Causal association of Electromagnetic Signals using the Cayley Menger Determinant
aa r X i v : . [ g r- q c ] A ug Causal association of electromagnetic signals using the Cayley Menger determinant
Samuel Picton Drake
Defence Science and Technology Organisation,Edinburgh, South Australia 5111, AustraliaandSchool of Chemistry and Physics,The University of Adelaide, Adelaide,South Australia 5005, Australia
Brian D. O. Anderson and Changbin Yu
Research School of Information Sciences and Engineering,Australian National University, Canberra,Australian Capital Territory 0200, AustraliaandNational ICT Australia (NICTA),Australian National University, Canberra,Australian Capital Territory 0200, Australia
In complex electromagnetic environments it can often be difficult to determine whether signalsreceived by an antenna array emanated from the same source. The failure to appropriately assignsignal reception events to the correct emission event makes accurate localization of the signal sourceimpossible. In this paper we show that as the received signal events must lie on the light-coneof the emission event the CayleyMenger determinate calculated from using the light-cone geodesicdistances between received signals must be zero. This result enables us to construct an algorithmfor sorting received signals into groups corresponding to the same far-field emission.
PACS numbers: 89.20.Dd, 89.70.-a, 89.90.+n, 02.40.Ky, 02.40.Dr
The deinterleaving of radar pulses is vital for the suc-cessful operation of radar warning receivers [1]. Modernradars can emit up to a million pulses per second, hencein a multi-emitter environment deinterleaving becomes asignificant problem. Deinterleaving can be simplified ifradars are distinguishable on the basis of other knownpulse parameters such as transmitted power, carrier fre-quency or pulse width, however in many situations theradar pulse parameters are unknown or vary in an unpre-dictable way. There are a number of algorithms that arequite successful at deinterleaving pulse trains if the pulserepetition interval (PRI) is constant and a sufficientlylong set of adjacent pulses is recorded [2, 3, 4, 5]. Mod-ern civilian marine radar however often have a pseudo-random PRI pattern to minimise the possible effects ofinterference; hence the techniques described in the abovereferences will be ineffectual in deinterleaving the re-ceived radar pulses. In this paper we propose an alterna-tive method for deinterleaving received radar pulses thatrequires no knowledge of the emitter radar characteristicsand can be applied to only several pulses; this method isbased on the concept of the light-cone used to describeevents in Minkowski space-time [6].While we use radar pulses as an example in this letterit is important to realise that the proposed algorithm isvalid for associating any events that lie on the same light-cone. Furthermore although the following treatment as-sumes four receivers in 2+1 space-time it can be extendedtrivially to five receivers in 3 + 1 space-time. The Cayley-Menger determinant for four points in Eu-clidean space is defined as [7] D ≡ det s s s s s s s s s s s s (1)where s ij is the squared Euclidean distance between anytwo points in the space.It has previously been noted that the Cayley-Mengerdeterminant can be used to determine if a surface is flat [8]. The Cayley-Menger determinate for a flat surfaceis zero if the distance between points on the surface is thegeodesic distance which, in general, is not the Euclideandistance. The proof of this more general result can beverified by following the proof in Euclidean geometry aspresented in section 3.1 of [9] and realising that the resultgeneralises so long as there exists a coordinate system inwhich the metric tensor has constant coefficients, whichis an equivalent definition of the flatness of a surface [6].It is immediately apparent from this definition that theCayley-Menger determinant is zero in 2 + 1 Minkowskispace-time if the space-time events lie on a plane. If theMinkowski metric is used to calculate the space-time in-terval then s ij may be positive or negative depending onthe signature of the Minkowski metric.As the Cayley-Menger determinant specified by (1)only contains terms that are cubic in s ij , for errorlessmeasurements D = ( O (cid:0) s ij (cid:1) If any points