Method for determining positions of targets by bistatic measurements using signals scattered by the targets

0001The present invention relates to a method for determining positions of targets by bistatic measurements using signals scattered by the targets. Also the velocities of the targets can be determined. The method comprises a rapid bistatic association method which is suitable for, for instance, a network of radar stations in the manner of AASR (Associative Aperture Synthesis Radar) although there may be further fields of application. AASR is described, inter alia, in the published patent application <patcit id="pcit0001" dnum="WO02093192A"><text>WO 02/093192</text></patcit>, which is published after the priority date of the present application. In the following, the description will be concentrated on the new method of associating by bistatic measurements only.

0002First the fundamental problem that is solved by the invention will be presented. N_s stations (for instance radar stations) are imagined to be set out in the space (R³). The stations are designated S_j, j=1, ...N_s and their position vectors are designated ρ_j, j=1, ... N_s. In addition to the stations, there are also N_t moving targets which are to be detected. They are designated t_i, i=1, ... N_t and have corresponding time-dependent position vectors r_i = r_i(t), i=1, ... N_t.

0003Each station is capable of measuring distances (up to a certain maximum distance) and radial speed for each target. Thus, the station s_j, 1 ≤ j ≤ N_s will, at a certain point of time, measure <maths id="math0001"><math display="block"><msub><mi mathvariant="normal">d</mi><mi mathvariant="normal">j</mi></msub><mfenced><mi mathvariant="normal">k</mi></mfenced><mo mathvariant="normal">=</mo><mfenced open="|" close="|"><msub><mi mathvariant="bold">r</mi><mi mathvariant="normal">k</mi></msub><mo mathvariant="normal">-</mo><msub><mi mathvariant="bold">ρ</mi><mi mathvariant="normal">j</mi></msub></mfenced><mo mathvariant="normal">,</mo><mspace width="2em" /><mi mathvariant="normal">k</mi><mo mathvariant="normal">=</mo><mn mathvariant="normal">1</mn><mo mathvariant="normal">,</mo><mn mathvariant="normal">2</mn><mo mathvariant="normal">,</mo><mo mathvariant="normal">…</mo><mspace width="1em" /><msub><mi mathvariant="normal">N</mi><mi>dj</mi></msub><mo mathvariant="normal">≤</mo><msub><mi mathvariant="normal">N</mi><mi mathvariant="normal">t</mi></msub></math><img file="EP1514134B1_D0001.tif" /></maths><maths id="math0002"><math display="block"><msub><mi mathvariant="normal">v</mi><mi mathvariant="normal">j</mi></msub><mfenced><mi mathvariant="normal">k</mi></mfenced><mo mathvariant="normal">=</mo><mfenced><mi mathvariant="normal">d</mi><mo mathvariant="normal">/</mo><mi>dt</mi></mfenced><mo>⁢</mo><mfenced open="|" close="|"><msub><mi mathvariant="bold">r</mi><mi mathvariant="normal">k</mi></msub><mo mathvariant="normal">-</mo><msub><mi mathvariant="bold">ρ</mi><mi mathvariant="normal">j</mi></msub></mfenced><mo mathvariant="normal">,</mo><mspace width="1em" /><mi mathvariant="normal">k</mi><mo mathvariant="normal">=</mo><mn mathvariant="normal">1</mn><mo mathvariant="normal">,</mo><mn mathvariant="normal">2</mn><mo mathvariant="normal">,</mo><mo mathvariant="normal">…</mo><mspace width="1em" /><msub><mi mathvariant="normal">N</mi><mi>dj</mi></msub><mo mathvariant="normal">≤</mo><msub><mi mathvariant="normal">N</mi><mi mathvariant="normal">t</mi></msub></math><img file="EP1514134B1_D0002.tif" /></maths>

