The purpose of this page is to describe a substantial advance in theory and methodology for high-performance location of RF emitters using multiple moving sensors based on statistically optimum aperture synthesis. This advance was developed at SSPI during the several years preceding 2010 as part of the work outlined on Page 12.1. I developed the Basel concept and mathematical formulation described below without SSPI support because of the lack of indirect cost budget for this work. The software implementation and testing with real data was carried out with full support by SSPI originating with government funded contracts.
What’s New About Bayesian Emitter Location Technology?
In this introductory discussion, key differences between the following alternative technologies are exposed: (1) classical TDOA/FDOA-based and AOA-based estimation of unknown (non-random) emitter location using multiple sensors and calculation of corresponding confidence regions using stochastic Stochastic | adjective Involving random variables, as in stochastic process which is a time-indexed sequence of random variables. process models of the received data at the collection system’s sensors and averaging over the sample space of all possible received data; (2) the relatively novel Bayesian formulation of estimation of random Random | adjectiveUnpredictable, but not necessarily modeled in terms of probability and not necessarily stochastic. emitter location coordinates using multiple sensors and calculation of Bayesian confidence regions conditioned on observation of the actual received data; and (3) ad hoc methods for RF Imaging of spatially discrete emitters, using multiple sensors, derived from VLBI (Very Long Baseline Interferometry) concepts and methods developed for Radio Astronomy.
A key concept in (2) and (3) is aperture synthesis and the production of RF images: the output image produced—in which bright spots or corresponding peaks in amplitudes are sought as the probable locations of sources—can be thought of as being somewhat analogous to an antenna pattern. The image is designed to exhibit peaks above the surface upon which the emitters reside (e.g., Earth) at whatever coordinates RF emitters are located, and the height of such peaks typically increase as the average transmitted power of the emitters increases, relative to background radiation and/or antenna and receiver noise.
Before proceeding, the topic to be addressed is put in perspective relative to more well-known methods of aperture synthesis.
Synthetic Aperture Photography (SAP) is analogous to Synthetic Aperture RF Imaging, (SARFI) neither of which is as strongly analogous to Synthetic Aperture Radar (SAR). The former two are passive and endeavor to image single or spatial arrangements of radiating energy sources whereas the latter is active and transmits energy at reflecting structures and synthesizes an image of the structure from the received reflections. To be more precise, SARFI normally images sources of RF energy, but can under certain circumstances be used to distinguish between sources and reflectors of sources and can thereby perform imaging without line of sight (given favorable characteristics of reflectors) and can mitigate multipath propagation degradation of images. In contrast, SAP typically images structures on the basis of the optical energy they reflect from the environment.
Consequently, these three types of aperture synthesis are quite distinct. In Medical Imaging, a CAT scan using computer aided tomography does perform a type of synthetic aperture imaging, but it is distinct from SAP, SARFI, and SAR. Somewhat more closely related is Radio Telescope Imaging (RTI), intended for imaging essentially contiguous spatially extended source of RF radiation, like stars. The assumption of a single RF point source to be located or a configuration of spatially discrete sources of RF energy to be jointly located justifies a unique mathematical model and statistical inference method.
I formulated such a model and developed a method of aperture synthesis that appears to have capability advantages beyond that of earlier RF imaging techniques for locating RF emitters on Earth using non-geostationary satellite receivers. I never received funding to compare performances of these two competing approaches, and I have no idea of what might have been done in this vein during the 13 years since I retired from that work. To my knowledge, any work in this area was likely classified by the US government on the basis of national defense. My internal (independent) R&D on this approach, performed at my company SSPI in the first decade of the 21st Century, was not classified and is summarized in what follows here.
In the Bayesian Aperture Synthesis for Emitter Location (BASEL)) system described herein, the Image is actually either the likelihood function or the posterior probability density function of emitter location coordinates, given the received data impinging on the array. This image can be displayed as either grey-level or multicolor intensity overlaid on a map of the geographical area seen by the sensors, or it can be displayed as the height of a surface above the relevant region of Earth’s surface.
Significant reductions in computational complexity can be realized by producing images that are only proportional to these statistical Statistical | adjective Of or having to do with Statistics, which are summary descriptions computed from finite sets of empirical data; not necessarily related to probability. functions or even only monotonic nonlinearly warped versions of these. Contours of constant elevation for such images produce probability containment regions, and can be computed exactly, unlike the classical location-error containment regions (typically ellipses or ellipsoids) used with TDOA/FDOA-based location systems which are only approximations and, in some cases, quite crude approximations.
In addition to the Image derived from the received data, explicit formulas for an ideal Image can be obtained by substituting a multiple-signal-plus-noise model for the received data in the image formula and then replacing the finite-time-average autocorrelation functions of the signals and the noise that appear in the formula with their idealized versions, which produces a sort of antenna pattern—but not in the usual sense. Rather, this produces an idealized (independent of specific received data) image for whatever multi-signal spatial scenario is modeled.