do not lie on a flat surface(2)In 2 + 1 space-time the light-cone is a two dimensionalsurface defined by all the possible paths of a photon emit-ted at a particular space-time point. Using the conditionstated in (2) we conclude that the Cayley-Menger deter-minant is zero for all space-time events that lie on thesame light-cone and hence are causally connected to thesame emission event.Evaluation of the Cayley-Menger determinant us-ing (1) requires knowledge of the geodesic distance be-tween points on the light-cone. Calculation of thegeodesic distance requires determining the metric ten-sor for the embedded conical surface and integrating thecorresponding geodesic equation [6]. This process can besimplified considerably by choosing an appropriate set ofcoordinates { ξ, ψ, ζ } so that the metric tensor is diago-nal and constant on the surface of the light-cone. If thiscan be done then the geodesic distance is the Euclideandistance in these coordinates.One such set of coordinates is given by ξ = r cos ( φ sin θ ) (3a) ψ = r sin ( φ sin θ ) (3b) ζ = θ (3c)and the corresponding inverse relations r = p ξ + ψ (4a) φ = 1sin ζ arctan (cid:18) ψξ (cid:19) (4b) θ = ζ (4c)where { r, θ, φ } are the spherical polar coordinates andare related to the 2 + 1 cartesian space-time coordinatesby r = p ( x − x ) + ( y − y ) + ( t − t ) (5a) φ = arctan (cid:18) y − y x − x (cid:19) (5b) θ = arctan t − t p ( x − x ) + ( y − y ) ! (5c)where time is measured in natural units, i.e., c = 1 and[ t , x , y ] is the space-time coordinate of the emissionevent. Note that the coordinates defined by (3) are notthe conical coordinates as defined by [10], they have beenformulated so that the metric tensor on the cone is diag-onal and constant.The infinitesimal Euclidean distance between two points in coordinates defined by (3) is ds = dξ + dψ + (cid:0) ξ + ψ (cid:1) (cid:18) ψξ (cid:19) cot ζ ! dζ +2 ψ arctan (cid:18) ψξ (cid:19) cot ζdξdζ − ξ arctan (cid:18) ψξ (cid:19) cot ζdψdζ If we constrain the path between any two points to be onthe light-cone so that ζ = π then the geodesic distancebetween any two points { ξ i , ψ i , ζ i = π } and { ξ j , ψ j , ζ j = π } is s ij = ( ξ i − ξ j ) + ( ψ i − ψ j ) (6)The ultimate aim of many signal association algo-rithms is to provide information so that the emitter maybe localised. Calculation of the Cayley-Menger determi-nant (1) using (6) requires the space-time coordinates ofthe emission event for insertion into (5) and hence wouldnot appear to be useful.In the far-field limit the geodesic distance on the light-cone can be approximated by the Euclidean distance i.e.,( ξ i − ξ j ) +( ψ i − ψ j ) ≈ ( x i − x j ) +( y i − y j ) +( t i − t j ) The far field approximation is valid if the difference be-tween the geodesic distance on the light-cone and theEuclidean distance is much less than the Euclidean dis-tance. It can be shown that this condition is equivalent tothe inter-antenna spatial distances being much less thanthe spatial distance to the emitter, i.e., p ( x i − x ) + ( y i − y ) ≫ q ( x i − x j ) + ( y i − y j ) as in a typical radar application.In realistic scenarios the measured signal time of ar-rival is noisy which means that the expected value of D from (1) will not be zero, even if all the points lie on aflat surface. To calculate the expected value of D in thepresence of noise we would add a noise term ǫ ti to eachof the receive events, calculate the square distance as afunction of the noiseless distances and the noise, substi-tute these distances into (1) and obtain an expression forthe Cayley-Menger determinant in terms of the noiselessspace-time distances and ǫ t .This rather complex procedure can be avoided and theeffect of noise on (1) can be approximated by a rathersimple expression once it is realised the Cayley-Mengerdeterminant contains only terms of the type s ij s kl s mn and hence, to lowest order, the effect of noise can be ap-proximated by s m σ t , where s m is the maximum squareinterval between events and σ t is the variance in the timeof arrival noise. Using this result it is possible to con-struct a hypothesis test: H : All recieve events are caused by the sameemission eventThe decision rule we use to test between this hypothesisis H = ( true If D ≤ . σ t ) s m false If D > . σ t ) s m (7)The factors 2 and 1 .
