweenthe two lines drawn from the point in question to thetwo generating ground stations (earth curvature isneglected). As pointed out previously in (7), the anglebetween the two lines is 2aj. For any point x', y' withstations located at xa, ya and Xb, yb, 0j for the jth hyper-bola is given by

As a matter of interest it should be noted that

1 / Ya - yI Yb yalli tan-' -,tan-'

2 XS, - xI Xb - XI/(13)

where the absolute value signs are put in for conveni-ence, since Kj is always taken positive. As stated previ-ously, it is assumed that the area of intersection is smallenough so that these angles may be assumed constantover the area.

Calling the coordinates of the point 0, the best esti-mate of position, xo, yo, the distance dj, the perpendicu-lar distance from xo, yo to the hyperbola, is given by

dj= xo sin j -yo cos 0j + KjArj. (14)

This expression for dj and the expression for Od1 givenin (8) may now be substituted into (5) to obtain anequation to be minimized (neglecting the one half sinceit is a constant)

m (xo sin 0j - yo cos 0j + KjArj)2minimize 2

J=1 Kj2orj3* (15)

Since the origin of the coordinate system is taken as thetrue position of the vehicle xo, which is the difference inthe x direction, and yo, which is the difference in the ydirection, specify the error.

Eq. (15) may be solved by expanding, simplifyingand differentiating. Expanding (15) and equating it toan arbitrary variable z,

m SKj2'ATj2 x02 sin2 ej yo2 COS2 Ojz = , + xosi2~+ co20

j=1 -K j2U'rj2 Kj2 G"j2 Kj2o-rj2KjArjxo sin O1 KjATjyo cos OJ

+2 - 2A? 2g.. .2 j r2

xoyo sin 0j cos Oj-- 2 .

Kj2c,7j2 _

Before minimizing z the followingmade for ease of handling:

(16)

substitutions are

m sin2 0ji K 2,yr j2

j-l Kja2B =m Cos2 Oj

3 1 K~j20T3m sin 0j cos 03

C = iE,K312K- 73

.2

(17)

(18)

(19)

MIaking these substitutions into (16) and cancelingout the Kj's in the first term,

m A'AT,2z - E + xo2A + yo2B - 2xoyoC

j= 1 JTj

oj=(tan-lY_Y+tnly2 Xa -X Xb - Y'

(12)m KjATJXo sin 0j

+ E-59 2~j=Z K 2 2Tj

"I KjATjyo cos j

31 Kj2orj2

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Marchand: Error Distributions at Best Position Estimate

The values of xo and yo for minimum z can now bedetermined by equating the partial of z with respect toxo to zero, and the partial of z with respect to yo to zero,and solving the simultaneous equations for xo and yo.By means of second derivatives these values are checkedto see that they are a minimum. Thus

1 FtmI -AB Kl~r

CcosOj sinO] (21)

Kj, 2(. Tj2

to be taken into account (an assumption normallymade), the only portion of xo2 and yo2 which changesfrom one ith term to the next is (ATj2)i. However, bydefinition

Il n

E (iA7-j2) i = 0,j.1n i=i

(26)

Now substituting (25) into (23a) and using (26) tosimplify,

aiid

Ye = B E K1jA cos 0j- C sinn0j

Kj2'f2

1-AR - C2B (27)

(22)

These values xo and yo are the errors for any one setof time readings. A population of errors, n in number forinstance, have to be analyzed and the variance of thispopulation of errors, o-,2 for the x direction and o-Y2 forthe y direction, is determined by

1nO--o2 Z (xo2)i (23a)

n

O'f 2 = E (y02), (23b)n i=1

where in a population of n sets of readings, (xO2), is thesquare of the x error for the ith set of readings and(yo2), is the square of the y error for the ith set of read-ings. Similarly, the x0, yo spatial correlation coefficient isgiven by px 0,y where

1nPxo,yO = E (Xoyo)t- (24)no-zoo.yo i=1

In addition to assuming that there is no correlation be-tween time readings at the various stations it is alsoassumed that all biased or fixed errors in the time meas-urements have been adjusted or calibrated out, or thatenough of a population has been taken that the meanerror is zero for each station.

Eq. (21) is now substituted into (23a) to obtain thevariance in the x direction. First determining x02 bysquaring (21),

1 - in (C cos Oj-B sin6j20O2=( - C)LZEKj2Ar12 (K2j.2)2j+ (other terms which drop out when summed over i). (25)

From inspection of (22) it can be seen that a similarexpression is obtained for ye2, the only difference beingin the numerator within the summation sign. This ex-pression for x02 is substituted into the first equation of(23), namely, (23a). When this is done each of the jterms in (25) is summed up over the population of n, xohaving a different value as i is varied from 1 to n.Assuming thai Oj and axj (which determines Kj) do notvary within the region of error by any amount that has

and similarly, for the variance in the y direction,

1O'O2 = A.