One of the keys to transitioning from TDOA/FDOA and AOA based RF-emitter location, to RF Imaging, is replacing the unknown parameters, TDOA, FDOA, and AOA, in data models with their functional dependence on unknown emitter location coordinate variables on Earth’s surface and known sensor locations overhead. This process is called geo-registration. This way, when sensors move along known trajectories during data collection and TDOA, FDOA, and AOA functions vary in a known manner, but the unknown emitter-location coordinates do not change as long as the emitter is not moving, long coherent integration over periods during which the sensor positions change substantially is enabled. This enables long integration over periods during which the sensor positions change substantially. These sensor trajectories increase the size of the synthesized aperture beyond that of the distance between fixed sensors.
In many applications, prior probability densities for emitter locations used in the Bayesian formulation of statistical inference are unknown, in which case they can be specified as uniform over the entire region of practically feasible locations of interest. This is not a weakness of this approach. It simply means that the location estimation optimization criterion is based on the likelihood function of candidate emitter coordinates over a specified region instead of the posterior probability density function of those coordinates. Moreover, when prior information is available, this Bayesian approach incorporates that information in a statistically optimum manner through the prior probability densities. Furthermore, these priors play a critical role in location updating through Bayesian Learning for which posteriors from one period of collection become priors for the next period of collection.
Also, starting with the posterior probability formulation leads to a natural choice of alternative to the classical (and often low-quality) approximate error containment regions for performance quantification, and this choice is the regions contained within contours of constant elevation of the emitter location posterior probability density function above the surface defined by the set of all candidate emitter coordinates.
This alternative metric for quantifying location accuracy has the important advantage of not averaging over all possible received data (the population) in a stochastic process model, but rather conditioning the probability of emitter location on the actual data received. It also has the advantage of not requiring the classical approximation for containment regions which is known to be inaccurate in important types of applications where attainable location accuracy is not high.
Besides this advantageous difference in location-accuracy quantification, the Bayesian approach produces optimal imaging algorithms that have aspects in common with now-classical Imaging systems for radio astronomy, referred to as VLBI, and also open the door to various methods for improving the capability of RF imaging for spatially discrete emitters. The improvements over ad hoc VLBI-like processing result from abandoning the interpretation in terms of imaging essentially continuous spatial distributions of radiation sources in favor of locating spatially discrete sources of radiation. In contrast to radio astronomy, where the sensors are on the surface of rotating Earth and the diffuse sources of radiating energy to be imaged are in outer space, in RF Imaging of manmade radio emitters, the sources are typically located on the surface of Earth and the sensors follow overhead trajectories.
In conclusion, BASEL technology improves significantly on both (1) traditional radar emitter location methods based on sets of TDOA and/or FDOA and/or AOA measurements applied to communications emitters and (2) more recent methods of RF Imaging for communications emitters derived from classical VLBI technology for radio astronomy. The improvement comes in the form of not only higher sensitivity and higher spatial resolution resulting from increases in coherent signal processing gain, but also novel types of capability that address several types of location systems impairments.
An especially productive change in the modeling of received data that is responsible for some of the unusual capability of BASEL technology, particularly coherent combining of statistics over multiple sensor pairs with unknown phase relationships, is the adoption of time-partitioned models for situations in which signals of interest are known in some subintervals of time and unknown in others. This occurs, for example, when receiver-training data is transmitted periodically, such as in the GSM cellular telephone standard. For such signals, the posterior PDF calculator takes on two distinct forms, one of which applies during known-signal time intervals and the other of which applies during unknown-signal time intervals. When the emitter location is not changing over a contiguous set of such intervals of time, these two distinct posterior PDFs (images) can be combined into a single image, which enjoys benefits accruing from both models. Thus, BASEL can perform two types of time partitioning when appropriate: (1) partitioning the data into known-signal time-intervals and unknown-signal time intervals, and (2) partitioning time into subintervals that are short enough to ensure the narrowband approximation to the Doppler effect due to sensor motion is accurate in each reduced subinterval. It doesn’t matter which partitioning results in shorter time intervals: The data from all intervals is optimally combined.
There is more than one way to beneficially combine RF images, depending on the benefits desired. For example, the presence of known portions of a signal in received data enables signal-selective geolocation. In addition, it also enables measurement of the phase offsets of the data from distinct sensor pairs. I have formulated a method for combining images that essentially equalizes these phases over multiple contiguous subintervals of time thereby enabling coherent combining of signal selective images over distinct pairs of moving sensors. This method combines CAFs (Cross Ambiguity Functions) from the data portions in which the signal is unknown with RCAFs (Radar CAFs) from the remaining data portions using a simple quadratic form. In this configuration, BASEL coherently combines images over both time and space, using narrowband RCAF and CAF measurements.
Another way the BASEL Processor provides innovative geolocation capability is through processing that mitigates multipath propagation and blocked line of sight. Highlights of this aspect of BASEL are listed below
SOME EXAMPLES OF BASEL CAPABILITY
The following material provides a few details about BASEL RF Imaging.
