1
SET MODE PASSIVE LOCATION IN TOA/TDOA MODES FIELD OF THE INVENTION
The present invention addresses the resolving of the problems associated with the passive location of targets in TOA (Time Of Arrival) or TDOA (Time Difference Of Arrival) mode. 5
For TOA, the invention makes it possible to locate any target by analyzing the arrival times, on one and the same receiver, of the waves transmitted by one or more transmitters and reflected by the target.
For TDOA, the invention makes it possible to locate a transmitting 10 target by measuring differences of arrival time of the transmitted wave on a number of receivers that are synchronized and scattered in space.
The present invention relates more particularly (but not exclusively) to the field of passive radars.
15
By its generic nature, the present invention also addresses all the mode combinations (multiple TOA, multiple TDOA, or even mixed TOAn-DOA).
20 CONTEXT OF THE INVENTION - PRIOR ART
The basic principle of the passive location methods, whether in TOA or TDOA mode, is to determine the positioning of targets by using the information supplied simultaneously by different information sources.
In TOA mode, or "Time Of Arrival" mode, interest is focused on the
25 signals transmitted by one or more transmitters and the same signals reflected by a target. The measurement, for a given transmitter, of the delay between the fonward path (transmitter -^ receiver) and the reflected path (transmitter -> target -^ receiver) is used to define a location curve (or a surface if the problem is dealt with in 3D) that takes the form of an ellipse or
30 an ellipsoid.
Thus, if a number of transmitters are analyzed simultaneously, and provided that there is the capability to receive the fon^/ard and reflected paths, it Is possible to detemiine the position of the target that is sought by determining

2
the mutual points/areas of intersections of the different location curves/surfaces.
In TDOA mode, or "Time Difference Of Arrival" mode, interest is 5 focused on the location of transmitting targets by means of a number of synchronized receivers (at least two), a main receiver and one (or more) secondary receivers. As in the TOA mode, the position of the transmitting target is then determined geometrically from location curves (hyperbolas or hyperboloids) established from the measurements of the delays between the 10 different signals originating from the target and arriving at different instants on each of the receivers.
One of the problems raised by passive location stems from the taking-into-account of the uncertainty of the measurement delivered by the receivers. In
15 practice, to produce an accurate location of the target, it is necessary to know as accurately as possible the location area that is compatible with the uncertainties affecting the measurements. Consequently, the location curves are in reality location areas, each area being situated between two extreme curves, the spacing of which depends on the accuracy of the receivers. That
20 way, the points of intersections of the different location curves ideally obtained by using a number of transmitting sources (TOA mode) or a number of receivers (TDOA mode) are replaced in practice by areas of intersection within which these points are situated. Then, the search for the location areas that are compatible with the
25 measurements firstly involves the mathematical characterization of said areas followed by the search for them in a space of interest (i.e. the space in which the presence of a target is sought).
The usual methods used to perform this search are generally grid methods 30 which involve finely meshing all the space in which the targets are sought, that is, a space that is vast enough to contain the area of uncertainty and systematically analyzing each mesh to check whether it belongs to the location area. There are also algebraic methods of the least-square type, or even probabilistic methods, the complexity (and therefore the complexity of 35 implementation) of which increases with the number of infonnation sources.

Apart from the grid methods, none of these methods provides a way of finely restoring the areas of uncertainties associated with the measurement errors (they give only error ellipses or ellipsoids). In addition, the grid methods require a large computation capability to process each mesh sufficiently 5 quickly and determine the location area sufficiently quickly.
DESCRIPTION OF THE INVENTION
One aim of the invention is to benefit from the advantages of the grid processing operations notably in terms of resolution, yet without suffering the 10 d rawbacks thereof.
To this end, the subject of the invention is a method of passively locating a target in TOA or TDOA mode that implements a successive subdivision into blocks of an initial space (in which a target is to be located). The set of blocks
15 is analyzed iteratively. On each iteration, each block of interest is subdivided into smaller identical subblocks. A block of interest is, according to the invention, a block in which at least one point belongs to the location area being sought. The iterative process is stopped when the size (resolution) of the subblocks obtained on the current iteration corresponds to the desired
20 resolution.
The set of blocks resulting from this process provides an approximation of the shape of the location area that is sought (if the latter exists in the initial space).
25 More specifically, its subject is a method of passively locating a target
in TOA or TDOA mode implementing a successive subdivision into blocks of an initial space (in which a target is to be located) and a search within each block for the presence of points belonging to the location area being sought. This invention is also characterized by the fact that the subdivision and the
30 search are performed in the form of iterative steps on a selection of candidates blocks, modified on each iteration, so that, on each iteration, the blocks of the selection obtained on completion of the preceding Iteration are searched to find the blocks containing at least one point belonging to the location area being sought. The blocks not containing any point are
35 subsequently excluded from the selection, whereas the blocks containing at

