Patent Publication Number: US-9885772-B1

Title: Geolocating wireless emitters

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     This application is a continuation of U.S. patent application Ser. No. 14/836,059 filed Aug. 26, 2015, entitled “Geolocating Wireless Emitters”, which claims the benefit of U.S. Provisional Patent Application Ser. No. 62/042,073 filed Aug. 26, 2014, entitled “Geolocating Wireless Emitters”. Each of the above applications, U.S. patent application Ser. No. 14/836,059 filed Aug. 26, 2015, entitled “Geolocating Wireless Emitters” and U.S. Provisional Patent Application Ser. No. 62/042,073 filed Aug. 26, 2014, entitled “Geolocating Wireless Emitters” is hereby incorporated by reference in its entirety. 
    
    
     STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT 
     The disclosed invention was made with government support under Contract No. HR011-09-C-0065 awarded by the Defense Advanced Research Projects Agency (DARPA). The government has certain rights in the invention. 
    
    
     FIELD OF THE INVENTION 
     The present invention relates to processes of geolocation in general, and specifically to geolocation processes that make use of reference signals and geolocation data to locate a target wireless emitter. 
     BACKGROUND 
     Determining the location or motion of a wireless emitter may be important in many applications; for example, military operations may need to determine the location of an enemy unit via tracking the location of a wireless emitter used by that unit. These wireless emitters may be referred to as “hostile” or “non-cooperative” emitters, as they do not intentionally broadcast geolocation information, and may be actively attempting to avoid detection. Non-cooperative emitters may also be narrow-band radio-frequency emitters, further complicating the problem of finding non-cooperative emitters. Methods and apparatuses that can locate such non-cooperative emitters may be significant in a wide range of applications. 
     BRIEF DESCRIPTION 
     The shortcomings of the prior art are overcome and additional advantages are provided through the provision, in one aspect, of a method including: obtaining at least one pair of reference channel values, a pair of target channel values, and a set of geolocation data; processing the at least one pair of reference channel values, the pair of target channel values, and the set of geolocation data to determine a phase-difference metric; and, processing the phase-difference metric to determine one or more sets of spatial location coordinates. 
     In another aspect, additional advantages may be provided through the provision of an apparatus including a signal processing circuit, the signal processing circuit being configured to perform a method, the method including: obtaining, from an input communications interface, at least one pair of reference channel values, a pair of target channel values, and a set of geolocation data; processing the at least one pair of reference channel values, the pair of target channel values, and the set of geolocation data to determine a phase-difference metric; and, processing the phase-difference metric to determine one or more sets of spatial location coordinates. 
     In another aspect, additional advantages may be provided through the provision of an apparatus including a signal processing circuit, the signal processing circuit being configured to: obtain, from a radio-frequency receiver, at least one reference signal and a target signal; process the target signal to determine a target channel value and the at least one reference signal to determine at least one reference channel value; and output, to a radio-frequency transmitter, the target channel value and the at least one reference channel value. 
     In another aspect, also provided is a computer program product, including one or more computer readable storage medium readable by one or more processors for execution by the one or more processors to perform a method, the method including: obtaining, from an input communications interface, at least one pair of reference channel values, a pair of target channel values, and a set of geolocation data; processing the at least one pair of reference channel values, the pair of target channel values, and the set of geolocation data to determine a phase-difference metric; and, processing the phase-difference metric to determine one or more sets of spatial location coordinates. 
     In yet another aspect, also provided is a system, the system including: at least one reference emitter capable of transmitting a reference signal, the at least one reference emitter having a known location; a target emitter capable of transmitting a target signal; a plurality of receiver nodes, the receiver nodes being operative to move, and the receiver nodes configured to receive the reference signal and the target signal, determine a pair of reference channel values and a pair of target channel values, and transmit the reference channel values and target channel values; and, a processing node, the processing node being configured to obtain the reference channel values and target channel values, and process the reference channel values and target channel values to determine a set of spatial location coordinates, wherein the spatial location coordinates correspond to a location of the target emitter. 
     Additional features and advantages are realized through the techniques disclosed herein. Other embodiments and aspects are described in detail herein and are considered a part of the claimed invention. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       One or more aspects are particularly pointed out and distinctly claimed as examples in the claims at the conclusion of the specification. The foregoing and other objects, features, and advantages are apparent from the following detailed description taken in conjunction with the accompanying drawings in which: 
         FIG. 1A  is a functional block diagram of an embodiment of a signal processing circuit of a node (apparatus) capable of receiving a plurality of signals and estimating complex channel values; 
         FIG. 1B  depicts the functional block diagram of  FIG. 1A , detailing further an exemplary embodiment of a signal processing circuit; 
         FIG. 1C  is a hardware schematic diagram of an exemplary embodiment of an apparatus including a signal processing circuit; 
         FIG. 1D  depicts examples of one or more components that may be included as part of a signal processing circuit as depicted by  FIG. 1C ; 
         FIG. 1E  is a functional block diagram of an embodiment of a processing node including a signal processing circuit capable of processing channel values to determine a set of spatial location coordinates; 
         FIGS. 2A-2B  depict an exemplary embodiment of a system including a plurality of receiver nodes that may be used to determine the location of a target emitter; 
         FIG. 3  is a block diagram depicting an exemplary embodiment of a method of determining a set of spatial location coordinates; and, 
         FIG. 4  depicts one embodiment of a computer program product, in accordance with one or more aspects of the invention. 
     
    
    