0004For stations that are sufficiently close to each other, also bistatic measurement information is obtained, i.e. transmitting from one station and registration at another. For the pair of stations (S_i, S_j) it means that the following is registered <maths id="math0003"><math display="block"><msub><mi mathvariant="normal">d</mi><mi>ij</mi></msub><mfenced><mi mathvariant="normal">k</mi></mfenced><mo mathvariant="normal">=</mo><mfenced open="|" close="|"><msub><mi mathvariant="bold">r</mi><mi mathvariant="normal">k</mi></msub><mo mathvariant="normal">-</mo><msub><mi mathvariant="bold">ρ</mi><mi mathvariant="normal">i</mi></msub></mfenced><mo>+</mo><mo mathvariant="normal">|</mo><msub><mi mathvariant="bold">r</mi><mi mathvariant="normal">k</mi></msub><mo mathvariant="normal">-</mo><msub><mi mathvariant="bold">ρ</mi><mi mathvariant="normal">j</mi></msub><mo mathvariant="normal">|</mo><mo mathvariant="normal">=</mo><msub><mi mathvariant="normal">d</mi><mi mathvariant="normal">i</mi></msub><mfenced><mi mathvariant="normal">k</mi></mfenced><mo mathvariant="normal">+</mo><msub><mi mathvariant="normal">d</mi><mi mathvariant="normal">j</mi></msub><mfenced><mi mathvariant="normal">k</mi></mfenced><mo mathvariant="normal">,</mo><mspace width="1em" /><mi mathvariant="normal">k</mi><mo mathvariant="normal">=</mo><mn mathvariant="normal">1</mn><mo mathvariant="normal">,</mo><mn mathvariant="normal">2</mn><mo mathvariant="normal">,</mo><mo mathvariant="normal">…</mo><mspace width="2em" /><msub><mi mathvariant="normal">N</mi><mi>dij</mi></msub><mo mathvariant="normal">≤</mo><msub><mi mathvariant="normal">N</mi><mi mathvariant="normal">t</mi></msub></math><img file="EP1514134B1_D0003.tif" /></maths><maths id="math0004"><math display="block"><msub><mi mathvariant="normal">v</mi><mi>ij</mi></msub><mfenced><mi mathvariant="normal">k</mi></mfenced><mo mathvariant="normal">=</mo><mfenced><mi mathvariant="normal">d</mi><mo>/</mo><mi>dt</mi></mfenced><mo>⁢</mo><mfenced open="|" close="|"><msub><mi mathvariant="bold">r</mi><mi mathvariant="normal">k</mi></msub><mo mathvariant="normal">-</mo><msub><mi mathvariant="bold">ρ</mi><mi mathvariant="normal">i</mi></msub></mfenced><mo>+</mo><mfenced><mi mathvariant="normal">d</mi><mo>/</mo><mi>dt</mi></mfenced><mo mathvariant="normal">|</mo><msub><mi mathvariant="bold">r</mi><mi mathvariant="normal">k</mi></msub><mo mathvariant="normal">-</mo><msub><mi mathvariant="bold">ρ</mi><mi mathvariant="normal">j</mi></msub><mo mathvariant="normal">|</mo><mo mathvariant="normal">=</mo><msub><mi mathvariant="normal">v</mi><mi mathvariant="normal">i</mi></msub><mfenced><mi mathvariant="normal">k</mi></mfenced><mo mathvariant="normal">+</mo><msub><mi mathvariant="normal">v</mi><mi mathvariant="normal">j</mi></msub><mfenced><mi mathvariant="normal">k</mi></mfenced><mo mathvariant="normal">,</mo><mspace width="1em" /><mi mathvariant="normal">k</mi><mo mathvariant="normal">=</mo><mn mathvariant="normal">1</mn><mo mathvariant="normal">,</mo><mn mathvariant="normal">2</mn><mo mathvariant="normal">,</mo><mspace width="3em" /><msub><mi mathvariant="normal">N</mi><mi>dij</mi></msub><mo mathvariant="normal">≤</mo><msub><mi mathvariant="normal">N</mi><mi mathvariant="normal">t</mi></msub></math><img file="EP1514134B1_D0004.tif" /></maths>

0005It is to be noted that with these designations, d_ii(k)=2d_i(k), v_ii(k)=2v_i(k), i=1,2, ... , k=1,2, ...

0006For each sensor (monostatic or bistatic geometry) targets are thus registered in respect of distance and Doppler. It is a priori not possible to know which registration from one sensor is associated with a certain registration from another sensor, i.e. originating from the same target. If registrations from different sensors are paired incorrectly, false targets, ghost targets, arise. The problem of association is to discriminate, among all conceivable possibilities of combining sensor data, corresponding to conceivable target candidates, between correct combinations (targets) and false combinations (ghosts).

0007The maybe most straight-forward method is to consider three neighbouring stations and their monostatic registrations, which for the sake of simplicity are assumed to be N in number. These measurements can be combined in N³ ways where each combination corresponds to a target position which is determined up to reflection in the plane containing the three stations. (Certain combinations can be incompatible, corresponding to false candidates.) These ~N³ candidates can then one by one be compared with the bistatic measurements and either be rejected or accepted. The problem of such a method is that it will be very slow if the number of targets, N, is large. For this reason, more efficient association algorithms have been developed.

0008Each target is to be determined in respect of position as well as velocity, i.e. they are to be positioned in a six-dimensional state space. The number of cells in the state space can be very large (~10¹⁸), which means that traditional projection methods will be irretrievably slow.

0009The above published patent application <patcit id="pcit0002" dnum="WO02093192A"><text>WO 02/093192</text></patcit> discloses a method of attacking the problem of association by designing a sensor network so that each target is registered by many sensors (monostatic and bistatic), i.e. a high degree of redundancy is obtained in the system. Then the state space is divided into a manageable number of relatively large cells.

0010If the cells are just large enough, it will be possible to reject many of them, i.e. they cannot contain any targets, for the following reasons. If the cell contains a target, all (or almost all) of the possible sensors that can register targets in the current cell, will indicate such a registration. On the other hand, if the cell is empty, some sensors, and yet not too many, will still indicate registrations (from other targets) that are compatible with the cell in question. Owing to redundancy, a sufficient number of sensors will indicate the cell as empty, and it can be rejected. When the number of cells is thus reduced, the surviving cells are divided into smaller cells and the process is repeated. The process is repeated until the cells in the state space have reached the desired size. As the cells are becoming smaller, fewer and fewer ghost targets will survive, so that, when interrupting the process, practically only real targets are left. What speaks in favour of this method is that it uses (but also requires) the redundancy of the sensor network. However, it is not yet quite clear how rapid the method may eventually be.

0011An alternative method is disclosed in the published patent application <patcit id="pcit0003" dnum="WO02093191A"><text>WO 02/093191</text></patcit>, which is published after the priority date of the present application, and implies that use is made of certain symmetries of the combination sensors - measurement data. Given two stations, the two monostatic measurements, together with the bistatic measurement, will share a symmetry, viz. that the three measuring geometries are all insensitive to rotation of the targets about the axis extending through the two stations. This means that it is possible to make an initial rapid screening of the candidates and delete a large number of false associations (ghosts). The subsequent final association will then be significantly more rapid. A drawback, however, is that the monostatic measurements will be important, which may be disadvantageous in connection with reconnaissance of stealth targets.