AB - C2(28)

The rms circular variance, noted as 0Jms, is obtainedby summing up (27) and (28) so that

A+2B0Jrm s2 =

AB - C2(29)

The spatial correlation coefficient Px0,,y is then obtainedby substituting (21) and (22) into (24) and simplifying

CPxo, o = VA- (30)

Thus (28), (29) and (30) give the two-dimensionalerror distribution in terms of the error distribution ofthe time-distance measurements from each station (orthe time difference distance difference measurementsfrom each possible pair of stations) and the actualspatial layout of the stations. The equation for theprobability density function can be obtained by substi-tuting (27), (28) and (29) into the standard equationfor the probability density function. Thus the proba-bility density of the error p(xo, yo) is, after simplifica-tion,

p(x0, yo)-VARB-C2,2= exp [-'(Axe2 - 2Cxoyo + ByO2]. (31)

27r2

ERROR ELLIPSEThe ellipse of constant error probability density (or

error ellipse) in two dimensions is determined by (31).The important factors are the magnitudes of the majorand minor axes of this ellipse and the angle which theaxes make with the originally chosen coordinates. Theseare obtained by rotating the original axes through anangle q to a new set of axes x', y' so that

XO' = xo cos 0 + yo sin 0

yo' = yo cos -xo sin ¢ (32)

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IEEE TRANSACTIONS ON AEROSPACE AND NAVIGATIONAL ELECTRONICS

where xo' and yo' are the magnitudes of the x and yerrors in the new coordinates. The correlation coefficientin the new coordinates pxo' .wo is given by

I1Pxo' o,io = (xo'yo')j.

flo-x0ofGy0' ij1(33)

To determine the angle of rotation 4 necessary toreduce the correlation coefficient to zero, and thusobtain the angle that the ellipse axis makes with theoriginal coordinate axes, pxo,,yo of (33) is equated tozero. Substituting (32) into (33) and simplifying bysumming up over the population n yields for the angle 4

1 -1 2pxo,yo7oxOoyO4) tan'

2 2 (34)2 f o2 eO>_

To obtain 4 in terms of the parameters A, B and C,(27), (28) and (30) are substituted into (34)

1 2C4 - tan- (35)

The deviations and variances for the errors in thenew coordinates are obtained by summing up the ex-pressions for the errors (32) over the population n. Car-rying out this summation,

UxoP2 = :o2 COS2 -+ 2Px0,y0OoxOyO sin 4 cos ±+LTYV2 sin2 4)

O', 2 = ,o2 cos2 2pxO,yoox0oy0 sin)

cos + oXo2 sin2 4). (36)

These variances, given in (36), are the variances of themajor and minor axes of the error ellipse. Substitutingfrom (27), (28), (30) and (35) into (36) and simplifying,

1 A±+B+V\(A -B)2±4C2(x

2 =B-C (37)

and

1 A+-B-V(A-B)2+4C2° 2 =-AB-C

2 AB - C(38)

CONCLUSIONSThus the error distributions of the best estimate of

position from a set of time difference-distance differencemeasurements used to obtain a set of hyberbolas iscompletely specified. The general expressions for theangles a and 0 for any position in the coordinates andany number and positions of stations can be obtainedsimply by analytic geometry and the complete plot oferrors can be obtained by use of a computer.The one item in question is the error distribution of

the time measurements. However, this distribution canbe made a function of the distance from the station(not necessarily linear), as well as the position of thestation. The error distribution of the time of arrival ofthe signals is dependent upon many items as pointedout in the Introduction.To check the results the general solution for only

two hyperbolas was obtained and the optimum positionturns out to be the intersection of the hyperbolas, asexpected, and the equations for the error distributionreduce to those given in a report on the range reliabilityand accuracy of a low-frequency loran system,3 whichsolves the problem for the limited case of two hyper-bolas.

ACKNOWLEDGAIENT

The author wishes to thank R. Kester and N. Braver-man of the Federal Aviation Agency for their very gen-erous help and criticism in the solution of this problem.

3 The Range Reliability and Accuracy of a Low FrequencyLoran System," Office of Chief Signal Officer, Pentagon, Washington,D. C., Rept. No. ORS-P-23; January, 1946.

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