least one point are re-subdivided into subblocks and replaced in the selection by the duly formed subblocks. The selection obtained on each iteration defines the location area with a resolution that increases on each iteration.
5 According to the invention, the iterations are stopped when the
subblocks forming the selection define the location area with the desired resolution.
The method according to the invention also comprises an initialization 10 step in which a first block [XQ] is defined that corresponds to an "a priori"
search space, this block constituting the initial selection.
According to a prefen^ed embodiment, the method according to the
invention mainly comprises:
15 - an initialization step in which a first block [XQ] Is defined that
corresponds to an a priori search space and that leads to the formation of an initial list Lo° formed from the block [Xg],
- an iterative step consisting in :
- subdividing each block [Xn] of the current list Lo'"^ into four
20 adjoining subblocks [Xn^]. [Xn^], [Xn^] and [Xn^],
- searching for and selecting from the duly constituted subblocks those containing at least one point forming part of the location area,
- updating the current list to obtain a list Lo' In which :
-a) the blocks [Xn] for which no subblock has been selected are
25 deleted,
-b) the other blocks are replaced by the selected subblock [Xn'],
the iterative step also includes an operation to compare the size A^o^^) of the blocks constituting the list Lo', the method being stopped when the size A'-'O^^^ 30 Is greater than the desired resolution objective A'^J.
According to this embodiment, a block [Xj] Is selected if at least one of Its points satisfies the criterion defined as follows :

xl
tr +
fb?
E-1
[cr
df
a b?
^^^ + ^Hr +
aF
i_k]
£_
cF
dF
0€j([x*])withj([xJ]) = 0 6j([x'])wrthj([x^])=
0€j([x'])withj([x'l) = 0€j([x'])withJ([x'])=

in "TOA 2D" mode,
in "TDOA 2D" mode,
-1
in "TOA 3D" mode,
[bF
-1
in "TDOA 3D" mode.

The invention relies on the set mode approach used to locate the target. The measurements deriving from the various sensors are modeled by intervals (i.e., they include a bounded error). Knowing these measurements, the invention uses an iterative process based on an ad hoc set mode criterion (i.e. dependent on the problem, TOA/TDOA single/multiple sensors) to find and approach with the desired resolution, ad the areas of the space that are likely, in light of the measurements, to contain a target. The location is set mode in the sense that the invention provides a solution set to the location problem (i.e., a set of target positions guaranteed to contain the true position of the target).
Unlike a grid method, there is no need to mesh all the space, the iterative process used in the invention makes it possible to concentrate directly on the regions of interest.
DESCRIPTION OF THE FIGURES
The features and benefits of the invention will be better appreciated from the description that follows, which explains the invention through a particular embodiment taken as a nonlimiting example and based on the appended figures, which represent:
- figure 1, a typical single-transmitter geometrical configuration for implementing the TOA mode,
- figure 2, a representation of the set of location curves ideally