0012A yet alternative method is disclosed in the <patcit id="pcit0004" dnum="US4499468A"><text>U.S. patent 4,499,468</text></patcit>. Differences of bistatic distances in one set of measurements are compared with differences of bistatic distances in another set of measurements on an element-by-element basis, which will lead to a heavy burden of calculation when there are many targets. There is also a discussion about how to eliminate false targets.

0013The present method according to the invention is based on using a rapid method where only bistatic measurements are utilised. Furthermore the method manages a certain dropout of sensors in a better way than the method that has been discussed in <patcit id="pcit0005" dnum="WO0209319A"><text>WO 02/09319</text></patcit>. The method solves the current problem of association by being designed in the manner as is evident from the independent claim. Advantageous embodiments of the invention are defined in the remaining claims.

0014Before a more detailed description of the invention, first a multistatic network of ground radar stations will be contemplated, in which each radar station transmits radar pulses that are scattered towards flying targets and are then received by the surrounding stations. There will then be a situation involving a large number of bistatic measurements (i.e. the transmitting and the receiving station are located in different positions) and also monostatic measurements which, however, are not used in the invention. The bistatic measurements contain information about the total distance transmitter-target-receiver and corresponding Doppler information. A coherent air situation image is then to be created from all these measurements. This problem, the problem of association, is non-trivial if there are a large number of targets.

0015For intuitive understanding of the invention, a simple case is taken into consideration, involving only one target, m₁, and four stations S₁, S₂, S₃, S₄. Now assume that the measurements d₁₂. d₃₄, d₁₃, d₂₄ are performed, where d_ij means the total distance s_i - m₁ - s_j. It will be appreciated that d₁₂ + d₃₄= d₁₃ + d₂₄ must be the case since both expressions mean the total distance from the target to the four stations.

0016If there are now N targets instead, the above observation can be used to correctly associate data in the following manner. All conceivable combinations of data of the type d₁₂ and d₃₄ are formed; they will be N² in number. In the same way, N² combinations of data of the type d₁₃ and d₂₄ are formed. These combinations are sorted and compared, and only sums from the two amounts that are equal (within a given tolerance) can correspond to real targets. The same discussion can be used about the Doppler velocities which thus give a further screening. In this way, quick and easy association of measurement data can be effected.

0017In general, the transmitters and receivers must be positioned and the range of the transmitters must be chosen so that a target at an arbitrary point within the position space can be measured via scattering in the target of at least four cooperating bistatic pairs of transmitters and receivers. The number of transmitters and receivers can be large. At least four such cooperating pairs are selected among these bistatic pairs to perform the association and the determination of the distance.

0018Below follows a more systematic presentation of the calculations. In order to obtain a simple description, the following (non-critical) assumptions are made. Assume that there are four stations and N targets, which all are seen by all sensors (monostatic as well as bistatic).

