On unique passive geolocation of multiple radio-frequency emitters

We derive necessary conditions for unique geolocation of multiple radio-frequency sources. These conditions iden- tify the maximum number of transmitters that can be lo- cated by multiple passive antenna arrays, often referred to as Resolution capacity (RC). Our derivations extend pre- viously published results in the field of Angle-Of-Arrival (AOA). We show that if each array consists of M elements and there are L arrays, the number of narrowband sources that can be located is (M i 1)L. This number is L times higher than the number of narrowband signals that can be resolved by each array if AOA is used. This observation leads to the conclusion that gelocation by AOA is subop- timal and other methods should be developed that can ex- ploit the information collected by all the arrays together. We also derive similar results for wideband signals. Nu- merical examples are used for demonstration of the math- ematical results. olocation is based on L distinct AOA measurements. The intermediate step of AOA estimation is the main cause of low resolution capacity as well as low accuracy under low signal to noise ratio conditions. According to the max- imum likelihood principle, geolocation should be based directly on the observed signals in all the available arrays. Assume that the emitter location is described by z coor- dinates (z can be 2 or 3). Then, if L > z the number of AOA estimates is larger than the number of emitter co- ordinates. But estimating more parameters than the min- imum needed is associated with accuracy loss. In other words, the AOA approach does not exploit the constraint that all AOA estimates are associated with a single source and thus all line of bearings must intersect. In this work we discuss the observed signals in all the spatially separated arrays together instead of discussing the signals at each array separately. Our observation mod- els agree with most of the models used for narrowband AOA and for wideband AOA. Obviously, one can apply any geolocation technique to the models and this include AOA, TDOA (Time Difference of Arrival) and DPD (Di- rect Position Determination, (6)) or any combination. The observation model lends itself to different assumptions re- garding the propagation channel (known/unknown atten- uation, time dependency, etc.,) the signal structure (Gaus- sian, constant modulus, etc.,) the array configuration (uni- form linear, circular, etc.) and the noise statistics. We derive necessary conditions for model identifiabil- ity based on the requirement that the number of equations should be equal or larger than the number of unknown pa- rameters. We discuss unknown channels, as is common in most geolocation systems and we also discuss known channels for pure academic interest since currently there are no reliable means to measure relative gain and phase between remote sites. As expected, the RC in the case of unknown channel is lower than the RC in the case of known channel. Notation: We use uppercase bold fonts to denote ma- trices and lowercase bold fonts for vectors. The super- scripts X H ,X T , stand for conjugate transpose, and trans- pose, respectively. We use IJ for the J ◊ J identity ma- trix . The symbol › stand for the Kronecker product, and diag(z) is a diagonal matrix with the vector z on its main diagonal. Finally, Diag(Z1,··· ,ZN) is a block diagonal matrix with the matrices Z1,··· ,ZN on its main diago- nal.