6
obtained in TOA mode with three transmitting sources,
- figure 3, a typical two-receiver geometrical configuration for Implementing the TDOA mode,
- figure 4, a representation of the set of location curves ideally obtained in TDOA mode with three secondary receivers,
- figure 5, the representation of a location area actually obtained in TOA mode with one transmitting source,
- figure 6, the representation of a location area actually obtained in TDOA mode with one receiver,
- figure 7, a theoretical flow diagram of the method according to the invention,
- figure 8, an illustration of the application of the method according to the Invention to the TOA 2D mode with a single-transmitter system,
- figure 9, an illustration of the application of the method according to the invention to the TDOA 2D mode with a single secondary receiver system,
- figure 10, an Illustration of the application of the method according the invention to the TOA 2D mode with a two-transmitter system, and
- figure 11, an Illustration of the application of the method according to the invention to the TDOA 2D mode with a two secondary receivers system.
DETAILED DESCRIPTION
Interest is focused initially on figure 1 which schematically shows the ideal operating principle of the TOA location mode. In the interest of clarity of the explanation, the model illustrated here is a two-dimensional model corresponding to the analysis of the signals obtained from a single transmitter (2D single-transmitter model).
As the figure illustrates, this "single transmitter" operating mode involves a transmitting source 12 (transmitter), a receiver 11, and a target 13 that is to be located. To locate the target, the receiver 11 measures the time delay that exists between the received wave originating directly from the transmitter 12 (forward wave), and the received wave originating from the reflection of the wave transmitted by the source 12, on the target 13 that is to be located (reflected wave).
In such a configuration, if the distance between the transmitting source 12

and the receiver 11 is denoted L, the distance between the source 12 and the target 13 RT and the distance between the target and the receiver RR, the location of the target 13 involves measuring the delay between the forward and reflected waves and determining from this delay, by any known appropriate method, the distance Rb traveled by the reflected wave, defined by:
The target 13 is then located as illustrated by the curve 21 of figure 2 on an ambiguity ellipse, the foci 22 and 23 of which are the position of the transmitting source 12 and that of the receiver 11, and the major axis (semi-major axis) of which has the length Rb/2.
Consequently if a Cartesian frame of reference xOy is defined that is
centered on the middle of the line segment [Rx, Tx] 14 linking the source 12
-> and the receiver 11, and the vector Ox of which is collinear to the vector
RyJx , the equation of this location ellipse 21 In the frame of reference xOy is expressed by the following equation :
a2 b2
with :
a = Rb/2
and b = Va2-L2/4
To refine the location of the target 13, it is obviously necessary to have a number of transmission sources. That way, for one and the same target 13, a location ellipse can be associated with each source, the intersections of these ellipses defining the possible positions of the source to be located. The curves 21 and 24 of figure 2 illustrate the results obtained with a "2D two-transmitter" configuration from which two location ellipses are obtained, the foci 22, 23 and 25 of which are respectively the receiver and the first source for the ellipse 21, and the receiver and the second source for the ellipse 24. The set of the possible places of location of the target 13 then comprises the

8
two points of intersection 26 and 27 of the two curves. Consequently, to determine the position of the target 13 without ambiguity, it is necessary to have at least one additional transmitting source ("2D multiple-transmitter" configuration), the intersection of the three location ellipses 21, 24 and 28 defining a single common point, the point 26 for example, on which the target 13 is situated.
Interest is then focused on figure 3 which schematically shows the ideal operating principle of the TDOA location mode. In the Interest of clarity of the explanation, the model illustrated here is, as for the TOA mode presentation, a two-dimensional model corresponding to the analysis of the signals obtained from a transmitting target 31 and received by two receivers 32 and 33 distant from each other, a reference receiver, called main receiver, and a so-called secondary receiver, synchronized on the reference receiver (2D single secondary receiver model).
As illustrated by figure 3, this "secondary single-receiver" operating mode Involves a transmitting target 31 (transmitter) that is to be located, a reference receiver 32, and a secondary receiver 33. To locate the target, the time delay that exists between the instant of reception by the reference receiver of the wave transmitted by the target 31 and the instant of reception of this same wave by the secondary receiver 33 is then analyzed.
In such a configuration, if the distance between the two receivers 32 and 33 is denoted L, the distance between the transmitting target 31 and reference receiver 32 is denoted RR and the distance between the transmitting target 31 and the secondary receiver 33 is denoted RRI, the location of the target 31 consists in measuring the delay between the waves received by the two receivers and detemiinlng from this delay, by any known appropriate method, the difference in distance traveled Rdi defined by : Rdi =RRI-RR
The target 31 is then located as illustrated by the curve 41 of figure 4 on a hyperbola having foci 42 and 43 which are the position of the reference receiver and that of the secondary receiver and for which the distance