0019Input data is thus (monostatic measurements) <maths id="math0005"><math display="block"><msub><mi mathvariant="normal">d</mi><mi mathvariant="normal">j</mi></msub><mfenced><mi mathvariant="normal">k</mi></mfenced><mo mathvariant="normal">=</mo><mfenced open="|" close="|"><msub><mi mathvariant="bold">r</mi><mi mathvariant="normal">k</mi></msub><mo mathvariant="normal">-</mo><msub><mi mathvariant="bold">ρ</mi><mi mathvariant="normal">j</mi></msub></mfenced><mo mathvariant="normal">,</mo><mspace width="2em" /><mi mathvariant="normal">k</mi><mo mathvariant="normal">=</mo><mn mathvariant="normal">1</mn><mo mathvariant="normal">,</mo><mn mathvariant="normal">2</mn><mo mathvariant="normal">,</mo><mo mathvariant="normal">…</mo><mspace width="1em" /><mi mathvariant="normal">N</mi><mo mathvariant="normal">,</mo><mspace width="1em" /><mi mathvariant="normal">j</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mn>4</mn></math><img file="EP1514134B1_D0005.tif" /></maths><maths id="math0006"><math display="block"><msub><mi mathvariant="normal">v</mi><mi mathvariant="normal">j</mi></msub><mfenced><mi mathvariant="normal">k</mi></mfenced><mo mathvariant="normal">=</mo><mfenced><mi mathvariant="normal">d</mi><mo mathvariant="normal">/</mo><mi>dt</mi></mfenced><mo>⁢</mo><mfenced open="|" close="|"><msub><mi mathvariant="bold">r</mi><mi mathvariant="normal">k</mi></msub><mo mathvariant="normal">-</mo><msub><mi mathvariant="bold">ρ</mi><mi mathvariant="normal">j</mi></msub></mfenced><mo mathvariant="normal">,</mo><mspace width="1em" /><mi mathvariant="normal">k</mi><mo mathvariant="normal">=</mo><mn mathvariant="normal">1</mn><mo mathvariant="normal">,</mo><mn mathvariant="normal">2</mn><mo mathvariant="normal">,</mo><mo mathvariant="normal">…</mo><mspace width="1em" /><mi mathvariant="normal">N</mi><mo mathvariant="normal">,</mo><mspace width="1em" /><mi mathvariant="normal">j</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mn>4</mn></math><img file="EP1514134B1_D0006.tif" /></maths> and (bistatic measurements) <maths id="math0007"><math display="block"><msub><mi mathvariant="normal">d</mi><mi>ij</mi></msub><mfenced><mi mathvariant="normal">k</mi></mfenced><mo mathvariant="normal">=</mo><mfenced open="|" close="|"><msub><mi mathvariant="bold">r</mi><mi mathvariant="normal">k</mi></msub><mo mathvariant="normal">-</mo><msub><mi mathvariant="bold">ρ</mi><mi mathvariant="normal">j</mi></msub></mfenced><mo>+</mo><mo mathvariant="normal">|</mo><msub><mi mathvariant="bold">r</mi><mi mathvariant="normal">k</mi></msub><mo mathvariant="normal">-</mo><msub><mi mathvariant="bold">ρ</mi><mi mathvariant="normal">j</mi></msub><mo mathvariant="normal">|</mo><mo mathvariant="normal">=</mo><msub><mi mathvariant="normal">d</mi><mi mathvariant="normal">i</mi></msub><mfenced><mi mathvariant="normal">k</mi></mfenced><mo mathvariant="normal">+</mo><msub><mi mathvariant="normal">d</mi><mi mathvariant="normal">j</mi></msub><mfenced><mi mathvariant="normal">k</mi></mfenced><mo mathvariant="normal">,</mo><mspace width="2em" /><mi mathvariant="normal">k</mi><mo mathvariant="normal">=</mo><mn mathvariant="normal">1</mn><mo mathvariant="normal">,</mo><mn mathvariant="normal">2</mn><mo mathvariant="normal">,</mo><mo mathvariant="normal">…</mo><mspace width="2em" /><mi mathvariant="normal">N</mi><mo>,</mo><mspace width="1em" /><mi mathvariant="normal">j</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mn>4</mn></math><img file="EP1514134B1_D0007.tif" /></maths><maths id="math0008"><math display="block"><msub><mi mathvariant="normal">v</mi><mi>ij</mi></msub><mfenced><mi mathvariant="normal">k</mi></mfenced><mo mathvariant="normal">=</mo><mfenced><mi mathvariant="normal">d</mi><mo>/</mo><mi>dt</mi></mfenced><mo>⁢</mo><mfenced open="|" close="|"><msub><mi mathvariant="bold">r</mi><mi mathvariant="normal">k</mi></msub><mo mathvariant="normal">-</mo><msub><mi mathvariant="bold">ρ</mi><mi mathvariant="normal">i</mi></msub></mfenced><mo>+</mo><mfenced><mi mathvariant="normal">d</mi><mo>/</mo><mi>dt</mi></mfenced><mo mathvariant="normal">|</mo><msub><mi mathvariant="bold">r</mi><mi mathvariant="normal">k</mi></msub><mo mathvariant="normal">-</mo><msub><mi mathvariant="bold">ρ</mi><mi mathvariant="normal">j</mi></msub><mo mathvariant="normal">|</mo><mo mathvariant="normal">=</mo><msub><mi mathvariant="normal">v</mi><mi mathvariant="normal">i</mi></msub><mfenced><mi mathvariant="normal">k</mi></mfenced><mo mathvariant="normal">+</mo><msub><mi mathvariant="normal">v</mi><mi mathvariant="normal">j</mi></msub><mfenced><mi mathvariant="normal">k</mi></mfenced><mo mathvariant="normal">,</mo><mspace width="1em" /><mi mathvariant="normal">k</mi><mo mathvariant="normal">=</mo><mn mathvariant="normal">1</mn><mo mathvariant="normal">,</mo><mn mathvariant="normal">2</mn><mo mathvariant="normal">,</mo><mo mathvariant="normal">…</mo><mspace width="2em" /><mi mathvariant="normal">N</mi><mo>,</mo><mspace width="1em" /><mi mathvariant="normal">j</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mn>4</mn></math><img file="EP1514134B1_D0008.tif" /></maths> where i = j in the bistatic case corresponds to monostatic measurements, i.e. i ≠ j can be assumed if desirable.

0020It is to be noted that, for instance, for station j, with the monostatic measurement d_j(k), k=1,2, ... N, it is not possible to know which measurement belongs to a certain target, i.e. the measurements are to be regarded as a set which is as a suggestion sorted according to distance. In this way, there is no connection between a certain index k which belongs to two different sensor registrations.

0021The method is now based on the following observation: For each registered target (not candidate, but real target) there must be a k, a k', an I and an I', all between 1 and N so that <maths id="math0009"><math display="block"><msub><mi mathvariant="normal">d</mi><mn mathvariant="normal">12</mn></msub><mfenced><mi mathvariant="normal">k</mi></mfenced><mo mathvariant="normal">+</mo><msub><mi mathvariant="normal">d</mi><mn mathvariant="normal">34</mn></msub><mfenced><mi mathvariant="normal">l</mi></mfenced><mo mathvariant="normal">=</mo><msub><mi mathvariant="normal">d</mi><mn mathvariant="normal">13</mn></msub><mfenced><mi mathvariant="normal">kʹ</mi></mfenced><mo mathvariant="normal">+</mo><msub><mi mathvariant="normal">d</mi><mn mathvariant="normal">24</mn></msub><mfenced><mi mathvariant="normal">lʹ</mi></mfenced></math><img file="EP1514134B1_D0009.tif" /></maths>