     DETAILED DESCRIPTION 
     Aspects of the present invention and certain features, advantages, and details thereof, are explained more fully below with reference to the non-limiting examples illustrated in the accompanying drawings. Descriptions of well-known materials, fabrication tools, processing techniques, etc., are omitted so as not to unnecessarily obscure the invention in detail. It should be understood, however, that the detailed description and the specific examples, while indicating aspects of the invention, are given by way of illustration only, and are not by way of limitation. Various substitutions, modifications, additions, and/or arrangements, within the spirit and/or scope of the underlying inventive concepts will be apparent to those skilled in the art from this disclosure. 
       FIG. 1A  is a functional block diagram of an embodiment of a node (apparatus)  10 ,  10 -N. Node  10 ,  10 -N, generally noted as node  10 , is represented specifically as nodes  10 - 1 ,  10 - 2 , etc. in  FIGS. 2A-2B . Node  10 ,  10 -N may generally be referred to as a “receiver node,” and may include a circuit, which may be signal processing circuit  100 , capable of obtaining a target signal and at least one reference signal, processing the target signal to determine a target channel value and the at least one reference signal to determine at least one reference channel value, and outputting the target channel value and at least one reference channel value, in accordance with one or more aspects of the present invention. The circuit may be configured to obtain multiple signals via a radio-frequency receiver  110  (receiver), such as a target signal put out by a target wireless emitter  30  (target emitter), and at least one reference signal output by at least one reference emitter  20 , as depicted in, for example,  FIGS. 2A-2B . The circuit may be further configured to output, via a radio-frequency transmitter  120  (transmitter), multiple channel values, such as a target channel value (corresponding to the target signal) and at least one reference channel value (corresponding to the reference signal). Channel values, including the target channel value and the at least one reference channel value, may be determined via a signal characterization circuit  130 , which may process the signals obtained from receiver  110  and output the channel values to transmitter  120 . Signal processing circuit  100  may also be configured to obtain a set of geolocation data, which may be geolocation data for the node  10 ,  10 -N, and output the set of geolocation data to transmitter  120 . Signal processing circuit  100  may thus also include a localization circuit  140  configured to obtain the set of geolocation data for the node. 
     In one example, receiver  110  may be an oscillator-based receiver, for example, a superheterodyne receiver. Receiver  110  may handle one or more signal processing functions typically associated with superheterodyne receivers, such as signal mixing, filtering, amplification, and de-modulation of signals, in receiving one or more signals and making those signals available to signal characterization circuit  130 . Similarly, in one example, transmitter  120  may be an oscillator-based transmitter, for example, a superheterodyne transmitter, and may handle one or more signal processing functions typically associated with superheterodyne transmitters, such as amplifying and filtering a signal, performing impedance matching, and modulating a carrier wave signal with a signal output by signal characterization circuit  130 . In another example, antenna  5  may pick up a signal and responsively output an electrical signal received by receiver  105 . Receiver  110  may be, in a simple form, a conductor coupling an antenna  5  to a signal characterization circuit  130 , so that receiver  110  may receive an electrical signal via antenna  5  and conduct the electrical signal to signal characterization circuit  120 . In one embodiment, receiver  110  may also amplify and filter a radio signal picked up by antenna  5 . Similarly, an antenna  5  coupled to transmitter  120  may emit a signal in response to an electrical signal output from transmitter  120 . Transmitter  120  may be, in a simple form, a conductor coupling cancellation signal processing circuit  120  to antenna  5 , so that transmitter  120  may receive an electrical signal from signal characterization circuit  130  and transmit the electrical signal to antenna  5 . In one embodiment, transmitter  120  may also amplify and/or filter a signal output by signal characterization circuit  130  prior to transmission of a signal to be emitted by antenna  5 . In one embodiment transmitter  120  may modulate a carrier wave signal with a signal output by signal characterization circuit  130 . It may be understood that receiver  110  and transmitter  120  need not be of similar types, and either receiver  110  or transmitter  120  may, in other embodiments, handle a portion of the functions described above, and may handle additional signal processing functions. 
     Signal characterization circuit  130  generally may be responsible for determining channel values for obtained signals. As described herein, channel values determined by signal characterization circuit  130  at node  10 ,  10 -N may be processed to determine a set of spatial location coordinates, which may correspond to a location of a target emitter emitting signals that are obtained at node  10 ,  10 -N. Signal characterization circuit  130  may obtain, from receiver  110 , a target signal and at least one reference signal. Obtaining either type of signal by signal characterization circuit  130  may include, for example, obtaining a baseband signal, such as an at least one baseband reference signal and a baseband target signal, or other signal having a reduced frequency (e.g., an intermediate frequency signal) as may be output by receiver  110  which can include a representation of the corresponding carrier wave signal. Signal characterization circuit  130  may process each signal to determine a channel value corresponding to the signal, and output a target channel value (corresponding to the target signal) and at least one reference channel value (corresponding to the at least one reference signal). Processing a signal may include, for example, estimating a complex channel value for a signal based on the signal&#39;s time of arrival at receiver node  10 ,  10 -N. In one embodiment, processing the target signal to obtain the target channel value may include extracting information about the at least one reference signal, and using the information in combination with the target signal to obtain the target channel value. Processing may also include filtering of multiple signals received by node  10 ,  10 -N to determine a “peak” target signal and a “peak” reference signal. In one embodiment, receiver node  10 ,  10 -N may only pick up one signal emitted by target emitter  30  or reference emitter  20 , as in  FIGS. 2A-2B , at any particular time. In another embodiment, such as in an environment that include numerous objects that may block or reflect signals, node  10 ,  10 -N may receive a signal not only along a direct-line path between the receiver node  10 ,  10 -N and the corresponding emitter, but may also receive the same signal at a different power reflected back by any number of such objects. The direct-line path may correspond to the one signal of the highest or “peak” power, out of the numerous similar signals received. Processing may also include extracting information about the at least one reference signal, and using the extracted information to adjusting a timing associated with the target signal, in order to more accurately estimate the target channel value. Signal characterization circuit  130  may also associate a time with the target channel value and at least one reference channel value. The time may correspond to the time at which the target signal and at least one reference signal arrive at node  10 ,  10 -N. Signal characterization circuit  130  may then output the target channel value and at least one reference channel value, to transmitter  120 . Signal characterization circuit  130  may output the at least one reference channel value and target channel value in the form of, for example, a signal that may be output to transmitter  120 . The output signal may, in one example, be a baseband signal or other signal having a reduced frequency, such as a baseband jamming signal or a baseband feedback signal, that transmitter  110  may use to modulate a carrier wave signal. 
     Signal processing circuit  100  may also include a localization circuit  140 . Localization circuit  140  may provide self-localization, geolocation data for node  10 ,  10 -N. Localization circuit  140  may include, for example, a global-positioning service (GPS) unit, an inertial navigational unit, or other hardware and/or software capable of determining geolocation data for node  10 ,  10 -N. Localization circuit  140  may be coupled with an antenna  5  to receive geolocation data. As described above, localization circuit  140  may be coupled with a shared antenna  5  that may be shared with other functional circuit blocks of signal processing circuit  100 , such as receiver  110  or transmitter  120 , or it may be a separate antenna  5  dedicated to localization circuit  140 . 
     Node  10 ,  10 -N may be configured to move over a time interval T 1  to T 2 , during which time node  10 ,  10 -N changes its location. Signal processing circuit  100  may obtain a first target signal and at least one first reference signal at time T 1 , and obtain geolocation data at time T 1 , and determine a first target channel value and at least one first reference channel value corresponding to time T 1 . Signal processing circuit  100  may then obtain a second target signal and at least one second reference signal at time T 2 , and obtain geolocation data at time T 2 , and determine a second target channel value and at least one second reference channel value corresponding to time T 1 . Signal processing circuit  100  may output the first target channel value and at least one first reference channel value, and subsequently output the second target channel value and at least one second reference channel value. The motion of receiver node  10 ,  10 -N over time may be significant to determining a location of a target emitter. By determining a first set of channel values and a second set of channel values, as well as any number of additional sets of channel values determined at different times and different locations of receiver node  10 ,  10 -N, it may be possible to accurately determine a set of spatial location coordinates, as described herein below, corresponding to a location of a target emitter. 
       FIG. 1B  depicts one example embodiment of a receiver node  10 ,  10 -N including signal processing circuit  100 , in which the functional block depicting receiver  110  is further detailed with additional components  250 ,  255 ,  260 , and  270  to describe additional functions that signal processing circuit  100  may perform. Components  250 ,  255 ,  260 , and  270  may, in one example, include hardware components corresponding to one or more parts of an analog signal processing circuit, as described further below in  FIG. 1C . In another example, one or more of components  250 ,  255 ,  260 , and  270  may include hardware or software components corresponding to one or more parts of a digital processing circuit, as described further below in  FIG. 1C . In the example illustrated, receiver  110  may include at least a first tuner (Tuner A)  250  and may also include a second tuner (Tuner B)  255 . Multiple, separate tuners may be advantageous where, for example, a target emitter and a reference emitter are transmitting signals in widely separated bandwidths, such as when the reference emitter is a wide-band signal transmitter and the target emitter is a narrow-band signal transmitter. In such cases, it may not be possible to use a single tuner  250  to receive and process both signals, and separate tuners may be used to pick up signals in each band. The separate tuners  250 ,  255  may be further coupled to separate antennae  5 . A receiver  110  including multiple tuners  250 ,  255  may also include a reference oscillator  260  coupled to each tuner. Reference oscillator  260  may, for example, enforce phase locking on signals picked up at tuners  250 ,  255 . Receiver  110  may also include a clock  270 . Clock  270  may provide time data to signal characterization circuit  130  for signals received through tuners  250 ,  255  to mark, for example, a time of arrival of each signal received at receiver node  10 ,  10 -N. Clock  270  may be coupled with reference oscillator  260 . Signal characterization circuit  130  may obtain the signals, including a target signal and at least one reference signal, from tuners  250 ,  255 , and may also obtain time data from clock  270 . 
     In the example described in  FIG. 1B  above, localization circuit  140  may also provide a pulse reference that may be used to correct for errors arising from either clock  270 , reference oscillator  260 , or both. Imprecise components, such as an analog reference oscillator  260  or a common digital clock  270 , may be used in a receiver  110 , such as a superheterodyne receiver, as such components may be relatively inexpensive to obtain and incorporate into a signal processing circuit  100 . However, such components may not be completely reliable in their output. A reference oscillator  260  set to provide a particular reference frequency may not always oscillate at the set frequency, but may vary above or below the set frequency as a result, for example, of variations in electric voltage supplied to reference oscillator  260 . Reference oscillator  260  may itself be a source of variation in a phase of the output reference frequency. Similarly, a clock  270  measuring time by analog means or digital means may not precisely measure the passage of time, and may measure time slightly faster or slower than the actual passage of time; clock  270  may thus provide imprecise measures of the times at which signals are received at node  10 ,  10 -N. The errors in a reference oscillator or clock may not be known for a particular receiver node  10 ,  10 -N, and the magnitude of errors may vary between any pair of receiver nodes  10 ,  10 -N. To account for and minimize the impact of errors caused by imprecise components, localization circuit  140  may provide a “pulse reference” which may be used to determine the average variation in either or both the reference oscillator  260  and clock  270 . A GPS unit, for example, may provide pulses at a rate of one pulse per second, which may correspond to a rate at which the device receives data from, for example, an external localization source such as a GPS satellite. Signal characterization circuit  130  may be configured to obtain pulse reference data from localization circuit  140  along with geolocation data obtained from localization circuit  140 , and signal characterization circuit may output pulse reference data to transmitter  120  in addition to other output, including, for example, the target channel value and at least one reference channel value. The pulse reference data may be correlated with a number of channel values obtained over a particular period of time in order to determine the average time error introduced by clock  270 . The average time error may be used to correct one or more time values associated with output channel values, such as a target channel value or a reference channel value. Similarly, it may be possible to correlate the pulse reference provided by localization circuit  140  with the frequency output of reference oscillator  260  to determine the average frequency error or deviation added to a signal, such as a target signal or a reference signal, by the reference oscillator. Accounting for such errors in the reference oscillator  260  and clock  270  at each receiver node  10 ,  10 -N may further increase the accuracy of phase-difference metrics that may be used to locate a target emitter  30 . Correlation of the number of channel values obtained with pulse reference data, as well as correlation of the frequency output of reference oscillator  260  with pulse reference data, may be performed, in one example, at a processing node  40 , as described herein below. 
       FIG. 1C  depicts a hardware schematic diagram of one embodiment of an apparatus  10 ,  10 -N,  40  including a signal processing circuit  100 , in which signal processing circuit  100  includes both a signal processing circuit in the form of an Analog Signal Processing Circuit  210  and a signal processing circuit in the form of a Digital Signal Processing Circuit  220  (DSPC). The apparatus as depicted in  FIG. 1C  may, in one embodiment, be a receiver node  10 ,  10 -N, as functionally described above in  FIG. 1A . The apparatus as depicted in  FIG. 1C  may, in another embodiment, be a node  40 , as depicted in  FIG. 1E . In the exemplary embodiment depicted, ASPC  210  may be configured to at least collect analog signals and make those signals available for processing, either within ASPC  210  or within DSPC  220 . ASPC may include hardware components configured to perform signal processing, for example a hardware oscillator or a hardware de-modulator, but at a minimum may be capable of receiving an analog signal and passing that signal to other signal processing circuit  100  components. Regardless of what signal processing ASPC  210  performs, its output is an analog signal that may then be passed to an analog-to-digital (A/D) converter  230 . The output of the A/D converter  230  may be one or more digital information signals, which may then be passed to DSPC  220 . In turn, DSPC  220  may be configured to process, via one or more digital hardware or software components, the digital information signals. DSPC  220  may be configured to perform digitally any signal processing functions, such as modulation, de-modulation, filtering, amplification, and so on. 
     Generally, either ASPC  210  or DSPC  220 , or a combination of both, may be configured to perform one or more signal characterizations. Signal characterization may include determination of channel values for obtained signals, including, for example, a target channel value for a target signal and at least one reference channel value for at least one reference signal, as described herein. Signal characterization may also include associating a time with the target channel value and the at least one reference channel value, where the time corresponds to a time at which the target signal and the at least one reference signal arrive at receiver node  10 ,  10 -N. In the embodiment of  FIG. 1C , the functions of function blocks  110 ,  120 ,  130 , and/or  140  as depicted in  FIG. 1A  and  FIG. 1B  may be performed by any designed division of labor scheme between ASPC  210  and DSPC  220 . Thus, in one example, DSPC  220  may be a circuit including a signal characterization circuit. In another example, DSPC  220  may be a circuit including a localization circuit. 
       FIG. 1D  illustrates exemplary embodiments of a digital processing circuit  220 , as may be implemented in one or more embodiments of apparatus  200  depicted in  FIG. 1C . Digital signal processing circuit  220  can include, in one embodiment, one or more of a field programmable gate array (FPGA)  221 , an application-specific integrated circuit (ASIC)  222 , or a processor system  223  comprising a processor provided by a central processing unit (CPU) in communication with a memory, each depicted in dashed form to highlight that each is an optional component of digital signal processing circuit  220 . In each case, there may be a plurality of such components present in processing circuit  220 . An FPGA  221 , if present, may further be coupled with a memory to allow for pre-configuration of one or more processing functions. An FPGA  221  and/or ASIC  222 , if present, may include a processor system provided in the manner of a processor system  223 . Digital signal processing circuit  220  may also include other circuit components external to an FPGA  221 , ASIC  222 , and/or processor system  223 . In one embodiment, signal processing circuit  100  may include only a subset of elements, for example, signal processing circuit  100  may include one element of the elements depicted in reference to any one of  FIGS. 1A-1D . In one example, such a subset of elements may be used with a legacy system having one or more remaining elements of signal processing circuit  100 . Reference emitter  20 , in a further embodiment, may be a node (apparatus), equipped with a radio-frequency transmitter, having one or more of the functional elements depicted in reference to  FIG. 1A . 
       FIG. 1E  is a functional block diagram depicting an embodiment of a node (apparatus)  40 . Node  40  may generally be referred to as a processing node or alternatively as a “fusion node,” and may include a circuit, which may be signal processing circuit  100 , that may include a DSPC  220  and an input communications interface  280 . In one embodiment, node  40  may be a processing system configured to obtain at least one pair of reference channel values, a pair of target channel values, and a set of geolocation data; process the at least one pair of reference channel values, the pair of target reference channel values, and the set of geolocation data to determine a phase-difference metric; and, process the phase-difference metric to determine one or more sets of spatial location coordinates. Where node  40  includes input communication interface  280 , node  40  can obtain the at least one pair of reference channel values, the pair of target channel values, and the set of geolocation data from input communication interface  280 . Input communication interface  280  may be a communications interface in wired or wireless communication with one or more receiver nodes  10 ,  10 -N, and/or be in wired or wireless communication with reference node  20 , so that in one embodiment input communication interface  280  obtains one or more of the at least one pair of reference channel values, the pair of target channel values, or the set of geolocation data from one or more of receiver nodes  10 ,  10 -N and/or reference node  20 . In one embodiment, DSPC  220  of signal processing circuit  100  of node  40  can include one or more of the elements as depicted for DPSC  220  in  FIG. 1D . In another embodiment, signal processing circuit  100  of node  40  can include one or more elements of signal processing circuit  100  as depicted in any one of  FIGS. 1A-1D . In another embodiment, node  40  may co-located at node  10 ,  10 -N and/or node  20 . In yet another embodiment, node  40  may be external to one or more of nodes  10 ,  10 -N, and node  20 . 
     In an exemplary embodiment of a system for locating a target emitter  30 , as described herein and as may be depicted in  FIGS. 2A-2B , the pair of target channel values may be the target channel values determined by receiver nodes  10 , 10 - 1  and  10 , 10 - 2  for target emitter  30 . Similarly, the at least one pair of reference channel values may correspond to the reference channel values determined by receiver nodes  10 ,  10 - 1  and  10 , 10 - 2  for reference emitter  20 . In other exemplary embodiments, more than two receiver nodes  10 ,  10 -N may be deployed to locate a target emitter; this may require determining multiple “pairs” of receiver nodes  10 ,  10 -N and may result in multiple pairs of target channel values and multiple pairs of reference channel values being processed through node  40 . The geolocation data may include geolocation data obtained by receiver nodes  10 ,  10 -N, and may also include geolocation data corresponding to reference emitter  20 , as well as geolocation data corresponding to multiple reference emitters if multiple reference emitters are available. In one embodiment DSPC  220  of signal processing circuit  100  of node  40  may obtain the pair of target channel values, the at least one pair of reference channel values, and the geolocation data from input communication interface  280 . 
     Determining a phase-difference metric may include “canceling out” one or more phase variations present in any of the signals transmitted from, for example, receiver nodes  10 ,  10 -N, a target emitter  30 , and at least one reference emitter  20 . Phase variations may be unknown quantities that, without cancellation, would otherwise need to be known in order to determine a phase-difference metric that may be used to determine one or more sets of spatial location coordinates for, for instance, the target emitter  30 . Phase variations in the target signals may be canceled out, for example, by forming a target conjugate product between the two target channel values of the pair of target channel values. Each target channel value, as described herein above, may correspond to a target channel value determined from a target signal obtained at a receiver node  10 ,  10 -N. Phase variations in the reference signals may similarly be canceled by forming, for example, a reference conjugate product between the two reference channel values of the at least one pair of reference channel values. Phase variations in the receiver node  10 ,  10 -N signals may then be canceled out by determining the phase-difference metric, which may be the conjugate product between the target conjugate product and the reference conjugate product. The phase-difference metric may then be processed to determine a set of spatial location coordinates, which may correspond to a location of the target emitter  30 . 
     The at least one pair of reference channel values may also be one pair of a plurality of pairs of reference channel values, and DSPC  220  of signal processing circuit  100  of node  40  may be further configured to process the plurality of pairs of reference channel values with the pair of target channel values and the set of geolocation data to determine the phase-difference metric. The plurality of pairs of reference channel values may correspond to a plurality of reference signals, in which each of the plurality of reference signals may be picked up by a receiver node  10 ,  10 -N. This may be the case where, for example, a plurality of reference emitters  20  may be available to broadcast reference signals. 
     In another exemplary embodiment, the at least one pair of reference channel values may be an at least one first pair of reference channel values, the pair of target channel values may be a first pair of target channel value, the set of geolocation data may be a first set of geolocation data, and the phase-difference metric may be a first phase-difference metric. The at least one first pair of reference channel values, the first pair of target channel values, and the first set of geolocation data may correspond to a time T 1 , where T 1  may be a time at which the at least one first reference signal and the first target signal were received at a receiver node  10 ,  10 -N. DSPC  220  of signal processing circuit  100  of node  40  may thus be further configured to obtain, from input communications interface  280 , at least one second pair of reference channel values, a second pair of target channel values and a second set of geolocation data; process the at least one second pair of reference channel values, the second pair of target channel values, and the second set of geolocation data to determine a second phase-difference metric; process the first phase-difference metric and the second phase-difference metric to determine a location metric; and, process the location metric to determine the one or more sets of spatial location coordinates. The second pair of target channel values, the at least one second pair of reference channel values, and the second set of geolocation data may correspond to a time T 2 , where T 2  may be a time at which the at least one second reference signal and the second target signal were received at a receiver node  10 ,  10 -N. Processing the first phase-difference metric and the second phase-difference metric may include, for example, forming a conjugate product of the first phase-difference metric and the second phase-difference metric to obtain the location metric. Obtaining the location metric may “cancel out” a time-dependent phase variation in the target signals and the reference signals, and may result in a location metric that may more accurately determine the set of spatial location coordinates than a single phase-difference metric may determine. 
     In yet another exemplary embodiment, DSPC  220  of signal processing circuit  100  of node  40  may be configured to process the one or more sets of spatial location coordinates to determine a velocity. The velocity may correspond to the speed and direction of motion of target emitter  30 . Processing the one or more sets of spatial location coordinates may include carrying out multiple iterations of determining a plurality of phase differences and determining a plurality of location metrics over a period of time, during which receiver nodes  10 ,  10 -N continue to receive target signals and reference signals at different locations along differing paths of motion and continue to transmit pairs of target channel values and pairs of reference channel values to processing node  40 . 
     DSPC  220  of signal processing circuit  100  of node  40  may further be configured to obtain pulse reference data, via input communications interface  280 , process the pulse reference data to determine an average time error, and process the average time error to correct one or more time values. The time values may be one or more time values associated with a target channel value or a reference channel value. The average time error may, for example, be an average time error of a clock  270 , the clock being a clock included as part of a signal processing circuit  100  of a receiver node  10 ,  10 -N. DSPC  220  of signal processing circuit  100  of node  40  may also be further configured to process the pulse reference data to determine an average frequency error, and process the average frequency error to correct a phase variation. The phase variation may be a phase variation in a signal, such as a target signal or an at least one reference signal, received at a receiver  10 ,  10 -N that includes a reference oscillator  260 , the phase variation being introduced into the signal by the reference oscillator  260 . 
     As previously described above, node  40  may be co-located with apparatus  10 ,  10 -N as depicted in  FIG. 1C , and may include signal processing circuit  100  as depicted in  FIG. 1C . In one embodiment, signal processing circuit  100  of node  40  may include an analog signal processing circuit  210  (ASPC) and a digital signal processing circuit  220  (DSPC). Functions of function blocks  280  and  130 , as depicted in  FIG. 1E , may be performed by any designed division of labor scheme between ASPC  210  and DSPC  220 . In one example, ASPC  220  may be a circuit including an input communications interface  280 . For instance, input communication interface  280  may include a radio-frequency receiver  110  that may be part of ASPC  210 . In another embodiment, signal processing circuit  100  of node  40  may only include digital signal processing circuit  220  without including an analog signal processing circuit  210 . In one example, node  40  may be an apparatus external to any of receiver nodes  10 ,  10 -N and reference emitter  20 . In another example, node  40  may be co-located with receiver node  10 ,  10 -N, and may share one or more components of receiver node  10 ,  10 -N, such as a signal characterization circuit  130 . In yet another example, node  40  may be co-located with reference emitter  20 . In another embodiment, one or more functions described as being performed by node  40  may be distributed between external nodes, for example, node  40 , nodes  10 ,  10 -N, and node  20  where nodes  40 ,  10 ,  10 -N, and  20  are nodes external to one another. 
       FIGS. 2A-2B  are illustrative schematics of a system and process for locating a target emitter  30 . In  FIG. 2A , a reference emitter  20  is deployed at a particular known location. A target emitter  30 , whose location is unknown and for which location coordinates are to be determined, is also present. Target emitter  30  may be stationary, as depicted in  FIGS. 2A-2B , or in alternative examples may be in motion, e.g., in motion on the ground, in motion in the air or in space. A first receiver node  10 ,  10 - 1  and a second receiver node  10 ,  10 - 2  are also deployed. Receiver node  10 ,  10 - 1  is in motion along path  11 - 1 , which may be in any particular direction and speed relative to reference emitter  20  and target emitter  30 . Similarly, receiver node  10 ,  10 - 2  is in motion in a different path  11 - 2 . Receiver nodes  10 ,  10 - 1  and  10 ,  10 - 2  may be, in one example, in motion along paths  11 - 1 ,  11 - 2  on ground; for instance, receiver nodes  10 ,  10 -N may be, for instance, ground-based nodes mounted on or carried in a ground vehicle capable of moving along the ground. In another example, receiver nodes  10 ,  10 - 1  and  10 ,  10 - 2  may be in motion along paths  11 - 1 ,  11 - 2  above ground, e.g., in the air or in space. For instance, receiver nodes  10 ,  10 -N may be mounted on or carried in an aircraft, or may be mounted in a satellite in orbit around the earth. In exemplary embodiments receiver nodes  10 ,  10 -N may be in common control with reference emitter  20 . Reference emitter  20  can be broadcasting a reference signal  21  picked up at receiver nodes  10 ,  10 - 1  and  10 ,  10 - 2 . Reference emitter  20  may, in one embodiment, be a ground-based emitter, e.g. a reference emitter with a fixed and known location. In another embodiment, reference emitter  20  may be a ground-based emitter or an air or space-based emitter that may be in motion, for which location information may be known at any time. Receiver nodes  10 ,  10 -N pick up these signals at particular points along their respective paths of motion  11 -N at a time T 1 . Similarly, target emitter  30  can be broadcasting a target signal  31  picked up at receiver nodes  10 ,  10 -N at time T 1 . In the example of  FIGS. 2A-2B  shown, for simplicity, only the direct-path signals  21 ,  31  between emitters  20 ,  30  and receiver nodes  10 ,  10 -N are shown. It may be understood that a signal from any emitter, such as the target emitter, may be received at a receiver node  10 ,  10 -N from multiple paths, and receiver node  10 ,  10 -N may determine which of the multiple signal instances corresponds to the direct-path signal, as described herein above. Receiver node  10 ,  10 - 1  may process signals  31  and  21  to determine a target channel value and at least one reference channel value, and may transmit the channel values via signal  12 - 1  to a processing node  40 . Receiver node  10 ,  10 - 1  may also transmit its own geolocation information via signal  12 - 1  to processing node  40 . Receiver node  10 ,  10 - 2  may also similarly process signals  21  and  31  to determine a target channel value and at least one reference channel value, and transmit the channel values via signal  12 - 2  to an input communications interface  280  of node  40 . Receiver nodes  10 ,  10 -N thus determine a pair of target channel values, at least one pair of reference channel values, and a set of geolocation data. A signal processing circuit  100 , e.g. a signal processing circuit  100  of node  40 , may obtain the pair of target channel values, the at least one pair of reference channel values, and the set and geolocation data, process the pairs of channel values and set of geolocation data to determine a phase-difference metric, and process the phase-difference metric to determine one or more sets of spatial location coordinates. The spatial location coordinates may correspond to possible locations of target emitter  30 . Node  40 , in the embodiment illustrated by  FIGS. 2A-2B , is depicted as being a separately located node from receivers  10 ,  10 -N and reference emitter  20 . It may be understood that in alternative embodiments node  40  may be co-located with any of receiver nodes  10 ,  10 -N or reference emitter  20 . 
       FIG. 2B  depicts the system of  FIG. 2A , in which receiver nodes  10 ,  10 -N have moved over time T 1  to T 2 , the receiver nodes  10 ,  10 -N located at new positions at T 2  along their respective paths of motion  11 -N. Receiver  10 ,  10 - 1  receives reference signal  21  and target signal  31  at its new position and determines a second target channel value and at least one second reference channel value based on the new position of receiver node  10 ,  10 - 1 . The second target channel value and at least one second reference channel value may be sent via signal  13 - 1  to node  40 . Receiver node  10 ,  10 - 1  may also obtain geolocation data at this new position, and send the geolocation data with signal  13 - 1 . Receiver  10 ,  10 - 2  similarly receives reference signal  21  and target signal  31  and sends a second target channel value and at least one second reference channel value, as described above, to processing node  40 . Receiver nodes  10 ,  10 -N thus determine a second pair of target channel values, at least one second pair of reference channel values, and a second set of geolocation data. Node  40  may process the second pair of reference channel values, the second pair of target channel values, and second set of geolocation data to determine a second phase-difference metric. Node  40  may further process the first phase-difference metric, determined in  FIG. 2A , and the second phase-difference metric to determine a location metric, and process the location metric to determine a set of spatial location coordinates. The set of spatial location coordinates may correspond to a location of target emitter  30 . The location metric may provide a more accurate set of spatial location coordinates for the target emitter than the one first phase-difference metric determined in  FIG. 2A , as the processing of the first phase-difference and second phase-difference metrics may eliminate time-varying phase differences of signals  21 ,  31  output by target emitter  30  and reference emitter  20 . 
       FIG. 3  is a block-diagram depicting an embodiment of a method  300  which be used to determine a set of spatial location coordinates. In one embodiment, method  300  may be performed by a signal processing circuit  100 , e.g. a signal processing circuit of node  40 . As described herein, the spatial location coordinates may correspond to a location of a target emitter. At least one pair of reference channel values, a pair of target channel values, and a set of geolocation data may be obtained by signal processing circuit  100  at block  310 . Where signal processing circuit  100  includes input communication interface  280 , DSPC  220  of signal processing circuit  100  may obtain the at least one pair of reference channel values, the pair of target channel values, and the set of geolocation data from input communication interface  280  at block  310 , which has previously obtained values and data from one or more of nodes  10 ,  10 -N, and/or  20 . As described herein above, the pair of target channel values may be the target channel values determined by receiver nodes  10 , 10 -N and the at least one pair of reference channel values may correspond to the reference channel values determined by receiver nodes  10 ,  10 -N. The geolocation data may include geolocation data obtained by receiver nodes  10 ,  10 -N, corresponding to locations of receiver nodes  10 ,  10 -N, and may also include geolocation data corresponding to reference emitter  20 . The at least one pair of reference channel values, pair of target channel values, and set of geolocation data may be processed by signal processing circuit  100  to determine a phase-difference metric, at block  320 . The processing by signal processing circuit  100  at block  320  may include one or more of forming a target conjugate product between the two target channel values of the pair of target channel values, forming a reference conjugate product between the two reference channel values of the at least one pair of reference channel values, and forming a conjugate product between the target conjugate product and the reference conjugate product to determine the phase-difference metric. The phase-difference metric may be processed by signal processing circuit  100  to determine the set of spatial location coordinates, at block  330 . 
     The method  300  above may include additional processing steps and obtaining additional data. In one exemplary embodiment of the above method  300  that can be performed by signal processing circuit  100 , e.g. of node  40 , the at least one pair of reference channel values may be an at least one first pair of reference channel values, the pair of target channel values may be a first pair of target channel value, the set of geolocation data may be a first set of geolocation data, and the phase-difference metric may be a first phase-difference metric. The method  300  may then further include obtaining at least one second pair of reference channel values, a second pair of target channel values and a second set of geolocation data; processing, by signal processing circuit  100 , the at least one second pair of reference channel values, the second pair of target channel values, and the second set of geolocation data to determine a second phase-difference metric; processing, by signal processing circuit  100 , the first phase-difference metric and the second phase-difference metric to determine a location metric; and, processing, by signal processing circuit  100 , the location metric to determine the one or more sets of spatial location coordinates. Where signal processing circuit  100  includes input communication interface  280 , DSPC  220  of signal processing circuit  100  may obtain the at least one second pair of reference channel values, the second pair of target channel values, and the second set of geolocation data from input communication interface  280 . Processing the first phase-difference metric and the second phase-difference metric may include, for example, forming a conjugate product of the first phase-difference metric and the second phase-difference metric to obtain the location metric. Obtaining the location metric may “cancel out” a time-dependent phase variation in the first and second target channel values, which may correspond to the time-dependent variation in a target signal, and the first and second reference channel values, which may correspond to the time-dependent variation in the at least one reference signal. 
     In yet another exemplary embodiment, the method  300  may further include processing, by signal processing circuit  100 , the one or more sets of spatial location coordinates to determine a velocity. The velocity may correspond to the speed and direction of motion of target emitter  30 . Processing the one or more sets of spatial location coordinates may include carrying out multiple iterations of determining a plurality of phase differences and determining a plurality of location metrics over a period of time. 
     In one embodiment, the method  300  as set forth in  FIG. 3  can be performed by and using signal processing circuitry of signal processing circuit  100  of node  40 , which can be external to one or more of nodes  10 ,  10 -N,  20  or co-located with one or more of nodes  10 ,  10 -N,  20 . In another embodiment, performance of the functions of method  300  can be distributed in accordance with a distributed processing scheme between externally disposed nodes. In the case where functions of method  300  are distributed a subset of functions of method  300  can be performed by and using signal processing circuitry of a signal processing circuit  100  of a first node, e,g. node  40 , and a subset of the functions of method  300  can be performed by and using signal processing circuitry of one or more second node, e,g. node  10 ,  10 -N,  20  which second node can be external to node  40 . An apparatus for use in performance of method  300  can include signal processing circuit  100  of a single node, e,g. node  40 ,  10 ,  10 -N,  20 , external from one or more other nodes set forth herein. An apparatus herein can include a first apparatus (node). An apparatus herein can include a combination of nodes (apparatus) as set forth herein. An apparatus for use in performance of method  300  can include signal processing circuitry of a signal processing circuit  100  of a first node and signal processing circuitry of and external second node in the case that method  300  is performed using a distributed processing scheme. A signal processing circuit herein can include a combination of signal processing circuitry of a signal processing circuit  100  of a first node and signal processing circuit  100  of an external second node, e.g., in the case in which a distributive processing scheme is used for performance of a method, e,g. method  300 . 
     Further details of methods for determining a set of spatial location coordinates, according to one or more embodiments described herein, as well as one or more embodiments of apparatuses and/or systems configured to determine a set of spatial location coordinates, according to one or more embodiments described herein, are set forth below. 
     To apply synthetic aperture phase-sensing methods, an estimate of the phase of the carrier of the signal may be obtained. For modern digital signals, this may be most readily accomplished by exploiting predictable signal components such as pilots, preambles, mid-ambles, and similar reference signals that are known in advance from the structure of the signal. As discussed further below, it is also possible to use copy-aided methods, in which the receiver node(s)  10 ,  10 -N estimate the symbols that were transmitted on-the-fly. In either case, the receiver nodes  10 ,  10 -N may construct a local synthetic copy of the signal as transmitted from each emitter  20 ,  30 . 
     For time segment n of receive aperture m (for this discussion, different antennas on each receive node  10 ,  10 -N each can be modeled as separate receiver nodes), the received data vector (after cancellation of higher levels signals) can be expressed as:
 