96 come from that fact that the timedifference variance is twice the time variance and that95% of time of arrival measurements lie between ± . σ t of the noiseless time of arrival.An algorithm for associating space-time measurementsto a particular far-field emission event in 2 + 1 space-timeis as follows:1. Select four signal reception events from a list ofpossibilities2. Determine the Euclidean space-time interval be-tween all combinations of event quartets3. Calculate the Cayley-Menger determinant us-ing (1). Use this to test H using (7). If H is truethen the events chosen in step 1. are associated tothe same emission event.4. Repeat steps 1 to 3 until combinations are ex-haustedAs an example of how this algorithm would work inpractice consider a four element antenna array with fourantennas placed at the four corners of a three metresquare, i.e., p = (cid:20) (cid:21) p = (cid:20) (cid:21) p = (cid:20) − (cid:21) p = (cid:20) − (cid:21) (8)Consider a situation in which three signals arrive at eachof the four antennas so that t = { . , . , . } (9a) t = { . , . , . } (9b) t = { . , . , . } (9c) t = { . , . , . } (9d)How many emissions are there and what times are asso-ciated with each emission?The times given in (9) were generated by adding zeromean Gaussian noise with 50 picosecond standard de-viation to the arrival time of signals generated by twoemitters located 5km from the centre of the antenna ar-ray. The third set of arrival times was generated by arandom number generator and could represent spuriousmeasurements. There are 3 ways in which the sam-ple data can be grouped into quartets, however of these81 possible combinations only two satisfy the hypothe-sis test as stated in (7). Using the algorithm outlinedabove we are able to correctly conclude that there are two emissions events whose corresponding receive events x = [ ct, x, y ] are x = [4 . , , T x = [3 . , , T x = [7 . , − , T x = [8 . , , − T and x = [7 . , , T x = [5 . , , T x = [2 . , − , T x = [4 . , , − T Each quartet of events can used to estimate the locationemitter using the time difference of arrival technique [11].The Defence Science and Technology Organisation(DSTO) of Australia is building an antenna array sys-tem to test the association algorithm outlined in thispaper. The antenna array will be constructed using aprecise time interval measuring unit such as the ATMD-GPX [12]. Once the evaluation system is built trials willbe conducted with two and three radars placed at about5km from the antenna array. The tests will be conductedat the DSTO signals testing facility in Adelaide, Aus-tralia. Results of the trial will be reported elsewhere. [1] R. G. Wiley,
Electronic intelligence : the analysis ofradar signals (Artech House, Dedham, MA, 1982).[2] T. Conroy and J. B. Moore, Signal Processing, IEEETransactions on , 3326 (1998).[3] A. Logothetis and V. Krishnamurthy, Signal Processing,IEEE Transactions on , 1344 (1998).[4] J. B. Moore and V. Krishnamurthy, Signal Processing,IEEE Transactions on , 3092 (1994).[5] D. J. Milojevic and B. M. Popovic, Radar and SignalProcessing, IEE Proceedings F , 98 (1992).[6] C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravi-tation (W. H. Freeman and Company, 1973).[7] L. M. Blumenthal,
Applications of Distance Geometry (Oxford University Press, Oxford, 1953). [8] S. Weinberg,
Gravitation and cosmology: principles andapplications of the general theory of relativity (Wiley,New York, 1972).[9] D. Michelucci and S. Foufou, in
ACM Symposium onSolid Modeling and Applications (ACM Press, Genova,Italy, 2004), pp. 285–290.[10] P. Moon and D. E. Spencer,
Field theory handbook : Incl.coordinate systems, differential equations and their solu-tions (Springer, Berlin, 1988), 3rd ed.[11] D. J. Torrieri, Aerospace and Electronic Systems, IEEETransactions on
AES-20 , 183 (1984).[12] acam mess electonic,
Atmd-gpx tdc-gpx eval-uation system: Datasheet , World WideWeb electronic publication (2005), URL ..