0022For the same k, k', I, I', the following is also applicable <maths id="math0010"><math display="block"><msub><mi mathvariant="normal">v</mi><mn mathvariant="normal">12</mn></msub><mfenced><mi mathvariant="normal">k</mi></mfenced><mo mathvariant="normal">+</mo><msub><mi mathvariant="normal">v</mi><mn mathvariant="normal">34</mn></msub><mfenced><mi mathvariant="normal">l</mi></mfenced><mo mathvariant="normal">=</mo><msub><mi mathvariant="normal">v</mi><mn mathvariant="normal">13</mn></msub><mfenced><mi mathvariant="normal">kʹ</mi></mfenced><mo mathvariant="normal">+</mo><msub><mi mathvariant="normal">v</mi><mn mathvariant="normal">24</mn></msub><mfenced><mi mathvariant="normal">lʹ</mi></mfenced></math><img file="EP1514134B1_D0010.tif" /></maths>

0023The reason is that if the target has the space vector r_t, it is applicable for the target that <maths id="math0011"><math display="block"><msub><mi mathvariant="normal">d</mi><mn mathvariant="normal">12</mn></msub><mfenced><mi mathvariant="normal">k</mi></mfenced><mo mathvariant="normal">+</mo><msub><mi mathvariant="normal">d</mi><mn mathvariant="normal">34</mn></msub><mfenced><mi mathvariant="normal">l</mi></mfenced><mo mathvariant="normal">=</mo><mfenced open="|" close="|"><msub><mi mathvariant="bold">r</mi><mi mathvariant="normal">t</mi></msub><mo mathvariant="normal">-</mo><msub><mi mathvariant="bold">ρ</mi><mn mathvariant="normal">1</mn></msub></mfenced><mo mathvariant="normal">+</mo><mfenced open="|" close="|"><msub><mi mathvariant="bold">r</mi><mi mathvariant="normal">t</mi></msub><mo mathvariant="normal">-</mo><msub><mi mathvariant="bold">ρ</mi><mn mathvariant="normal">2</mn></msub></mfenced><mo mathvariant="normal">+</mo><mfenced open="|" close="|"><msub><mi mathvariant="bold">r</mi><mi mathvariant="normal">t</mi></msub><mo mathvariant="normal">-</mo><msub><mi mathvariant="bold">ρ</mi><mn mathvariant="normal">3</mn></msub></mfenced><mo mathvariant="normal">+</mo><mfenced open="|" close="|"><msub><mi mathvariant="bold">r</mi><mi mathvariant="normal">t</mi></msub><mo mathvariant="normal">-</mo><msub><mi mathvariant="bold">ρ</mi><mn mathvariant="normal">4</mn></msub></mfenced></math><img file="EP1514134B1_D0011.tif" /></maths> and in the same way that <maths id="math0012"><math display="block"><msub><mi mathvariant="normal">d</mi><mn mathvariant="normal">13</mn></msub><mfenced><mi mathvariant="normal">k</mi></mfenced><mo mathvariant="normal">+</mo><msub><mi mathvariant="normal">d</mi><mn mathvariant="normal">24</mn></msub><mfenced><mi mathvariant="normal">l</mi></mfenced><mo mathvariant="normal">=</mo><mfenced open="|" close="|"><msub><mi mathvariant="bold">r</mi><mi mathvariant="normal">t</mi></msub><mo mathvariant="normal">-</mo><msub><mi mathvariant="bold">ρ</mi><mn mathvariant="normal">1</mn></msub></mfenced><mo mathvariant="normal">+</mo><mfenced open="|" close="|"><msub><mi mathvariant="bold">r</mi><mi mathvariant="normal">t</mi></msub><mo mathvariant="normal">-</mo><msub><mi mathvariant="bold">ρ</mi><mn>3</mn></msub></mfenced><mo mathvariant="normal">+</mo><mfenced open="|" close="|"><msub><mi mathvariant="bold">r</mi><mi mathvariant="normal">t</mi></msub><mo mathvariant="normal">-</mo><msub><mi mathvariant="bold">ρ</mi><mn>2</mn></msub></mfenced><mo mathvariant="normal">+</mo><mfenced open="|" close="|"><msub><mi mathvariant="bold">r</mi><mi mathvariant="normal">t</mi></msub><mo mathvariant="normal">-</mo><msub><mi mathvariant="bold">ρ</mi><mn mathvariant="normal">4</mn></msub></mfenced></math><img file="EP1514134B1_D0012.tif" /></maths> so that they are equal. The argument for the velocities is identical. The suggested method now is as follows.

0024<b>Step 1</b>. Form the N² sums <maths id="math0013"><math display="block"><msub><mi mathvariant="normal">d</mi><mn mathvariant="normal">12</mn></msub><mfenced><mi mathvariant="normal">k</mi></mfenced><mo mathvariant="normal">+</mo><msub><mi mathvariant="normal">d</mi><mn mathvariant="normal">34</mn></msub><mfenced><mi mathvariant="normal">l</mi></mfenced><mo mathvariant="normal">,</mo><mspace width="2em" /><mn mathvariant="normal">1</mn><mo mathvariant="normal">≤</mo><mi mathvariant="normal">l</mi><mo mathvariant="normal">,</mo><mspace width="1em" /><mi mathvariant="normal">k</mi><mo mathvariant="normal">≤</mo><mi mathvariant="normal">N</mi></math><img file="EP1514134B1_D0013.tif" /></maths>