 r   m,n   =X   n   a   m,n   +n   m,n   (1)
 
where X n  contains copy data for the K modeled signals incident on the aperture for the geolocation set, and a m,n  is the set of stacked channel responses for receiver node  10 ,  10 -N m at the specified time index n. Noise and un-modeled interference is included in n m,n . The copy-data matrix, X has the form:
 
 X   n   =[X   0,−P,n   ,□,X   0,P,n   ,□,X   K−1,−P,n   ,□,X   K−1,P,n ]  (2)
 
where X k,p,n  is an N s ×L matrix containing the copy data for user k, Doppler offset p during time window n. The copy data can be obtained in a variety of ways including the use of invariant fields and exploitation of redundant data. In dense co-channel environments, non-repeated data is typically estimated using Multi-User Detection (MUD). Data that cannot be accurately estimated may be modeled as noise.
 
     For the data received by receive node  10 ,  10 -N m at time index n, the Maximum Likelihood estimate of the channels for the K users may be obtained using
 
 â   m,n =( X   n   H   R   nn   −1   X   n ) −1   X   n   H   R   nn   −1   r   m,n   (3)
 
where R nn  is the noise covariance matrix (this can be modeled as a scaled identity matrix if the sample rate is comparable to the noise bandwidth). The joint channel estimate â m,n  has a form:
 
 â   m,n   =[h   0,m,−P,n   T   ,□,h   0,m,P,n   T   ,□,h   K−1,m,−P,n   T   ,□,h   K−1,m,P,n   T ] T   (2)
 
where h k,m,p,n  is the L×1 vector channel estimate corresponding to user k, aperture m, Doppler offset p, and time window n. Together the collection of vectors {h k,m,p,n } for a given (k,m,n) tuple form the Ambiguity Function (AF) estimate as a function of delay and Doppler shift. If the signal bandwidth is sufficiently large and SNR is adequate, the first-arriving multipath component can be identified from the ambiguity function and a combination of TDOA, FDOA, and DOA methods can be used to estimate the location of user k. If the frequency offsets are chosen properly, the channel estimates for each frequency index p can be calculated independently, reducing computational complexity.
 