9
between peaks has the value Rcii/2.
Consequently, if a Cartesian frame of reference xOy is defined that is
centered on the middle of the line segment [Rx, Rxi] 34 linking the reference
receiver 32 and the secondary receiver 33, and the vector Ox of which is
5 collinear to the vector RxRxi. the equation for this location hyperbola 41 in the frame of reference xOy is expressed :
x2 v2
^-\ = 1 [2]
C2 d2 ^ ^
with :
10 c = Rdi/2
and d = yjl^/A-c^
To refine the location of the target 31, it Is obviously necessary to have a
15 number of secondary receivers. That way, for one and the same target 31, a location hyperbola can be associated with each reference receiver/secondary receiver pairing. The intersections of these hyperbolas then define the possible positions of the target. The curves 41 and 44 of figure 4 Illustrate the results obtained with a "2D two secondary receivers" configuration from
20 which two tocation hyperbola are obtained, the foci 42, 43 and 45 of which are respectively the reference receiver Rx and the secondary receiver Rxi for the hyperbola 41 and the reference receiver Rx and a second secondary receiver Rx2 for the hyperbola 44. The set of possible places of location of the target 31 then comprises the points of intersection 46 and 47 of the two
25 curves.
Consequently, to determine the position of the target 31 without ambiguity it is necessary to have at least one additional secondary receiver ("2D multiple secondary receivers" configuration), the intersection of the three location hyperbola 41, 44 and 48 defining a single common point, the point 46 in the
30 example, on which the target 31 is situated.
The theoretical determination principle explained in the preceding paragraphs through a two-dimensional tocation ("2D" location) can naturally be extended to a location in space (I.e. In 3D location).

10
In TOA mode, the location ellipse in "single-transmitter" mode is replaced by an ellipsoid in space, an ellipsoid which can be represented by the following equation :
x2 , y2 z2
a2 b2 b2
5 with :
a = Rb/2
and b = Va2-L2/4
10 Similarly, in TDOA mode, the location hyperbola in "single secondary
receiver" mode is replaced by a hyperboloid in space, a hyperboloid which can be represented by;
X2 v2 22
^—y__±_=i [4]
C2 d2 d2 ^ ^
15 with
c = Rdi/2 and d = VL^/4-c2 .
20 Interest is now focused on figures 5 and 6 which illustrate, through
simple location scenarios in TOA 2D mode (figure 5) and in TDOA 2D mode (figure 6), the problem raised by the accuracy of the real measurements obtained with the receivers.
25 Like any measuring device, the receivers used by the passive location
systems provide measurements that are marred by a certain inaccuracy that can be assumed to be bounded. This inaccuracy means that the distance measurements performed in TOA or TDOA mode are represented, no longer by exact values, but by intervals, the size of which corresponds to the
30 maximum measurement error. These intervals [x] are defined by the following equation:
[x] = [x-,x+]={xe9?/x- RCII "*"^RdJ' such as that
illustrated in figure 6. This surface 63 can be described by the following parametrical form :

(x,y,z)6 9?2

1e

[c]' [d]^

[9]

with

[cl = [Rdil/2

0 and [d] = yll^/4-[cf
In the case of a location in space ("TDOA 3D"), the taking into account of the bounded errors on the measurement of Rdi leads to the location of the target concerned in a volume defined by two confocal hyperboloids, a volume that 15 can be described, in a similar manner, by the following parametrical form :

(x,y,z)e9^3

1e

[cP [dP Wl

[10]

Interest Is now focused on figure 7 which schematically shows the 20 principle of the method according to the invention. In order to make the explanation of the operating principle of the method according to the invention clearer, this principle is described here in detail for the particular case of the search for an area of a location by means of location systems of "single-transmitter" type (location in TOA mode) or "single secondary 25 receiver" type (location in TDOA mode).