0025Sort them according to the total distance and designate them <maths id="math0014"><math display="block"><msub><mi mathvariant="normal">d</mi><mrow><mn mathvariant="normal">12</mn><mo mathvariant="normal">+</mo><mn mathvariant="normal">34</mn></mrow></msub><mfenced><mi mathvariant="normal">m</mi></mfenced><mo mathvariant="normal">,</mo><mspace width="2em" /><mn mathvariant="normal">1</mn><mo mathvariant="normal">≤</mo><mi mathvariant="normal">m</mi><mo mathvariant="normal">≤</mo><msup><mi mathvariant="normal">N</mi><mn mathvariant="normal">2</mn></msup></math><img file="EP1514134B1_D0014.tif" /></maths>

0026<b>Step 2</b>. Proceed in the same way with <maths id="math0015"><math display="block"><msub><mi mathvariant="normal">d</mi><mn mathvariant="normal">13</mn></msub><mfenced><mi mathvariant="normal">kʹ</mi></mfenced><mo mathvariant="normal">+</mo><msub><mi mathvariant="normal">d</mi><mn mathvariant="normal">24</mn></msub><mfenced><mi mathvariant="normal">lʹ</mi></mfenced><mo mathvariant="normal">,</mo><mspace width="2em" /><mn mathvariant="normal">1</mn><mo mathvariant="normal">≤</mo><mi mathvariant="normal">lʹ</mi><mo mathvariant="normal">,</mo><mspace width="1em" /><mi mathvariant="normal">kʹ</mi><mo mathvariant="normal">≤</mo><mi mathvariant="normal">N</mi></math><img file="EP1514134B1_D0015.tif" /></maths> so that the following will also be obtained (sorted) <maths id="math0016"><math display="block"><msub><mi mathvariant="normal">d</mi><mrow><mn mathvariant="normal">13</mn><mo mathvariant="normal">+</mo><mn mathvariant="normal">24</mn></mrow></msub><mfenced><mi mathvariant="normal">mʹ</mi></mfenced><mo mathvariant="normal">,</mo><mspace width="2em" /><mn mathvariant="normal">1</mn><mo mathvariant="normal">≤</mo><mi mathvariant="normal">mʹ</mi><mo mathvariant="normal">≤</mo><msup><mi mathvariant="normal">N</mi><mn mathvariant="normal">2</mn></msup></math><img file="EP1514134B1_D0016.tif" /></maths>

0027<b>Step 3</b>. Associate targets from {d₁₂₊₃₄(m)}m=_1,2...N² with targets from {d₁₃₊₂₄(m')}_m'1,2... N² if <maths id="math0017"><math display="block"><mfenced open="|" close="|"><msub><mi mathvariant="normal">d</mi><mrow><mn mathvariant="normal">12</mn><mo mathvariant="normal">+</mo><mn mathvariant="normal">34</mn></mrow></msub><mfenced><mi mathvariant="normal">m</mi></mfenced><mo mathvariant="normal">-</mo><msub><mi mathvariant="normal">d</mi><mrow><mn mathvariant="normal">13</mn><mo mathvariant="normal">+</mo><mn mathvariant="normal">24</mn></mrow></msub><mfenced><mi mathvariant="normal">mʹ</mi></mfenced></mfenced><mo mathvariant="normal">&lt;</mo><mi>suitable tolerance</mi></math><img file="EP1514134B1_D0017.tif" /></maths>

0028<b>Step 4</b>. Investigate, and keep associated targets if they also satisfy <maths id="math0018"><math display="block"><mfenced open="|" close="|"><msub><mi mathvariant="normal">v</mi><mrow><mn mathvariant="normal">12</mn><mo mathvariant="normal">+</mo><mn mathvariant="normal">34</mn></mrow></msub><mfenced><mi mathvariant="normal">m</mi></mfenced><mo mathvariant="normal">-</mo><msub><mi mathvariant="normal">v</mi><mrow><mn mathvariant="normal">13</mn><mo mathvariant="normal">+</mo><mn mathvariant="normal">24</mn></mrow></msub><mfenced><mi mathvariant="normal">mʹ</mi></mfenced></mfenced><mo mathvariant="normal">&lt;</mo><mi>suitable tolerance</mi></math><img file="EP1514134B1_D0018.tif" /></maths>

0029"Suitable tolerance" in Step 3 is determined, inter alia, by the transmitted signal bandwidth, the purpose of the processing and hypotheses about size and number of the targets. Usually it is from about one meter to some twenty or thirty meters. Correspondingly, "suitable tolerance" in Step 4 is usually a few meters/second.

0030To establish that this really results in a rapid method, the following rough estimate may be used. Assume that there are many targets so that they are of the same magnitude as the number of distance bins and the number of Doppler bins. This common number is again designated N. It may then be estimated that, since the total number of cells (= number of distance bins by the number of Doppler bins) is the same as the number of candidates in for instance {d₁₂₊₃₄(m)}_m=1,2...N², each such candidate will be paired with typically a false candidate from {d₁₃₊₂₄(m')}_m'=_1,2... N². The number of candidates according to the above procedure thus is ~ N² (fewer with fewer targets), which is a great reduction compared with N³. Further processing can then take place by comparing with the remaining bistatic geometry {d₁₄₊₂₃(m")}_m"=_1,2...N², the mono static measurements or measurements involving other stations.