     If a single signal is incident at each receiver node  10 ,  10 -N and the length of the data set N s  is too small to allow resolution of frequency-shifted components, then we can make some simplifications. If the length of the channel is assumed to be N c , then a m  has elements corresponding to the value of the channel at uniformly spaced delays pT c  for p=[0, N c −1]. We form the N s ×N c  matrix X from the known samples of the transmitted signal x(kT s −pT c )=x k,p   
                   X   =     [           x     0   ,   0             x     0   ,   1           …         x     0   ,       N   c     -   1                   x     1   ,   0             x     1   ,   1           …         x     1   ,       N   c     -   1                   x     2   ,   0             x     2   ,   1           …         x     2   ,       N   c     -   1                   x     3   ,   0             x     3   ,   1           …         x     3   ,       N   c     -   1                 ⋮       ⋮       ⋱       ⋮             x         N   s     -   1     ,   0             x         N   s     -   1     ,   1           …         x         N   s     -   1     ,       N   c     -   1               ]             (   5   )                 r   m   =Xa   m   +n   m   (6)
 
{circumflex over ( a )}=( X   H   R   nn   −1   X ) −1   X   H   R   nn   −1   r   (7)
 
When R nn   −1 =σ n   2 I, this becomes:
 
{circumflex over ( a )}=( X   H   X ) −1   X   H   r   (8)
 
If the samples in x(kT s ) are nearly uncorrelated, this value can be approximated by:
 
                       a   ^     p     =         ∑     n   =   0         N   s     -   1       ⁢       x     n   -   p     *     ⁢     r   n             ∑     n   =   0         N   s     -   1       ⁢            x   n          2                 (   9   )               
This may be termed the cross-correlation approach to channel estimation.
 
     Multiple paths, or multipath, affects interferometric localization in several important ways. The Line-of-Sight (LOS) component has the information that is usable for interferometric localization. The goal is to extract the first arriving resolvable “cell”. This cell may not be the strongest component, in which case, it may be necessary to take care to detect the first arriving component. A cross-correlation between the received signal and an ideal copy of what was transmitted may produce a superposition of pulses. The first cell can be resolved in different ways. For example, one may define it as the first component for which d/dt of the cross correlation amplitude is zero within ½ cell width of the first energy detected above the Constant False Alarm Rate (CFAR) threshold. If the slope of the cross correlation amplitude does not go to zero, then the first component within ½ cell of delay at which the CFAR threshold is exceeded is used. 
     There may be several multipath components that are unresolvable in the first cell. These components have a similar affect as noise with some important differences. First, the multipath typically scales with the LOS component, and second the multipath may or may not add coherently as the receiver node  10 ,  10 -N moves. For example, at 900 MHz, with a speed of 1 m/s, the Doppler rate is about 3 Hz, leading to a coherence time on the order of 300 ms. For an aperture of 10 seconds, there are approximately 
     
       
         
           
             
               
                 
                   
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     independent integration intervals, in this case 30 intervals. This means that the specular component with have a gain of approximately 10 log 10 (N i ), or 14.8 dB in this example, with respect to any multipath. Thus, if the direct-to-multipath ratio (also referred to as the Rice factor) is 0 dB measured at any single point (when γ n,e (t k ) is estimated), and the noise level is low, then the “effective” integrated direct-to-multipath ratio will be 14.8 dB. 
     For a similar example with a speed of 0.03 m/s, the Doppler rate is approximately 0.1 Hz, leading to a coherence time similar to the capture time. In this case, the multipath will not “de-cohere” over the capture so the integrated direct-to-multipath ratio will be equal to the raw direct-to-multipath ratio, or 0 dB in this example. 
     The above examples demonstrate that the specific nature of the multipath may be very important. If a multipath is temporally resolvable, and it is not the peak detected, then it will minimally impact the localization algorithm. If the multipath is not temporally resolvable from the detected path, then its effect on phase-based localization will depend on how strong it is relative to the line-of-sight path, as well as its Doppler spread. As illustrated in the examples above, even in the case of a non-resolvable multipath, and even if the Rice factor is not large, if the Doppler spread of the component is large, then the multipath component may not significantly degrade localization. 
     For interferometric localization to be effective, it may be important that a line-of-sight component between the emitter and several receiver nodes be present. After integration, the line-of-sight component must have a dominant contribution to the selected peak of the ambiguity function. 
     In the interferometric approach, the estimated complex value of the channel for the first-arriving multipath component may be denoted as γ k,m (nT s ) which is the value of the Ambiguity Function (AF) for a first arriving (presumably direct path) component. The metric γ k,m (nT s ) is equivalent to the complex channel envelope for emitter k, receive node  10 ,  10 -N m at time index n, 
                         γ     k   ,   m       ⁡     (     nT   s     )       ∝       h   k     ⁡     (     nT   s     )         =                h     k   ,   0   ,   m       ⁡     (     nT   s     )            ⁢     e       -   j     ⁢           ⁢       2   ⁢   π   ⁢           ⁢       d     k   ,   m       ⁡     (     nT   s     )         λ           +       ∑     p   =   1       N   -   1       ⁢         h     k   ,   p   ,   m       ⁡     (     nT   s     )       ⁢     e     j   ⁢           ⁢   2   ⁢           ⁢   π   ⁢           ⁢     f     k   ,   p   ,   m       ⁢     nT   s                       (   11   )               
where h k,p,m (nT s ) is the complex amplitude of the pth multipath component for emitter k, and receiver node  10 ,  10 -N m where T s  represents the sample duration. The term f k,p,m  represents the frequency offset of each multipath component due to Doppler. The phase of the p=0 component varies with distance d k,m (nT s ) and wavelength, λ. For simplicity, in the remainder of this discussion, the subscript k is omitted and the metrics are operated for one emitter at a time so that
 
     
       
         
           
             
               
                 
                   
                     
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     The term interferometry refers to methods in which interference patterns between waves are used to extract information. The relationship between classical interferometry and phase-based localization can be understood by understanding that, via the principle of reciprocity, EQ. 12 would equally describe the narrowband complex envelope if the roles of the target emitter  30  and receiver  10 ,  10 -N were swapped so that receiver node  10 ,  10 -N m transmits and target emitter  30  receives. In this case, if a signal were emitted from each receiver node  10 ,  10 -N m with conjugate of phase of the metric in EQ. 12, properly accounting for time n, and all of the waves were superimposed, the contributions from the direct path components would add constructively at the location of target. In the cases described herein, where target emitter  30  emits signals and receiver nodes  10 ,  10 -N receive, localization is performed by measuring the quantity in EQ. 12 at each receiver node  10 ,  10 -N m at time index k. After the data has been collected from all of the receiver nodes  10 ,  10 -N, the system can compute where hypothetical waves launched as described in the previous paragraph would combine with maximum amplitude, indicating the location of the target. Thus, carrier-phase-based localization methods that operate in this way use interferometric concepts. 
     More precisely, using EQ. 12, a likelihood function can be constructed relating the metrics {γ m,n } to potential target emitter  30  locations (x,y,z). It may be assumed, for example, that the location of receiver node  10 ,  10 -N m is known at each time n, and that the effects of noise, multipath, phase noise, and other receiver imperfections are modeled by the metric variance σ m   2 . If a coherent phase reference is available at all receiver nodes  10 ,  10 -N, a likelihood function is obtained, 
                     L   ⁡     (     x   ,   y   ,   z     )       =       -       ∑     m   =   0         N   m     -   1       ⁢       ∑     n   =   0         N   n     -   1       ⁢     log   ⁡     (     π   ⁢           ⁢     σ   m   2       )             +       1     σ   m   2       ⁢              γ     m   ,   n       -       ρ     x   ⁢     x   ~         ⁢     h     0   ,   m   ,   n       ⁢     e       -   j     ⁢           ⁢   2   ⁢   π   ⁢           ⁢         d     m   ,   n       ⁡     (     x   ,   y   ,   z     )       /   λ                  2                 (   13   )               
where ρ x{tilde over (x)}  represents the cross-correlation between the local synthetic copy of the signal and the received signal.
 
     It may not always be practical to provide a coherent phase reference at different receiver nodes  10 ,  10 -N; however, it is often reasonable to provide a coherent phase reference for different measurements along a track for a single receiver node  10 ,  10 -N, particularly over the course of several seconds. In this case, a different likelihood expression is obtained, 
                       L   ⁡     (     x   ,   y   ,   z     )       =         ∑     m   =   0         N   m     -   1       ⁢     (         -     N   n       ⁢     log   ⁡     (     π   ⁢           ⁢     σ   m   2       )         -       ∑     n   =   0         N   n     -   1       ⁢                γ     m   ,   n            2     +              ρ     x   ⁢     x   ~         ⁢     h     0   ,   m   ,   n              2         σ   m   2           )       +       ∑     m   =   0         N   m     -   1       ⁢     (     log   (       I   o     (       2   ⁢           ⁢       ∑     n   =   0         N   n     -   1       ⁢       γ     m   ,   n       ⁢     ρ     x   ⁢     x   ~         ⁢     h     0   ,   m   ,   n     *     ⁢     e       -   j     ⁢           ⁢   2   ⁢   π   ⁢           ⁢         d     m   ,   n       ⁡     (     x   ,   y   ,   z     )       /   λ                 σ   m   2       )     )     )           ,           (   14   )               
where I 0 (z) is the 0 th  order modified Bessel function of the first kind.
 
     The likelihood function for all of the possible target emitter  30  locations over the search space may be calculated using the estimated channel for terminal k as the collection of receiver nodes move, collecting data at different time indices n. 
     In practice, because the carrier frequency of the source target emitter  30  is often not well behaved, a differential metric can be used that cancels the contribution of unknown transmitter phase variation. In this case, the complex envelope is measured synchronously at multiple moving nodes. Pairs of receiver nodes are selected and differential metrics are formed:
 
δ m,m′,n =γ m,n /γ m′,n   (15)
 
A metric surface can be formed using:
 
                       M   ⁡     (     x   ,   y   ,   z     )       =       ∑     b   =   0         N   p     -   1       ⁢       δ     m   ,     m   ′     ,   n       ⁢     e       -   j     ⁢           ⁢   2   ⁢       π   ⁡     (         d     m   ,   n       ⁡     (     x   ,   y   ,   z     )       -       d       m   ′     ,   n       ⁡     (     x   ,   y   ,   z     )         )       /   λ               ,           (   16   )               
The peak of this surface corresponds to the location of a target emitter  30 .
 
     Besides the practical constraints that complicate use of a common phase reference at different receiver nodes  10 ,  10 -N, it may be beneficial to make selective use of coherent and non-coherent combiners. One benefit is that non-coherent combining can be used to reduce the number of nearby false peaks in the likelihood surface. The use of non-coherent combining can also widen the peak associated with the correct location estimate, allowing the search space to be sampled more coarsely, which reduces computational complexity. 
     It is possible to exploit a combination of delay-based methods and interferometric methods jointly. The AF approach can be used to limit the search space, and then interferometric methods can be used to improve localization accuracy. One exemplary “error model” that captures these features is illustrated below. In this model, the received signal is modeled as: 
                       r   m     ⁡     (       nT   w     +     kT   s       )       =         (           h     0   ,   m   ,   n       ⁢     e     j   ⁢           ⁢   2   ⁢   π   ⁢           ⁢       f     0   ,   m   ,   n       ⁡     (       nT   w     +     kT   s       )           ⁢     e       -   j     ⁢           ⁢       2   ⁢   π   ⁢           ⁢       d     m   ,   n       ⁡     (     x   ,   y     )         λ             ︷     Direct   ⁢           ⁢   Path         +         h     1   ,   m   ,   n       ⁢     e     j   ⁢           ⁢   2   ⁢   π   ⁢           ⁢       f     1   ,   m   ,   n       ⁡     (       nT   w     +     kT   s       )               ︷     Multipath   ⁢           ⁢   Component           )     ×       x   (       nT   w     +     kT   s       )       ︷     Modulating   Signal         ⁢       e     j   ⁢           ⁢       ζ     m   ,   n       ⁡     (       nT   w     +     kT   s       )             ︷     Phase   ⁢           ⁢   Errors           +         n   m     ⁡     (       nT   w     +     kT   s       )         ︸   Noise       +       I   ⁡     (       nT   w     +     kT   s       )         ︸   Interference                 (   17   )               
The complex channel envelope can then be modeled as:
 
γ m,n   =h   0,m,n   e   −j2πd     m,n     (x,y)/λ   R   x,{tilde over (x)} Ψ( f   0,m,n   −f   m,n   r ) e   j(φ     m     +ξ     m,n     )   +h   1,m,n   R   x,{tilde over (x)} Ψ( f   1,m,n   −f   m,n   r ) e   j(φ     m     +ξ     m,n     ) +η m,n   (18)
 
where:
         h 0,m,n —Complex multipath component amplitude. The direct path (p=0) is real.   f p,m,n —Frequency offset of the p th  multipath component received at receiver node  10 ,  10 -N (m,n).   (x,y)—Location of the target emitter  30     λ—Wavelength   f m,n   r —Receiver frequency offset at receiver node  10 ,  10 -N (m,n).   d m,n (x,y)—Distance from target emitter  30  to receiver node  10 ,  10 -N (m,n)   R x,{tilde over (x)} —Cross-correlation between received signal x(nT w +kT s ) and synthesized signal {tilde over (x)}(nT w +kT s ), R x,{tilde over (x)} =E└x(nT w +kT s ) {tilde over (x)}*(nT w +kT s )┘   Ψ(f p,m,n −f m,n   r )—Coherence due to frequency offset:       

     
       
         
           
             
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             φ m —Phase offset of receiver node  10 ,  10 -N m relative to other nodes 
             τ m,n —Phase offset of receiver node  10 ,  10 -N m at window n. 
             η m,n —Noise and (Gaussian-modeled) interference at receiver node  10 ,  10 -N m at window n.
 