30

The basic principle of the method involves a progressive refining of the location area. It consists in subdividing the space that is to be analyzed (i.e., the initial search block [xo]x[yo] into adjoining subblocks in which the presence of a target is evaluated by means of an ad hoc criterion. The subblocks in which the presence of a target Is confirmed are in turn subdivided and the others are rejected. The resulting iterative process is repeated as long as the

13

presence of a target is confirmed in the blocks currently being analyzed and a stop criterion (corresponding to a block width objective) is not reached.
To this end, the method according to the invention comprises a number of 5 steps:
- an initialization step 71,
- an iterative calculation step 72.
The initialization step 71 consists in defining an initial block, [XQ],
10 corresponding to an a priori search space. In 2D mode, the following thus applies:
[Xol=klx[yo].
and in 3D mode, the following applies :
15
[Xo]=[xo]x[yo]>^[zol-
The step 71 also involves initializing a list LQ, of a size that varies during the
implementation of the method, comprising the list of blocks to be studied. The 20 content of U is initialized with [XQ].
The step 71 then consists in defining a stop criterion for the method. This stop criterion is given here by the "objective" resolution A°''i with which it is desired to ultimately characterize the location area. This resolution is 25 naturally limited by the accuracy of the measurements supplied by the receivers, but it can be arbitrarily set within this limit. In 2D location mode, it is perhaps defined In a coordinate system xOy by Ao"' = (A^',A°y''^).
The step 71 is followed by a step 72 canying out an iterative 30 processing operation that includes two nested processing loops, a main loop 73 and a secondary loop 74.
The main loop 73 consists in updating the list U established on initialization
with the results of the processing carried out by the second loop. Thus, on 35 each iteration i of the main loop, there is a re-updated list LQ available.

14
denoted VQ .
On each iteration, each block [Xi3j^J = L'o(k) forming the list UQ is subdivided
into N adjoining subblocks (N=4 in 2D mode, N=8 in 3D mode) and grouped together in a list L^.
5
As for the secondary loop 74, this consists in eliminating from L^, the blocks
that are incompatible with the location area being sought (that is, the blocks that do not validate the criterion).
10 Consequently, the element [Xi,,^ J of L'Q is replaced, in U^, by the list L^.
According to the invention, the method is chosen to determine whether a subblock [X'] includes a portion of the location area consists In determining whether one or more points of the subblock belong to that area. 15 To do this in the "single-transmitter" or "single secondary receiver" systems, simply the quantity J([X']) is considered, which Is defined according to the location mode concerned by:

in "TOA 2D" mode: jjx'l)^

[br

[11]

- In "TOA 3D" mode: j([xj])=

yiM
bf

+#t-1

[12]

in "TDOA 2D" mode: 4x'I)=

£_
p]
r_i2

il2

-1

[13]

y
c 2 2 d 2 2 zi d 2
I J
in"TDOA3D"mode: 4x'I)= fi—ril-"

[14]

26

with [Xi] = [xJ]x [yJ] for the "2D" modes.
and [Xi]= [xi]x[yi]x[zi] for the "3D" modes.

15
Therefore, to verify if at least one point of a block [Xi] belongs to the location
area, it is sufficient, according to the invention, to check whether:
0€J([XJ]).
5 The main loop is intrinsically an endless loop for which it is necessary
to determine a stop condition and an operation to test for the appearance of this stop condition. According to the invention, this stop condition is initialized in the step 71 and relates to the resolution A°*'J with which it is desired to define the location area. Thus, the main loop 73 includes a test operation 75 10 executed at the end of processing on each iteration i. This condition is defined by the following equation :
/^L'od) < l^obi

A'»'" £A°'^=-

and [11]

15 in which A'"'O^^ represents the resolution of the first block forming the list UQ ,
each block of the list having an klentical resolution at this stage. Thus, as long as the condition of the equation [11] is found to be verified, the iterative calculation step 72 is repeated. The iteratbn stops only when the desired resolution is reached.
20
Thus, when applied to a two-dimensional TOA or TDOA location processing operation, and for an iterative subdivision of each block into P == 4 subblocks, the method according to the invention can be described by the following sequence of actions :
25
1. Definition of the initial size of an analysis cell (block):
[Xo] = [xo]x[yo].
2. Initialization of tiie list U of the analyzed blocks: L^.
3. Definition of the stop criterion: Ao^J = (Af ^A^"^). 30 4. /beginning of main loop/: (Formation of UQ )