0031It is also to be noted that it is possible to involve {d,4+23(m")}=_1,2...N² from the beginning. This gives a possibility of having a redundancy, i.e. a possibility of managing a certain dropout in registrations, in the following way. The condition that | d₁₂₊₃₄(m)-d₁₃₊₂₄(m')| &lt; "suitable tolerance" can be seen as if both d₁₂₊₃₄(m) and d₁₃₊₂₄(m') are to be close to a certain given value. By requiring instead that two of d₁₂₊₃₄(m), d₁₃₊₂₄(m') and d₁₄₊₂₃(m") should be close to the indicated value (for some values of m, m' and m") there will still be a discrimination between false candidates (ghosts) and targets. However, it may be accepted that one of the measurements drops out.

0032The calculations in their entirety require O(N² log N) operations, and there are simple methods of really obtaining position and velocity from the candidates, i.e. after processing four bistatic distances are known for a certain candidate as follows: <maths id="math0019"><math display="block"><mfenced open="|" close="|"><mi mathvariant="bold">r</mi><mo mathvariant="normal">-</mo><msub><mi mathvariant="bold">ρ</mi><mn mathvariant="normal">1</mn></msub></mfenced><mo mathvariant="normal">+</mo><mfenced open="|" close="|"><mi mathvariant="bold">r</mi><mo mathvariant="normal">-</mo><msub><mi mathvariant="bold">ρ</mi><mn mathvariant="normal">2</mn></msub></mfenced><mo>=</mo><msub><mi mathvariant="normal">d</mi><mn>12</mn></msub></math><img file="EP1514134B1_D0019.tif" /></maths><maths id="math0020"><math display="block"><mfenced open="|" close="|"><mi mathvariant="bold">r</mi><mo mathvariant="normal">-</mo><msub><mi mathvariant="bold">ρ</mi><mn>3</mn></msub></mfenced><mo mathvariant="normal">+</mo><mfenced open="|" close="|"><mi mathvariant="bold">r</mi><mo mathvariant="normal">-</mo><msub><mi mathvariant="bold">ρ</mi><mn>4</mn></msub></mfenced><mo>=</mo><msub><mi mathvariant="normal">d</mi><mn>34</mn></msub></math><img file="EP1514134B1_D0020.tif" /></maths><maths id="math0021"><math display="block"><mfenced open="|" close="|"><mi mathvariant="bold">r</mi><mo mathvariant="normal">-</mo><msub><mi mathvariant="bold">ρ</mi><mn>1</mn></msub></mfenced><mo mathvariant="normal">+</mo><mfenced open="|" close="|"><mi mathvariant="bold">r</mi><mo mathvariant="normal">-</mo><msub><mi mathvariant="bold">ρ</mi><mn>3</mn></msub></mfenced><mo>=</mo><msub><mi mathvariant="normal">d</mi><mn>13</mn></msub></math><img file="EP1514134B1_D0021.tif" /></maths><maths id="math0022"><math display="block"><mfenced open="|" close="|"><mi mathvariant="bold">r</mi><mo mathvariant="normal">-</mo><msub><mi mathvariant="bold">ρ</mi><mn>2</mn></msub></mfenced><mo mathvariant="normal">+</mo><mfenced open="|" close="|"><mi mathvariant="bold">r</mi><mo mathvariant="normal">-</mo><msub><mi mathvariant="bold">ρ</mi><mn>4</mn></msub></mfenced><mo>=</mo><msub><mi mathvariant="normal">d</mi><mn>24</mn></msub></math><img file="EP1514134B1_D0022.tif" /></maths>

0033It is, of course, interesting to know the value of <b>r</b> (position of the target), i.e. a method of solving the above system of equations. (<b>ρ</b>_i, i=1,2,3,4, are the known positions/ position vectors of the stations and d₁₂, d₃₄, d₁₃, d₂₄ are the measured bistatic distances.) Generally seen, intersections of ellipsoids cause relatively complicated algebraic systems of equations, but in this case the system of equations can be solved by simpler methods.

0034If the system of equations is regarded as a 4x4 system, it is obvious that it is degenerated. At the same time the condition d₁₂ + d₃₄ = d₁₃ + d₂₄ guarantees that there is a parameter solution. By selecting the origin of coordinates in <b>ρ</b>₄ so that |<b>r</b> - <b>ρ</b>₄| = |<b>r</b>| = r and introducing r as a parameter, the following equations are obtained <maths id="math0023"><math display="block"><mfenced open="|" close="|"><mi mathvariant="bold">r</mi><mo mathvariant="normal">-</mo><msub><mi mathvariant="bold">ρ</mi><mn>1</mn></msub></mfenced><mo>=</mo><msub><mi mathvariant="normal">d</mi><mn>12</mn></msub><mo>-</mo><msub><mi mathvariant="normal">d</mi><mn mathvariant="normal">24</mn></msub><mo mathvariant="normal">+</mo><mi mathvariant="normal">r</mi></math><img file="EP1514134B1_D0023.tif" /></maths><maths id="math0024"><math display="block"><mfenced open="|" close="|"><mi mathvariant="bold">r</mi><mo mathvariant="normal">-</mo><msub><mi mathvariant="bold">ρ</mi><mn>2</mn></msub></mfenced><mo>=</mo><msub><mi mathvariant="normal">d</mi><mn mathvariant="normal">24</mn></msub><mo>-</mo><mi mathvariant="normal">r</mi></math><img file="EP1514134B1_D0024.tif" /></maths><maths id="math0025"><math display="block"><mfenced open="|" close="|"><mi mathvariant="bold">r</mi><mo mathvariant="normal">-</mo><msub><mi mathvariant="bold">ρ</mi><mn>3</mn></msub></mfenced><mo>=</mo><msub><mi mathvariant="normal">d</mi><mn>34</mn></msub><mo>-</mo><mi mathvariant="normal">r</mi></math><img file="EP1514134B1_D0025.tif" /></maths>