The Fisher information matrix corresponding to the likelihood function in EQ. 16 is:
 
           
         
       
    
                     I   ⁡     (     x   ,   y     )       =     -     E   ⁡     [       ∑     m   =   0         N   m     -   1       ⁢       1     K   m       ⁢         I   1     ⁡     (     K   m     )           I   0     ⁡     (     K   m     )         ⁢       (       2   ⁢   π     λ     )     2     ⁢       ∑       n   ′     =   0         N   n     -   1       ⁢       ∑     n   =   0         N   n     -   1       ⁢       F     m   ,   n       ⁢     v   ⁡     (     θ     m   ,     n   ′         )       ⁢       (       v   ⁡     (     θ     m   ,   n       )       -     v   ⁡     (     θ     m   ,     n   ′         )         )     H               ]                 (   19   )               
where,
 
                     F     m   ,   n       =     Re   ⁢     {       4     σ     T   ,   m     4       ⁢     γ     m   ,   n       ⁢     γ     m   ,     n   ′       *     ⁢     h     0   ,   m   ,   n       ⁢     h     o   ,   m   ,     n   ′         ⁢     χ     m   ,   n       ⁢     χ     m   ,     n   ′       *     ⁢     e     j   ⁢           ⁢       2   ⁢   π     λ     ⁢     (         d     m   ,   n       ⁡     (     x   ,   y     )       -       d     m   ,     n   ′         ⁡     (     x   ,   y     )         )           }               (   20   )               
The error sources listed above are all taken into account in this expression.
 
σ T,m   2 =2| h   0,m,n χ 0,m,n | 2 σ ξ,m   2 +σ h,m   2 |χ 1,m,n | 2 +σ η,m   2   (21)
 
     
       
         
           
             
               
                 
                   
                     
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                     v   ⁡     (     θ     m   ,   n       )       =     [           cos   ⁢           ⁢     θ     m   ,   n                   sin   ⁢           ⁢     θ     m   ,   n               ]             (   23   )               
Then the position error resulting from the SWCAL expression is bounded by:
 
var( x )+var( y )≧trace( J )  (24)
 
where J(x,y)=I −1 (x,y).
 
     Interferometric localization, as described herein, presents a number of challenges in practice. First, variation of local oscillators at the emitter  30  and the receiver node  10 ,  10 -N can complicate interferometric methods. Second, computation of the solution using the likelihood function is computationally complicated since the peak of the surface can be very narrow. As a result, if brute-force methods are used to search the likelihood surface, it may be necessary to finely grid the search area, resulting in a large number of hypothesized locations that need to be evaluated. Triple Difference Interferometry methods, as described previously and in further detail below, may overcome these challenges and problems with interferometric localization. 
     Referring again to  FIGS. 2A and 2B , using training sequences, pilots, copy-aided methods or other techniques, it may be assumed that the channel between emitter e ( 20 ,  30 ), and receiver node n ( 10 , 10 - 1 ,  10 , 10 - 2 ) can be estimated at time t k . The complex value of the first arriving path can be represented as: 
     
       
         
           
             
               
                 
                   
                     
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     Here A n,e  is a scalar constant, β e (t) is the time varying phase at emitter e ( 20 ,  30 ), α n (t) is the time varying phase at receiver node n ( 10 , 10 - 1 ,  10 , 10 - 2 ), d n,e  (t) is the time-varying distance from emitter e ( 20 ,  30 ) to receiver node n ( 10 , 10 - 1 ,  10 , 10 - 2 ), n n,e (t) is the noise affecting the measurement, and δ n  is the time error at receiver node n ( 10 , 10 - 1 ,  10 , 10 - 2 ). To localize a target emitter  30 , measurements at several pairs of receiver nodes  10 ,  10 -N may be required. The conjugate product between measurements at two different receiver nodes (effectively the difference in phases) is: 
                               δ   ^         n   1     ,     n   2     ,     e   1         ⁡     (     t   k     )       =       ⁢           γ   ^         n   1     ,     e   1         ⁡     (     t   k     )       ·         γ   ^         n   2     ,     e   1       *     ⁡     (     t   k     )                     =       ⁢       A       n   1     ,     e   1     ,   s       ⁢     A       n   2     ,     e   1     ,   s       ⁢     exp   [     j   (         β     e   1       ⁡     (       t   k     +     δ     n   1         )       -       β     e   1       (       t   k     +                                 ⁢     δ     n   2       )     -         2   ⁢   π       λ     e   1         ⁢     (         d       n   1     ,     e   1         ⁡     (       t   k     +     δ     n   1         )       -       d       n   2     ,     e   1         (       t   k     +                                     ⁢     δ     n   2       )     )     )     ]     ×     exp   [     j   (         α     n   1       ⁡     (       t   k     +     δ     n   1         )       -       α     n   2       ⁡     (       t   k     +     δ     n   2         )         )     ]                 ≈       ⁢     K   ⁢           ⁢     exp   [     j   (         β     e   1       ⁡     (     t   k     )       -       β     e   1       ⁡     (     t   k     )       -         2   ⁢   π       λ     e   1         ⁢     (     d       n   1     ,     e   1                                       ⁢       (       t   k     +     δ     n   1         )     -       d       n   2     ,     e   1         ⁡     (       t   k     +     δ     n   2         )         )     +       α     n   1       ⁡     (       t   k     +     δ     n   1         )       -                     ⁢       α     n   2       ⁡     (       t   k     +     δ     n   2         )       )     ]               =       ⁢     K   ⁢           ⁢     exp   [     j   (       -       2   ⁢   π       λ     e   1           ⁢     (         d       n   1     ,     e   1         ⁡     (       t   k     -     δ     n   1         )       -     d       n   2     ,     e   1                                           ⁢     (       t   k     +     δ     n   2         )     )     +       α     n   1       ⁡     (       t   k     +     δ     n   1         )       -       α     n   2       ⁡     (       t   k     +     δ     n   2         )         )     ]                 (   26   )               
In EQ. 26 it is assumed that dβ e     1   /dt&lt;&lt;(2π|δ n     1   −δ n     2   |). This step effectively removes the effect of transmitter  20 ,  30  frequency offset from the metric. This assumption introduces requires on the time alignment between receiver nodes n 1  ( 10 ,  10 - 1 ) and n 2  ( 10 ,  10 - 2 ) based on the expected phase variation at the emitter  20 ,  30 .
 
     Next, another ratio, the second phase difference, is formed: 
                         ρ   ^         n   1     ,     n   2     ,     e   1     ,     e   2         ⁡     (     t   k     )       =             δ   ^         n   1     ,     n   2     ,     e   1         ⁡     (     t   k     )       ·         δ   ^         n   1     ,     n   2     ,     e   2       *     ⁡     (     t   k     )         ≈       A       n   1     ,     e   1         ⁢     A       n   2     ,     e   1         ⁢     exp   ⁡     [     j   ⁡     (         -       2   ⁢   π       λ     e   1           ⁢     (         d       n   1     ,     e   1         ⁡     (       t   k     +     δ     n   1         )       -       d       n   2     ,     e   1         ⁡     (       t   k     +     δ     n   2         )         )       +       α     n   1       ⁡     (       t   k     +     δ     n   1         )       -       α     n   2       ⁡     (       t   k     +     δ     n   2         )         )       ]       ×     A       n   1     ,     e   2         ⁢     A       n   2     ,     e   2         ⁢     exp   ⁡     [     -     j   ⁡     (         -       2   ⁢   π       λ     e   2           ⁢     (         d       n   1     ,     e   2         ⁡     (       t   k     +     δ     n   1         )       -       d       n   2     ,     e   2         ⁡     (       t   k     +     δ     n   2         )         )       +       α     n   1       ⁡     (       t   k     +     δ     n   1         )       -       α     n   2       ⁡     (       t   k     +     δ     n   2         )         )         ]         ≈     K   ⁢           ⁢     exp   ⁡     [     -     j   ⁡     (           2   ⁢   π       λ     e   1         ⁢     (         d       n   1     ,     e   1         ⁡     (       t   k     +     δ     n   1         )       -       d       n   2     ,     e   1         ⁡     (       t   k     +     δ     n   2         )         )       -         2   ⁢   π       λ     e   2         ⁢     (         d       n   1     ,     e   2         ⁡     (       t   k     +     δ     n   1         )       -       d       n   2     ,     e   2         ⁡     (       t   k     +     δ     n   2         )         )         )         ]                   (   27   )               
If ∂(d n,e (t))/∂t&lt;&lt;λ e /2π|δ n     1   −δ n     2   | −1 , then
 
                         ρ   ^         n   1     ,     n   2     ,     e   1     ,     e   2         ⁡     (     t   k     )       =     K   ⁢           ⁢     exp   ⁡     [     -     j   ⁡     (           2   ⁢   π       λ     e   1         ⁢     (         d       n   1     ,     e   1         ⁡     (     t   k     )       -       d       n   2     ,     e   1         ⁡     (     t   k     )         )       -         2   ⁢   π       λ     e   2         ⁢     (         d       n   1     ,     e   2         ⁡     (     t   k     )       -       d       n   2     ,     e   2         ⁡     (     t   k     )         )         )         ]                 (   28   )               
If the location of reference emitter e 2  ( 20 ) and the location of each receiver node ( 10 ,  10 -N) are known, a corrected value {circumflex over (ρ)} n     1     ,n     2     ,e     1     c , (t k ) may be formed:
 
                               ρ   ^         n   1     ,     n   2     ,     e   1       c     ⁡     (     t   k     )       =       ⁢           ρ   ^         n   1     ,     n   2     ,     e   1     ,     e   2         ⁡     (     t   k     )       ·     exp   [       -   j     ⁢           ⁢       2   ⁢   π       λ     e   2         ⁢     (           d   ^         n   1     ,     e   2         ⁡     (     t   k     )       -                               ⁢         d   ^         n   2     ⁢     e   2         ⁡     (     t   k     )       )     ]               ≈       ⁢     K   ⁢           ⁢     exp   [         -   j     ⁢           ⁢       2   ⁢   π       λ     e   1         ⁢     (         d       n   1     ,     e   1         ⁡     (     t   k     )       -       d       n   2     ,     e   1         ⁡     (     t   k     )         )       +                           ⁢         2   ⁢   π       λ     e   2         ⁢     (         d       n   1     ,     e   2         ⁡     (     t   k     )       -       d       n   2     ,     e   2         ⁡     (     t   k     )         )       ]     ×                   ⁢     exp   ⁡     [     j   ⁡     (       -       2   ⁢   π       λ     e   2           ⁢     (           d   ^         n   1     ,     e   2         ⁡     (     t   k     )       -         d   ^         n   2     ,     e   2         ⁡     (     t   k     )         )       )       ]                   ≈       ⁢     K   ⁢           ⁢     exp   ⁡     [       -   j     ⁢           ⁢       2   ⁢   π       λ     e   1         ⁢     (         d       n   1     ,     e   1         ⁡     (     t   k     )       -       d       n   2     ,     e   1         ⁡     (     t   k     )         )       ]                       (   29   )               
This quantity, which may be called a double-difference metric or phase-difference metric can be used to locate target emitter e 1  ( 30 ). However, the error performance surface associated with this quantity has many local minima and the correct peak is very narrow, making it challenging to search.
 
     To control the shape of the error surface to simplify the search process, a further value may be formed, called the third phase difference or location metric, by forming the conjugate product between values of {circumflex over (ρ)} n     1     ,n     2     ,e     1     ,e     2   (t k ) at two points in time: 
                               η   ^         n   1     ,     n   2     ,     e   1     ,     e   2         ⁡     (       t   k     ,     t     k   -   L         )       =       ⁢           ρ   ^         n   1     ,     n   2     ,     e   1     ,     e   2         ⁡     (     t   k     )       ·         ρ   ^         n   1     ,     n   2     ,     e   1     ,     e   2       *     ⁡     (     t     k   -   L       )                     =       ⁢     K   ⁢           ⁢     exp   [         -   j     ⁢           ⁢       2   ⁢   π       λ     e   1         ⁢     (         d       n   1     ,     e   1         ⁡     (     t   k     )       -       d       n   2     ,     e   1         ⁡     (     t   k     )         )       +                           ⁢     j   ⁢       2   ⁢   π       λ     e   2         ⁢     (         d       n   1     ,     e   2         ⁡     (     t   k     )       -       d       n   2     ,     e   2         ⁡     (     t   k     )         )       ]     ×                   ⁢     exp   [       j   ⁢           ⁢       2   ⁢   π       λ     e   1         ⁢     (         d       n   1     ,     e   1         ⁡     (     t     k   -   L       )       -       d       n   2     ,     e   1         ⁡     (     t     k   -   L       )         )       -                       ⁢     j   ⁢           ⁢       2   ⁢   π       λ     e   2         ⁢     (         d       n   1     ,     e   2         ⁡     (     t     k   -   L       )       -       d       n   2     ,     e   2         ⁡     (     t     k   -   L       )         )       ]                 (   30   )               
For two-dimensional receiver node  10 ,  10 -N motion in the x-y plane, if the source is in the far field of each receiver node track  11 - 1 ,  11 - 2 , for a duration t k −t k−L , and the receiver nodes  10 ,  10 -N travel in straight lines, this quantity may be approximated with:
 
                         η   ^         n   1     ,     n   2     ,     e   1     ,     e   2         ⁡     (       t   k     ,     t     k   -   L         )       ≈     K   ⁢           ⁢     exp   ⁡     [       -   j     ⁢           ⁢   2   ⁢   π   ⁢           ⁢       LT   s     ⁡     (         v     x   ,     n   1         ⁡     (         cos   ⁢           ⁢     θ       n   1     ,     e   1             λ     e   1         -       cos   ⁢           ⁢     θ       n   1     ,     e   2             λ     e   2           )       +       v     y   ,     n   1         (           ⁢         sin   ⁢           ⁢     θ       n   1     ,     e   1             λ     e   1         -       sin   ⁢           ⁢     θ       n   1     ,     e   2             λ     e   2           )       )         ]       ×     exp   ⁡     [     j   ⁢           ⁢   2   ⁢   π   ⁢           ⁢       LT   s     (           ⁢         v     x   ,     n   2         ⁡     (         cos   ⁢           ⁢     θ       n   2     ,     e   1             λ     e   1         -       cos   ⁢           ⁢     θ       n   2     ,     e   2             λ     e   2           )       +       v     y   ,     n   2         ⁡     (         sin   ⁢           ⁢     θ       n   2     ,     e   1             λ     e   1         -       sin   ⁢           ⁢     θ       n   2     ,     e   2             λ     e   2           )         )       ]                 (   31   )               
where v x,n  and v y,n  are the velocity components of receiver node n ( 10 ,  10 -N) in the x and y directions and θ n,e  is the direction to emitter e ( 20 ,  30 ) from receiver node n ( 10 ,  10 -N), measured from the x-axis. This simplification highlights the relationship between TDI approaches and Frequency Difference of Arrival (FDOA) methods.
 