5. For n varying from 1 to the size of the list UQ"^
6. subdivision of each block n Into P=4 adjoining subblocks :

16
[Xi]=[x-.(x--.x^)/2].[y-.(y-+y^)/2]
K^]=[x-.(x-.x^)/2].[(y-.yi/2.yi
5 [x,^]=[(x-+x^)/2.xi.[y-.(y--.y^)/2]
k]=[(x-.x^)/2,xi.[(y-.y^)/2,yi
7. creation of the list U\ = {[X,'], [Xn^], [Xp^l, [Xn']}
10 8. /beginning of secondary loop/
9. For j varying from 1 to 4
10. Elimination of the subblocks [x|,J, that do not satisfying the selection
criterion 0 e j([xi, J): Formation of U\.
11. - if U] is not empty : replacement, in L'5'', of the block [X^] by
15 the list L"],
- If U\ is empty: elimination of the block [Xn] from the list U^""
12. /end of secondary loop/
13. /end of main loop/ (Fomiation of UQ )
14. Calculation of the resolution of the first element of U':

20 A'°*"= <,„-x

'-0'" y

15. - if A^'o^^^ < A°^i: return to /start of main loop/
- else : end of procedure
16. /end of procedure/
25 The principle of implementation of the method according to the invention can be advantageously illustrated by figures 8 and 9.
Figure 8 illustrates the implementation of the method according to the invention on a location in TOA mode, with a "2D single-transmitter" type system.
30 As can be seen the figure, the implementation of the method according to the invention is physically embodied in a breakdown into blocks of the space that includes the location area 81. This subdivision forms a mesh of the space, a mesh in which each cell is analyzed to determine whether or not it includes a

17
portion of the location area. The duly formed mesh is refined iteratively. During the successive iterations, certain cells 83 formed by the division of a larger cell 82 including a portion 84 of the location area no longer include such portions. According to the invention, the process of refining these 5 "empty" areas then ceases, which makes it possible advantageously to avoid continuing, for such cells, an analysis that is sterile and costly in computation workload and to concentrate the refining on the cells where it presents a benefit for determining as accurately as possible the location area 81. An irregular mesh is thus advantageously obtained, in which the tight mesh is 10 only centered on the location area.
Figure 9 illustrates in the same way the implementation of the method according to the invention on a location in TDOA mode, with a system of the "2D single secondary receiver" type. The implementation of the method 15 according to the invention is embodied in the same way by a subdivision into blocks of the space that includes the kication area 91. Areas 93 are also obtained for which the refining is not carried out and areas 94 for which it is continued.
20 The method according to the invention, detailed in the preceding
paragraphs for the particular case of systems of 'TOA single-transmitter" or 'TDOA single secondary receiver" types can obviously be applied to the more complex location systems of "TOA multiple transmitter" or "TDOA multiple secondary receiver" types in 2D or in 3D. These types of systems
25 can be used, by considering the intersections of the different location areas, to refine the location of the target being sought by limiting the location area to the intersections of the different areas obtained.
Regarding these modes, the method according to the invention, as described in the foregoing, remains applicable in principle. Only the criterion for
30 selection of a subblock used in the secondary loop of the step 72 of the iterative processing operation is modified. Thus, If the system includes a total number N of transmitters (location in "TOA multiple transmitter" mode) or receivers (location in "TDOA multiple secondary receiver" mode), a subblock [X] will be selected if the following equation is verified :
35


15 and in which Ji([X}) represents, for each individual system (Rx, T,^) or (Rx, Rxj), the quantity J({X]) defined previously ([11,12,13,14]).
The figures 10 illustrate the implementation of the method in the case of a location system of "TOA two-transmitters" type. As can be seen in the figure,
20 the mesh produced using the method according to the invention is once again an irregular mesh for which the tight cells, of greater resolution, are located on the location areas 101 and 102 defined by each individual system, and more specifically on the portions 103 and 104 of these areas that constitute intersections and that represents the possible location areas of the
25 target.
Turning to figure 11, this illustrates in a similar manner the Implementation of the method in the case of a location system of the "TDOA two secondary receivers" type. As for the preceding case, it can be seen that the mesh 30 produced by the method according to the invention comprises tight cells only in the places of the areas 113 and 114 representing the intersections of the location areas 111 and 112 defined by each individual system.

19
Since no system can be perfect, a certain tolerance can be accepted as to the number of basic criteria Jn([X]) that are simultaneously satisfied. This
tolerance is reflected by an acceptance of an interval [X] if A>M (withM