0035It is here possible to square the three equations, in which case r² can be deleted, and obtain the following (for some a,b,c,α,β,γ) <maths id="math0026"><math display="block"><mi mathvariant="bold">r</mi><mo>⋅</mo><msub><mi mathvariant="bold">ρ</mi><mn>1</mn></msub><mo>=</mo><mi>ar</mi><mo>+</mo><mi mathvariant="normal">α</mi></math><img file="EP1514134B1_D0026.tif" /></maths><maths id="math0027"><math display="block"><mi mathvariant="bold">r</mi><mo>⋅</mo><msub><mi mathvariant="bold">ρ</mi><mn>2</mn></msub><mo>=</mo><mi>br</mi><mo>+</mo><mi mathvariant="normal">β</mi></math><img file="EP1514134B1_D0027.tif" /></maths><maths id="math0028"><math display="block"><mi mathvariant="bold">r</mi><mo>⋅</mo><msub><mi mathvariant="bold">ρ</mi><mn>1</mn></msub><mo>=</mo><mi>cr</mi><mo>+</mo><mi mathvariant="normal">γ</mi></math><img file="EP1514134B1_D0028.tif" /></maths>

0036The latter system of equations can then be solved in a fairly straight-forward way. However, there will be two different cases in dependence on whether {<b>ρ</b>_i}_i=1,₂,₃ is linearly dependent or not.

0037The case of the velocities is similar, the following system of equations will be obtained <maths id="math0029"><math display="block"><msub><mover><mi mathvariant="italic">u</mi><mo>^</mo></mover><mn>1</mn></msub><mo mathvariant="italic">⋅</mo><mover><mi mathvariant="italic">v</mi><mo>‾</mo></mover><mo mathvariant="italic">+</mo><msub><mover><mi mathvariant="italic">u</mi><mo>^</mo></mover><mn>2</mn></msub><mo mathvariant="italic">⋅</mo><mover><mi mathvariant="italic">v</mi><mo>‾</mo></mover><mo>=</mo><msub><mi>v</mi><mn>12</mn></msub></math><img file="EP1514134B1_D0029.tif" /></maths><maths id="math0030"><math display="block"><msub><mover><mi mathvariant="italic">u</mi><mo>^</mo></mover><mn>3</mn></msub><mo mathvariant="italic">⋅</mo><mover><mi mathvariant="italic">v</mi><mo>‾</mo></mover><mo mathvariant="italic">+</mo><msub><mover><mi mathvariant="italic">u</mi><mo>^</mo></mover><mn>4</mn></msub><mo mathvariant="italic">⋅</mo><mover><mi mathvariant="italic">v</mi><mo>‾</mo></mover><mo>=</mo><msub><mi>v</mi><mn>34</mn></msub></math><img file="EP1514134B1_D0030.tif" /></maths><maths id="math0031"><math display="block"><msub><mover><mi mathvariant="italic">u</mi><mo>^</mo></mover><mn>1</mn></msub><mo mathvariant="italic">⋅</mo><mover><mi mathvariant="italic">v</mi><mo>‾</mo></mover><mo mathvariant="italic">+</mo><msub><mover><mi mathvariant="italic">u</mi><mo>^</mo></mover><mn>3</mn></msub><mo mathvariant="italic">⋅</mo><mover><mi mathvariant="italic">v</mi><mo>‾</mo></mover><mo>=</mo><msub><mi>v</mi><mn>13</mn></msub></math><img file="EP1514134B1_D0031.tif" /></maths><maths id="math0032"><math display="block"><msub><mover><mi mathvariant="italic">u</mi><mo>^</mo></mover><mn>2</mn></msub><mo mathvariant="italic">⋅</mo><mover><mi mathvariant="italic">v</mi><mo>‾</mo></mover><mo mathvariant="italic">+</mo><msub><mover><mi mathvariant="italic">u</mi><mo>^</mo></mover><mn>4</mn></msub><mo mathvariant="italic">⋅</mo><mover><mi mathvariant="italic">v</mi><mo>‾</mo></mover><mo>=</mo><msub><mi>v</mi><mn>24</mn></msub></math><img file="EP1514134B1_D0032.tif" /></maths> where <maths id="math0033"><math display="inline"><msub><mover><mi>u</mi><mo>^</mo></mover><mi>i</mi></msub><mo>=</mo><mfrac><mrow><mover><mi>r</mi><mo>‾</mo></mover><mo>-</mo><msub><mover><mi>r</mi><mo>‾</mo></mover><mi>i</mi></msub></mrow><mfenced open="|" close="|"><mover><mi>r</mi><mo>‾</mo></mover><mo>-</mo><msub><mover><mi>r</mi><mo>‾</mo></mover><mi>i</mi></msub></mfenced></mfrac><mo>,</mo></math><img file="EP1514134B1_D0033.tif" /></maths><maths id="math0034"><math display="inline"><mover><mi mathvariant="italic">V</mi><mo>‾</mo></mover><mo>=</mo><mover><mover><mi mathvariant="italic">r</mi><mo>‾</mo></mover><mo>˙</mo></mover><mo>,</mo></math><img file="EP1514134B1_D0034.tif" /></maths><i>i</i>=1,2,3,4. The system of equations can be processed in the same fundamental way as the previous system of equations.

0038The invention can be implemented in high-level languages which are suitable for calculations, such as MatLab, C, Pascal, Fortran etc.