     As performed previously when calculating {circumflex over (ρ)} n     1     ,n     2     ,e     1     c (t k ), a version of the location metric may be formed that removes the dependency on the reference emitter ( 20 ) location: 
                               η   ^         n   1     ,     n   2     ,     e   1       c     ⁡     (       t   k     ,     t     k   -   L         )       =       ⁢           ρ   ^         n   1     ,     n   2     ,     e   1       c     ⁡     (     t   k     )       ×         ρ   ^         n   1     ,     n   2     ,     e   1         c   *       ⁡     (     t     k   -   L       )                     ≈       ⁢     K   ⁢           ⁢     exp   ⁡     [       -   j     ⁢           ⁢       2   ⁢   π       λ     e   1         ⁢     (         d       n   1     ,     e   1         ⁡     (     t   k     )       -       d       n   2     ,     e   1         ⁡     (     t   k     )         )       ]       ×                     ⁢     exp   ⁡     [     j   ⁢           ⁢       2   ⁢   π       λ     e   1         ⁢     (         d       n   1     ,     e   1         ⁡     (     t     k   -   L       )       -       d       n   2     ,     e   1         ⁡     (     t     k   -   L       )         )       ]                   =       ⁢     K   ⁢           ⁢     exp   [       -   j     ⁢           ⁢       2   ⁢   π       λ     e   1         ⁢     (         d       n   1     ,     e   1         ⁡     (     t   k     )       -       d       n   2     ,     e   1         ⁡     (     t   k     )       -                               ⁢         d       n   1     ,     e   1         ⁡     (     t     k   -   L       )       +       d       n   2     ,     e   1         ⁡     (     t     k   -   L       )         )     ]                 (   32   )               
As before, the far-field simplification may be computed as:
 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             
                               η 
                               ^ 
                             
                             
                               
                                 n 
                                 1 
                               
                               , 
                               
                                 n 
                                 2 
                               
                               , 
                               
                                 e 
                                 1 
                               
                             
                             c 
                           
                           ⁡ 
                           
                             ( 
                             
                               
                                 t 
                                 k 
                               
                               , 
                               
                                 t 
                                 
                                   k 
                                   - 
                                   L 
                                 
                               
                             
                             ) 
                           
                         
                         = 
                           
                         ⁢ 
                         
                           K 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             exp 
                             ⁡ 
                             
                               [ 
                               
                                 
                                   - 
                                   j 
                                 
                                 ⁢ 
                                 
                                   
                                     2 
                                     ⁢ 
                                     π 
                                   
                                   
                                     λ 
                                     
                                       e 
                                       1 
                                     
                                   
                                 
                                 ⁢ 
                                 
                                   ( 
                                   
                                     
                                       
                                         d 
                                         
                                           
                                             n 
                                             1 
                                           
                                           , 
                                           
                                             e 
                                             1 
                                           
                                         
                                       
                                       ⁡ 
                                       
                                         ( 
                                         
                                           t 
                                           k 
                                         
                                         ) 
                                       
                                     
                                     - 
                                     
                                       
                                         d 
                                         
                                           
                                             n 
                                             2 
                                           
                                           , 
                                           
                                             e 
                                             1 
                                           
                                         
                                       
                                       ⁡ 
                                       
                                         ( 
                                         
                                           t 
                                           k 
                                         
                                         ) 
                                       
                                     
                                   
                                   ) 
                                 
                               
                               ] 
                             
                           
                           × 
                         
                       
                     
                   
                   
                     
                       
                           
                         ⁢ 
                         
                           exp 
                           ⁡ 
                           
                             [ 
                             
                               j 
                               ⁢ 
                               
                                 
                                   2 
                                   ⁢ 
                                   π 
                                 
                                 
                                   λ 
                                   
                                     e 
                                     1 
                                   
                                 
                               
                               ⁢ 
                               
                                 ( 
                                 
                                   
                                     
                                       d 
                                       
                                         
                                           n 
                                           1 
                                         
                                         , 
                                         
                                           e 
                                           1 
                                         
                                       
                                     
                                     ⁡ 
                                     
                                       ( 
                                       
                                         t 
                                         
                                           k 
                                           - 
                                           L 
                                         
                                       
                                       ) 
                                     
                                   
                                   - 
                                   
                                     
                                       d 
                                       
                                         
                                           n 
                                           2 
                                         
                                         , 
                                         
                                           e 
                                           1 
                                         
                                       
                                     
                                     ⁡ 
                                     
                                       ( 
                                       
                                         t 
                                         
                                           k 
                                           - 
                                           L 
                                         
                                       
                                       ) 
                                     
                                   
                                 
                                 ) 
                               
                             
                             ] 
                           
                         
                       
                     
                   
                   
                     
                       
                         ≈ 
                           
                         ⁢ 
                         
                           K 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             exp 
                             [ 
                             
                               
                                 - 
                                 j 
                               
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               2 
                               ⁢ 
                               π 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 
                                   LT 
                                   s 
                                 
                                 ( 
                                 
                                   
                                     
                                       v 
                                       
                                         x 
                                         , 
                                         
                                           n 
                                           1 
                                         
                                       
                                     
                                     ⁡ 
                                     
                                       ( 
                                       
                                         
                                           cos 
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           
                                             θ 
                                             
                                               
                                                 n 
                                                 1 
                                               
                                               , 
                                               
                                                 e 
                                                 1 
                                               
                                             
                                           
                                         
                                         
                                           λ 
                                           
                                             e 
                                             1 
                                           
                                         
                                       
                                       ) 
                                     
                                   
                                   + 
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         
                           
                               
                             ⁢ 
                             
                               
                                 v 
                                 
                                   y 
                                   , 
                                   
                                     n 
                                     1 
                                   
                                 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   
                                     sin 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       θ 
                                       
                                         
                                           n 
                                           1 
                                         
                                         , 
                                         
                                           e 
                                           1 
                                         
                                       
                                     
                                   
                                   
                                     λ 
                                     
                                       e 
                                       1 
                                     
                                   
                                 
                                 ) 
                               
                             
                             ) 
                           
                           ] 
                         
                         × 
                       
                     
                   
                   
                     
                       
                           
                         ⁢ 
                         
                           exp 
                           [ 
                           
                             j 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             2 
                             ⁢ 
                             π 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               
                                 LT 
                                 s 
                               
                               ( 
                               
                                 
                                   
                                     v 
                                     
                                       x 
                                       , 
                                       
                                         n 
                                         2 
                                       
                                     
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       
                                         cos 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         
                                           θ 
                                           
                                             
                                               n 
                                               2 
                                             
                                             , 
                                             
                                               e 
                                               1 
                                             
                                           
                                         
                                       
                                       
                                         λ 
                                         
                                           e 
                                           1 
                                         
                                       
                                     
                                     ) 
                                   
                                 
                                 + 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         
                             
                           ⁢ 
                           
                             
                               v 
                               
                                 y 
                                 , 
                                 
                                   n 
                                   2 
                                 
                               
                             
                             ⁡ 
                             
                               ( 
                               
                                 
                                   sin 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     θ 
                                     
                                       
                                         n 
                                         2 
                                       
                                       , 
                                       
                                         e 
                                         1 
                                       
                                     
                                   
                                 
                                 
                                   λ 
                                   
                                     e 
                                     1 
                                   
                                 
                               
                               ) 
                             
                           
                           ) 
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   33 
                   ) 
                 
               
             
           
         
       
     
     The triple difference metric of EQ. 32 may be expressed as a function of the unknown target emitter ( 30 ) location:
 
{circumflex over (η)} n     1     ,n     2     ,e   c ( t   k   ,t   k−L )= f   m ( x   e   ,y   e   ,z   e )= f   m ( w )  (34)
 
where w=[x e  y e  z e ] T . The measurements ρ n     1     ,n     2     ,e   c (t k , t k−L ) may be “stacked” for all values of k and selected pairs {n 1 ,n 2 } in a vector h where h=f(w)+n.
 
                     f   ⁡     (   w   )       =     [             f   0     ⁡     (   w   )               ⋮               f     N     m   -   1         ⁡     (   w   )             ]             (   35   )               
In other words, the m th  row of h contains the pair-wise measurements corresponding to pair {n 1 ,n 2 } for time samples t k  and t k−L .
 
     Given an initial estimate of the location for emitter e, w 0 =[x 0  y 0  z 0 ] T , the function of the correct location may be expressed using the first two terms of the Taylor series:
 
 f ( w )= f ( w   0 )+ G ( w   0 )( w−w   0 )  (36)
 
and G(w 0 ) is
 
     
       
         
           
             
               
                 
                   
                     G 
                     ⁡ 
                     
                       ( 
                       
                         w 
                         0 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       [ 
                       
                         
                           
                             
                               
                                 ∂ 
                                 
                                   
                                     f 
                                     0 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     w 
                                     ) 
                                   
                                 
                               
                               
                                 ∂ 
                                 
                                   w 
                                   0 
                                 
                               
                             
                           
                           
                             … 
                           
                           
                             
                               
                                 ∂ 
                                 
                                   
                                     f 
                                     0 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     w 
                                     ) 
                                   
                                 
                               
                               
                                 ∂ 
                                 
                                   w 
                                   
                                     
                                       N 
                                       w 
                                     
                                     - 
                                     1 
                                   
                                 
                               
                             
                           
                         
                         
                           
                             ⋮ 
                           
                           
                             
                                 
                             
                           
                           
                             ⋮ 
                           
                         
                         
                           
                             
                               
                                 ∂ 
                                 
                                   
                                     f 
                                     
                                       N 
                                       
                                         m 
                                         - 
                                         1 
                                       
                                     
                                   
                                   ⁡ 
                                   
                                     ( 
                                     w 
                                     ) 
                                   
                                 
                               
                               
                                 ∂ 
                                 
                                   w 
                                   0 
                                 
                               
                             
                           
                           
                             … 
                           
                           
                             
                               
                                 ∂ 
                                 
                                   
                                     f 
                                     
                                       N 
                                       
                                         m 
                                         - 
                                         1 
                                       
                                     
                                   
                                   ⁡ 
                                   
                                     ( 
                                     w 
                                     ) 
                                   
                                 
                               
                               
                                 ∂ 
                                 
                                   w 
                                   
                                     
                                       N 
                                       w 
                                     
                                     - 
                                     1 
                                   
                                 
                               
                             
                           
                         
                       
                       ] 
                     
                     ⁢ 
                     
                       | 
                       
                         w 
                         = 
                         
                           w 
                           0 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   37 
                   ) 
                 
               
             
           
         
       
     
     
       
         
           
             
               
                 
                   
                     
                       
                         ∂ 
                         
                           
                             f 
                             m 
                           
                           ⁡ 
                           
                             ( 
                             w 
                             ) 
                           
                         
                       
                       
                         ∂ 
                         
                           x 
                           e 
                         
                       
                     
                     ⁢ 
                     
                       | 
                       
                         w 
                         = 
                         
                           w 
                           0 
                         
                       
                     
                   
                   = 
                   
                     
                       
                         
                           ∂ 
                           
                             
                               
                                 η 
                                 ^ 
                               
                               
                                 
                                   n 
                                   1 
                                 
                                 , 
                                 
                                   n 
                                   2 
                                 
                                 , 
                                 e 
                               
                               C 
                             
                             ⁡ 
                             
                               ( 
                               w 
                               ) 
                             
                           
                         
                         
                           ∂ 
                           
                             x 
                             e 
                           
                         
                       
                       ⁢ 
                       
                         | 
                         
                           w 
                           = 
                           
                             w 
                             0 
                           
                         
                       
                     
                     = 
                     
                       j 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
                           
                             2 
                             ⁢ 
                             π 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             K 
                           
                           
                             λ 
                             e 
                           
                         
                         ⁡ 
                         
                           [ 
                           
                             
                               
                                 
                                   
                                     x 
                                     
                                       n 
                                       1 
                                     
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       t 
                                       k 
                                     
                                     ) 
                                   
                                 
                                 - 
                                 
                                   x 
                                   0 
                                 
                               
                               
                                 
                                   
                                     d 
                                     ^ 
                                   
                                   
                                     
                                       n 
                                       1 
                                     
                                     , 
                                     e 
                                   
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     t 
                                     k 
                                   
                                   ) 
                                 
                               
                             
                             - 
                             
                               
                                 
                                   
                                     x 
                                     
                                       n 
                                       2 
                                     
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       t 
                                       k 
                                     
                                     ) 
                                   
                                 
                                 - 
                                 
                                   x 
                                   0 
                                 
                               
                               
                                 
                                   
                                     d 
                                     ^ 
                                   
                                   
                                     
                                       n 
                                       2 
                                     
                                     , 
                                     e 
                                   
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     t 
                                     k 
                                   
                                   ) 
                                 
                               
                             
                             - 
                             
                               
                                 
                                   
                                     x 
                                     
                                       n 
                                       1 
                                     
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       t 
                                       
                                         k 
                                         - 
                                         L 
                                       
                                     
                                     ) 
                                   
                                 
                                 - 
                                 
                                   x 
                                   0 
                                 
                               
                               
                                 
                                   
                                     d 
                                     ^ 
                                   
                                   
                                     
                                       n 
                                       1 
                                     
                                     , 
                                     e 
                                   
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     t 
                                     
                                       k 
                                       - 
                                       L 
                                     
                                   
                                   ) 
                                 
                               
                             
                             + 
                             
                               
                                 
                                   
                                     x 
                                     
                                       n 
                                       2 
                                     
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       t 
                                       
                                         k 
                                         - 
                                         L 
                                       
                                     
                                     ) 
                                   
                                 
                                 - 
                                 
                                   x 
                                   0 
                                 
                               
                               
                                 
                                   
                                     d 
                                     ^ 
                                   
                                   
                                     
                                       n 
                                       2 
                                     
                                     , 
                                     e 
                                   
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     t 
                                     
                                       k 
                                       - 
                                       L 
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                           ] 
                         
                       
                       ⁢ 
                       
                         
                           
                             η 
                             ^ 
                           
                           
                             
                               n 
                               1 
                             
                             , 
                             
                               n 
                               2 
                             
                             , 
                             e 
                           
                           C 
                         
                         ⁡ 
                         
                           ( 
                           
                             w 
                             0 
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   38 
                   ) 
                 
               
             
           
         
       
     
     
       
         
           
             
               
                 
                   
                     
                       
                         ∂ 
                         
                           
                             f 
                             m 
                           
                           ⁡ 
                           
                             ( 
                             w 
                             ) 
                           
                         
                       
                       
                         ∂ 
                         
                           y 
                           e 
                         
                       
                     
                     ⁢ 
                     
                       | 
                       
                         w 
                         = 
                         
                           w 
                           0 
                         
                       
                     
                   
                   = 
                   
                     
                       
                         
                           ∂ 
                           
                             
                               
                                 η 
                                 ^ 
                               
                               
                                 
                                   n 
                                   1 
                                 
                                 , 
                                 
                                   n 
                                   2 
                                 
                                 , 
                                 e 
                               
                               C 
                             
                             ⁡ 
                             
                               ( 
                               w 
                               ) 
                             
                           
                         
                         
                           ∂ 
                           
                             y 
                             e 
                           
                         
                       
                       ⁢ 
                       
                         | 
                         
                           w 
                           = 
                           
                             w 
                             0 
                           
                         
                       
                     
                     = 
                     
                       j 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
                           
                             2 
                             ⁢ 
                             π 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             K 
                           
                           
                             λ 
                             e 
                           
                         
                         ⁡ 
                         
                           [ 
                           
                             
                               
                                 
                                   
                                     y 
                                     
                                       n 
                                       1 
                                     
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       t 
                                       k 
                                     
                                     ) 
                                   
                                 
                                 - 
                                 
                                   y 
                                   0 
                                 
                               
                               
                                 
                                   
                                     d 
                                     ^ 
                                   
                                   
                                     
                                       n 
                                       1 
                                     
                                     , 
                                     e 
                                   
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     t 
                                     k 
                                   
                                   ) 
                                 
                               
                             
                             - 
                             
                               
                                 
                                   
                                     y 
                                     
                                       n 
                                       2 
                                     
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       t 
                                       k 
                                     
                                     ) 
                                   
                                 
                                 - 
                                 
                                   y 
                                   0 
                                 
                               
                               
                                 
                                   
                                     d 
                                     ^ 
                                   
                                   
                                     
                                       n 
                                       2 
                                     
                                     , 
                                     e 
                                   
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     t 
                                     k 
                                   
                                   ) 
                                 
                               
                             
                             - 
                             
                               
                                 
                                   
                                     y 
                                     
                                       n 
                                       1 
                                     
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       t 
                                       
                                         k 
                                         - 
                                         L 
                                       
                                     
                                     ) 
                                   
                                 
                                 - 
                                 
                                   y 
                                   0 
                                 
                               
                               
                                 
                                   
                                     d 
                                     ^ 
                                   
                                   
                                     
                                       n 
                                       1 
                                     
                                     , 
                                     e 
                                   
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     t 
                                     
                                       k 
                                       - 
                                       L 
                                     
                                   
                                   ) 
                                 
                               
                             
                             + 
                             
                               
                                 
                                   
                                     y 
                                     
                                       n 
                                       2 
                                     
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       t 
                                       
                                         k 
                                         - 
                                         L 
                                       
                                     
                                     ) 
                                   
                                 
                                 - 
                                 
                                   y 
                                   0 
                                 
                               
                               
                                 
                                   
                                     d 
                                     ^ 
                                   
                                   
                                     
                                       n 
                                       2 
                                     
                                     , 
                                     e 
                                   
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     t 
                                     
                                       k 
                                       - 
                                       L 
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                           ] 
                         
                       
                       ⁢ 
                       
                         
                           
                             η 
                             ^ 
                           
                           
                             
                               n 
                               1 
                             
                             , 
                             
                               n 
                               2 
                             
                             , 
                             e 
                           
                           C 
                         
                         ⁡ 
                         
                           ( 
                           
                             w 
                             0 
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   39 
                   ) 
                 
               
             
           
         
       
     
                         ∂       f   m     ⁡     (   w   )           ∂     z   e         ⁢     |     w   =     w   0           =           ∂         η   ^         n   1     ,     n   2     ,   e     C     ⁡     (   w   )           ∂     z   e         ⁢     |     w   =     w   0           =     j   ⁢           ⁢         2   ⁢   π   ⁢           ⁢   K       λ   e       ⁡     [             z     n   1       ⁡     (     t   k     )       -     z   0             d   ^         n   1     ,   e       ⁡     (     t   k     )         -           z     n   2       ⁡     (     t   k     )       -     z   0             d   ^         n   2     ,   e       ⁡     (     t   k     )         -           z     n   1       ⁡     (     t     k   -   L       )       -     z   0             d   ^         n   1     ,   e       ⁡     (     t     k   -   L       )         +           z     n   2       ⁡     (     t     k   -   L       )       -     z   0             d   ^         n   2     ,   e       ⁡     (     t     k   -   L       )           ]       ⁢         η   ^         n   1     ,     n   2     ,   e     C     ⁡     (     w   0     )                   (   40   )               
where {circumflex over (d)} n,e (t) is computed at w 0 .
 
     A cost function is formed from the squared error between the measured values and the modeled values:
 
 J ( w )=∥ h−f ( w )∥ 2   (41)
 
And solving for ∇ w J(w)=0 gives:
 
 ŵ=w   0 +( G ( w   0 ) H   G ( w   0 )) −1   G ( w   0 ) H ( h−f ( w   0 ))  (42)
 
     If the target emitter  30  is moving during the period of measurements, it is possible to extend the model to handle this case w=[x e  y e  z e  v xe  v ye  v ze ] T . With w 0 =[x 0  y 0  z 0  v x0  v y0  v z0 ] T , 
     
       
         
           
             
               
                 
                   
                     
                       
                         ∂ 
                         
                           
                             f 
                             m 
                           
                           ⁡ 
                           
                             ( 
                             w 
                             ) 
                           
                         
                       
                       
                         ∂ 
                         
                           x 
                           e 
                         
                       
                     
                     ⁢ 
                     
                       | 
                       
                         w 
                         = 
                         
                           w 
                           0 
                         
                       
                     
                   
                   = 
                   
                     
                       
                         
                           ∂ 
                           
                             
                               
                                 η 
                                 ^ 
                               
                               
                                 
                                   n 
                                   1 
                                 
                                 , 
                                 
                                   n 
                                   2 
                                 
                                 , 
                                 e 
                               
                               C 
                             
                             ⁡ 
                             
                               ( 
                               w 
                               ) 
                             
                           
                         
                         
                           ∂ 
                           
                             x 
                             e 
                           
                         
                       
                       ⁢ 
                       
                         | 
                         
                           w 
                           = 
                           
                             w 
                             0 
                           
                         
                       
                     
                     = 
                     
                       j 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
                           
                             2 
                             ⁢ 
                             π 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             K 
                           
                           
                             λ 
                             e 
                           
                         
                         ⁡ 
                         
                           [ 
                           
                             
                               
                                 
                                   
                                     x 
                                     
                                       n 
                                       1 
                                     
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       t 
                                       k 
                                     
                                     ) 
                                   
                                 
                                 - 
                                 
                                   x 
                                   0 
                                 
                                 - 
                                 
                                   
                                     v 
                                     x0 
                                   
                                   ⁢ 
                                   
                                     t 
                                     k 
                                   
                                 
                               
                               
                                 
                                   
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                                   ( 
                                   
                                     t 
                                     k 
                                   
                                   ) 
                                 
                               
                             
                             - 
                             
                               
                                 
                                   
                                     x 
                                     
                                       n 
                                       2 
                                     
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
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                                     ) 
                                   
                                 
                                 - 
                                 
                                   x 
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                                 - 
                                 
                                   
                                     v 
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                                   ⁢ 
                                   
                                     t 
                                     k 
                                   
                                 
                               
                               
                                 
                                   
                                     d 
                                     ^ 
                                   
                                   
                                     
                                       n 
                                       2 
                                     
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                                     e 
                                   
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     t 
                                     k 
                                   
                                   ) 
                                 
                               
                             
                             - 
                             
                               
                                 
                                   
                                     x 
                                     
                                       n 
                                       1 
                                     
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       t 
                                       
                                         k 
                                         - 
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                                     ) 
                                   
                                 
                                 - 
                                 
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                                   0 
                                 
                                 - 
                                 
                                   
                                     v 
                                     x0 
                                   
                                   ⁢ 
                                   
                                     t 
                                     
                                       k 
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                                     d 
                                     ^ 
                                   
                                   
                                     
                                       n 
                                       1 
                                     
                                     , 
                                     e 
                                   
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     t 
                                     
                                       k 
                                       - 
                                       L 
                                     
                                   
                                   ) 
                                 
                               
                             
                             + 
                             
                               
                                 
                                   
                                     x 
                                     
                                       n 
                                       2 
                                     
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       t 
                                       
                                         k 
                                         - 
                                         L 
                                       
                                     
                                     ) 
                                   
                                 
                                 - 
                                 
                                   x 
                                   0 
                                 
                                 - 
                                 
                                   
                                     v 
                                     x0 
                                   
                                   ⁢ 
                                   
                                     t 
                                     
                                       k 
                                       - 
                                       L 
                                     
                                   
                                 
                               
                               
                                 
                                   
                                     d 
                                     ^ 
                                   
                                   
                                     
                                       n 
                                       2 
                                     
                                     , 
                                     e 
                                   
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     t 
                                     
                                       k 
                                       - 
                                       L 
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                           ] 
                         
                       
                       ⁢ 
                       
                         
                           
                             η 
                             ^ 
                           
                           
                             
                               n 
                               1 
                             
                             , 
                             
                               n 
                               2 
                             
                             , 
                             e 
                           
                           C 
                         
                         ⁡ 
                         
                           ( 
                           
                             w 
                             0 
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   43 
                   ) 
                 
               
             
           
         
       
     
     
       
         
           
             
               
                 
                   
                     
                       
                         ∂ 
                         
                           
                             f 
                             m 
                           
                           ⁡ 
                           
                             ( 
                             w 
                             ) 
                           
                         
                       
                       
                         ∂ 
                         
                           y 
                           e 
                         
                       
                     
                     ⁢ 
                     
                       | 
                       
                         w 
                         = 
                         
                           w 
                           0 
                         
                       
                     
                   
                   = 
                   
                     
                       
                         
                           ∂ 
                           
                             
                               
                                 η 
                                 ^ 
                               
                               
                                 
                                   n 
                                   1 
                                 
                                 , 
                                 
                                   n 
                                   2 
                                 
                                 , 
                                 e 
                               
                               C 
                             
                             ⁡ 
                             
                               ( 
                               w 
                               ) 
                             
                           
                         
                         
                           ∂ 
                           
                             y 
                             e 
                           
                         
                       
                       ⁢ 
                       
                         | 
                         
                           w 
                           = 
                           
                             w 
                             0 
                           
                         
                       
                     
                     = 
                     
                       j 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
                           
                             2 
                             ⁢ 
                             π 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             K 
                           
                           
                             λ 
                             e 
                           
                         
                         ⁡ 
                         
                           [ 
                           
                             
                               
                                 
                                   
                                     y 
                                     
                                       n 
                                       1 
                                     
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       t 
                                       k 
                                     
                                     ) 
                                   
                                 
                                 - 
                                 
                                   y 
                                   0 
                                 
                                 - 
                                 
                                   
                                     v 
                                     y0 
                                   
                                   ⁢ 
                                   
                                     t 
                                     k 
                                   
                                 
                               
                               
                                 
                                   
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                                     ^ 
                                   
                                   
                                     
                                       n 
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                                     , 
                                     e 
                                   
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
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                                   ) 
                                 
                               
                             
                             - 
                             
                               
                                 
                                   
                                     y 
                                     
                                       n 
                                       2 
                                     
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
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                                   ⁢ 
                                   
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                                     ^ 
                                   
                                   
                                     
                                       n 
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                                     , 
                                     e 
                                   
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
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                                     k 
                                   
                                   ) 
                                 
                               
                             
                             - 
                             
                               
                                 
                                   
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                                       n 
                                       1 
                                     
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
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                                   ⁢ 
                                   
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                                     d 
                                     ^ 
                                   
                                   
                                     
                                       n 
                                       1 
                                     
                                     , 
                                     e 
                                   
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     t 
                                     
                                       k 
                                       - 
                                       L 
                                     
                                   
                                   ) 
                                 
                               
                             
                             + 
                             
                               
                                 
                                   
                                     y 
                                     
                                       n 
                                       2 
                                     
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
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                                         k 
                                         - 
                                         L 
                                       
                                     
                                     ) 
                                   
                                 
                                 - 
                                 
                                   y 
                                   0 
                                 
                                 - 
                                 
                                   
                                     v 
                                     y0 
                                   
                                   ⁢ 
                                   
                                     t 
                                     
                                       k 
                                       - 
                                       L 
                                     
                                   
                                 
                               
                               
                                 
                                   
                                     d 
                                     ^ 
                                   
                                   
                                     
                                       n 
                                       2 
                                     
                                     , 
                                     e 
                                   
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     t 
                                     
                                       k 
                                       - 
                                       L 
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                           ] 
                         
                       
                       ⁢ 
                       
                         
                           
                             η 
                             ^ 
                           
                           
                             
                               n 
                               1 
                             
                             , 
                             
                               n 
                               2 
                             
                             , 
                             e 
                           
                           C 
                         
                         ⁡ 
                         
                           ( 
                           
                             w 
                             0 
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   44 
                   ) 
                 
               
             
           
         
       
     
     
       
         
           
             
               
                 
                   
                     
                       
                         ∂ 
                         
                           
                             f 
                             m 
                           
                           ⁡ 
                           
                             ( 
                             w 
                             ) 
                           
                         
                       
                       
                         ∂ 
                         
                           z 
                           e 
                         
                       
                     
                     ⁢ 
                     
                       | 
                       
                         w 
                         = 
                         
                           w 
                           0 
                         
                       
                     
                   
                   = 
                   
                     
                       
                         
                           ∂ 
                           
                             
                               
                                 η 
                                 ^ 
                               
                               
                                 
                                   n 
                                   1 
                                 
                                 , 
                                 
                                   n 
                                   2 
                                 
                                 , 
                                 e 
                               
                               C 
                             
                             ⁡ 
                             
                               ( 
                               w 
                               ) 
                             
                           
                         
                         
                           ∂ 
                           
                             z 
                             e 
                           
                         
                       
                       ⁢ 
                       
                         | 
                         
                           w 
                           = 
                           
                             w 
                             0 
                           
                         
                       
                     
                     = 
                     
                       j 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
                           
                             2 
                             ⁢ 
                             π 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             K 
                           
                           
                             λ 
                             e 
                           
                         
                         ⁡ 
                         
                           [ 
                           
                             
                               
                                 
                                   
                                     z 
                                     
                                       n 
                                       1 
                                     
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       t 
                                       k 
                                     
                                     ) 
                                   
                                 
                                 - 
                                 
                                   z 
                                   0 
                                 
                                 - 
                                 
                                   
                                     v 
                                     z0 
                                   
                                   ⁢ 
                                   
                                     t 
                                     k 
                                   
                                 
                               
                               
                                 
                                   
                                     d 
                                     ^ 
                                   
                                   
                                     
                                       n 
                                       1 
                                     
                                     , 
                                     e 
                                   
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     t 
                                     k 
                                   
                                   ) 
                                 
                               
                             
                             - 
                             
                               
                                 
                                   
                                     x 
                                     
                                       n 
                                       2 
                                     
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
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                                       k 
                                     
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                                 - 
                                 
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                                   0 
                                 
                                 - 
                                 
                                   
                                     v 
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                                   ⁢ 
                                   
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                                     k 
                                   
                                 
                               
                               
                                 
                                   
                                     d 
                                     ^ 
                                   
                                   
                                     
                                       n 
                                       2 
                                     
                                     , 
                                     e 
                                   
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     t 
                                     k 
                                   
                                   ) 
                                 
                               
                             
                             - 
                             
                               
                                 
                                   
                                     z 
                                     
                                       n 
                                       1 
                                     
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
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                                         k 
                                         - 
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                                     ) 
                                   
                                 
                                 - 
                                 
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                                   0 
                                 
                                 - 
                                 
                                   
                                     v 
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                                   ⁢ 
                                   
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                                       k 
                                       - 
                                       L 
                                     
                                   
                                 
                               
                               
                                 
                                   
                                     d 
                                     ^ 
                                   
                                   
                                     
                                       n 
                                       1 
                                     
                                     , 
                                     e 
                                   
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     t 
                                     
                                       k 
                                       - 
                                       L 
                                     
                                   
                                   ) 
                                 
                               
                             
                             + 
                             
                               
                                 
                                   
                                     z 
                                     
                                       n 
                                       2 
                                     
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       t 
                                       
                                         k 
                                         - 
                                         L 
                                       
                                     
                                     ) 
                                   
                                 
                                 - 
                                 
                                   z 
                                   0 
                                 
                                 - 
                                 
                                   
                                     v 
                                     z0 
                                   
                                   ⁢ 
                                   
                                     t 
                                     
                                       k 
                                       - 
                                       L 
                                     
                                   
                                 
                               
                               
                                 
                                   
                                     d 
                                     ^ 
                                   
                                   
                                     
                                       n 
                                       2 
                                     
                                     , 
                                     e 
                                   
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     t 
                                     
                                       k 
                                       - 
                                       L 
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                           ] 
                         
                       
                       ⁢ 
                       
                         
                           
                             η 
                             ^ 
                           
                           
                             
                               n 
                               1 
                             
                             , 
                             
                               n 
                               2 
                             
                             , 
                             e 
                           
                           C 
                         
                         ⁡ 
                         
                           ( 
                           
                             w 
                             0 
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   45 
                   ) 
                 
               
             
           
         
       
     
     
       
         
           
             
               
                 
                   
                     
                       
                         ∂ 
                         
                           
                             f 
                             m 
                           
                           ⁡ 
                           
                             ( 
                             w 
                             ) 
                           
                         
                       
                       
                         ∂ 
                         
                           v 
                           
                             x 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             e 
                           
                         
                       
                     
                     ⁢ 
                     
                       
                         
                            
                           
                             w 
                             = 
                             
                               w 
                               0 
                             
                           
                         
                         ⁢ 
                         
                           = 
                           
                             
                               ∂ 
                               
                                 
                                   
                                     η 
                                     ^ 
                                   
                                   
                                     
                                       n 
                                       1 
                                     
                                     , 
                                     
                                       
                                         n 
                                         2 
                                       
                                       ⁢ 
                                       e 
                                     
                                   
                                   C 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   w 
                                   ) 
                                 
                               
                             
                             
                               ∂ 
                               
                                 v 
                                 xe 
                               
                             
                           
                         
                          
                       
                       
                         w 
                         = 
                         
                           w 
                           0 
                         
                       
                     
                   
                   = 
                   
                     j 
                     ⁢ 
                     
                       
                         
                           2 
                           ⁢ 
                           π 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           K 
                         
                         
                           λ 
                           e 
                         
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             
                               t 
                               k 
                             
                             ⁢ 
                             
                               
                                 
                                   
                                     x 
                                     
                                       n 
                                       1 
                                     
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       t 
                                       k 
                                     
                                     ) 
                                   
                                 
                                 - 
                                 
                                   x 
                                   0 
                                 
                                 - 
                                 
                                   
                                     v 
                                     
                                       x 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       0 
                                     
                                   
                                   ⁢ 
                                   
                                     t 
                                     k 
                                   
                                 
                               
                               
                                 
                                   
                                     d 
                                     ^ 
                                   
                                   
                                     
                                       n 
                                       1 
                                     
                                     , 
                                     e 
                                   
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     t 
                                     k 
                                   
                                   ) 
                                 
                               
                             
                           
                           - 
                           
                             
                               t 
                               k 
                             
                             ⁢ 
                             
                               
                                 
                                   
                                     x 
                                     
                                       n 
                                       2 
                                     
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       t 
                                       k 
                                     
                                     ) 
                                   
                                 
                                 - 
                                 
                                   x 
                                   0 
                                 
                                 - 
                                 
                                   
                                     v 
                                     
                                       x 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       0 
                                     
                                   
                                   ⁢ 
                                   
                                     t 
                                     k 
                                   
                                 
                               
                               
                                 
                                   
                                     d 
                                     ^ 
                                   
                                   
                                     
                                       n 
                                       2 
                                     
                                     , 
                                     e 
                                   
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     t 
                                     k 
                                   
                                   ) 
                                 
                               
                             
                           
                           - 
                           
                             
                               t 
                               
                                 k 
                                 - 
                                 L 
                               
                             
                             ⁢ 
                             
                               
                                 
                                   
                                     x 
                                     
                                       n 
                                       1 
                                     
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       t 
                                       
                                         k 
                                         - 
                                         L 
                                       
                                     
                                     ) 
                                   
                                 
                                 - 
                                 
                                   x 
                                   0 
                                 
                                 - 
                                 
                                   
                                     v 
                                     
                                       x 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       0 
                                     
                                   
                                   ⁢ 
                                   
                                     t 
                                     
                                       k 
                                       - 
                                       L 
                                     
                                   
                                 
                               
                               
                                 
                                   
                                     d 
                                     ^ 
                                   
                                   
                                     
                                       n 
                                       1 
                                     
                                     , 
                                     e 
                                   
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     t 
                                     
                                       k 
                                       - 
                                       L 
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                           + 
                           
                             
                               t 
                               
                                 k 
                                 - 
                                 L 
                               
                             
                             ⁢ 
                             
                               
                                 
                                   
                                     x 
                                     
                                       n 
                                       2 
                                     
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       t 
                                       
                                         k 
                                         - 
                                         L 
                                       
                                     
                                     ) 
                                   
                                 
                                 - 
                                 
                                   x 
                                   0 
                                 
                                 - 
                                 
                                   
                                     v 
                                     
                                       x 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       0 
                                     
                                   
                                   ⁢ 
                                   
                                     t 
                                     
                                       k 
                                       - 
                                       L 
                                     
                                   
                                 
                               
                               
                                 
                                   
                                     d 
                                     ^ 
                                   
                                   
                                     
                                       n 
                                       2 
                                     
                                     , 
                                     e 
                                   
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     t 
                                     
                                       k 
                                       - 
                                       L 
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                         
                         ] 
                       
                     
                     ⁢ 
                     
                       
                         
                           η 
                           ^ 
                         
                         
                           
                             n 
                             1 
                           
                           , 
                           
                             n 
                             2 
                           
                           , 
                           e 
                         
                         C 
                       
                       ⁡ 
                       
                         ( 
                         
                           w 
                           0 
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   46 
                   ) 
                 
               
             
           
         
       
     
     
       
         
           
             
               
                 
                   
                     
                       
                         ∂ 
                         
                           
                             f 
                             m 
                           
                           ⁡ 
                           
                             ( 
                             w 
                             ) 
                           
                         
                       
                       
                         ∂ 
                         
                           v 
                           
                             y 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             e 
                           
                         
                       
                     
                     ⁢ 
                     
                       
                         
                            
                           
                             w 
                             = 
                             
                               w 
                               0 
                             
                           
                         
                         ⁢ 
                         
                           = 
                           
                             
                               ∂ 
                               
                                 
                                   
                                     η 
                                     ^ 
                                   
                                   
                                     
                                       n 
                                       1 
                                     
                                     , 
                                     
                                       
                                         n 
                                         2 
                                       
                                       ⁢ 
                                       e 
                                     
                                   
                                   C 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   w 
                                   ) 
                                 
                               
                             
                             
                               ∂ 
                               
                                 v 
                                 ye 
                               
                             
                           
                         
                          
                       
                       
                         w 
                         = 
                         
                           w 
                           0 
                         
                       
                     
                   
                   = 
                   
                     j 
                     ⁢ 
                     
                       
                         
                           2 
                           ⁢ 
                           π 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           K 
                         
                         
                           λ 
                           e 
                         
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             
                               t 
                               k 
                             
                             ⁢ 
                             
                               
                                 
                                   
                                     y 
                                     
                                       n 
                                       1 
                                     
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       t 
                                       k 
                                     
                                     ) 
                                   
                                 
                                 - 
                                 
                                   y 
                                   0 
                                 
                                 - 
                                 
                                   
                                     v 
                                     
                                       y 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       0 
                                     
                                   
                                   ⁢ 
                                   
                                     t 
                                     k 
                                   
                                 
                               
                               
                                 
                                   
                                     d 
                                     ^ 
                                   
                                   
                                     
                                       n 
                                       1 
                                     
                                     , 
                                     e 
                                   
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     t 
                                     k 
                                   
                                   ) 
                                 
                               
                             
                           
                           - 
                           
                             
                               t 
                               k 
                             
                             ⁢ 
                             
                               
                                 
                                   
                                     y 
                                     
                                       n 
                                       2 
                                     
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       t 
                                       k 
                                     
                                     ) 
                                   
                                 
                                 - 
                                 
                                   y 
                                   0 
                                 
                                 - 
                                 
                                   
                                     v 
                                     
                                       y 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       0 
                                     
                                   
                                   ⁢ 
                                   
                                     t 
                                     k 
                                   
                                 
                               
                               
                                 
                                   
                                     d 
                                     ^ 
                                   
                                   
                                     
                                       n 
                                       2 
                                     
                                     , 
                                     e 
                                   
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     t 
                                     k 
                                   
                                   ) 
                                 
                               
                             
                           
                           - 
                           
                             
                               t 
                               
                                 k 
                                 - 
                                 L 
                               
                             
                             ⁢ 
                             
                               
                                 
                                   
                                     y 
                                     
                                       n 
                                       1 
                                     
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       t 
                                       
                                         k 
                                         - 
                                         L 
                                       
                                     
                                     ) 
                                   
                                 
                                 - 
                                 
                                   y 
                                   0 
                                 
                                 - 
                                 
                                   
                                     v 
                                     
                                       y 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       0 
                                     
                                   
                                   ⁢ 
                                   
                                     t 
                                     
                                       k 
                                       - 
                                       L 
                                     
                                   
                                 
                               
                               
                                 
                                   
                                     d 
                                     ^ 
                                   
                                   
                                     
                                       n 
                                       1 
                                     
                                     , 
                                     e 
                                   
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     t 
                                     
                                       k 
                                       - 
                                       L 
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                           + 
                           
                             
                               t 
                               
                                 k 
                                 - 
                                 L 
                               
                             
                             ⁢ 
                             
                               
                                 
                                   
                                     y 
                                     
                                       n 
                                       2 
                                     
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       t 
                                       
                                         k 
                                         - 
                                         L 
                                       
                                     
                                     ) 
                                   
                                 
                                 - 
                                 
                                   y 
                                   0 
                                 
                                 - 
                                 
                                   
                                     v 
                                     
                                       y 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       0 
                                     
                                   
                                   ⁢ 
                                   
                                     t 
                                     
                                       k 
                                       - 
                                       L 
                                     
                                   
                                 
                               
                               
                                 
                                   
                                     d 
                                     ^ 
                                   
                                   
                                     
                                       n 
                                       2 
                                     
                                     , 
                                     e 
                                   
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     t 
                                     
                                       k 
                                       - 
                                       L 
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                         
                         ] 
                       
                     
                     ⁢ 
                     
                       
                         
                           η 
                           ^ 
                         
                         
                           
                             n 
                             1 
                           
                           , 
                           
                             n 
                             2 
                           
                           , 
                           e 
                         
                         C 
                       
                       ⁡ 
                       
                         ( 
                         
                           w 
                           0 
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   47 
                   ) 
                 
               
             
           
         
       
     
                         ∂       f   m     ⁡     (   w   )           ∂     v     z   ⁢           ⁢   e           ⁢              w   =     w   0         ⁢     =       ∂         η   ^         n   1     ,       n   2     ⁢   e       C     ⁡     (   w   )           ∂     v   ze                  w   =     w   0           =     j   ⁢         2   ⁢   π   ⁢           ⁢   K       λ   e       ⁡     [         t   k     ⁢           z     n   1       ⁡     (     t   k     )       -     z   0     -       v     z   ⁢           ⁢   0       ⁢     t   k               d   ^         n   1     ,   e       ⁡     (     t   k     )           -       t   k     ⁢           z     n   2       ⁡     (     t   k     )       -     z   0     -       v     z   ⁢           ⁢   0       ⁢     t   k               d   ^         n   2     ,   e       ⁡     (     t   k     )           -       t     k   -   L       ⁢           z     n   1       ⁡     (     t     k   -   L       )       -     z   0     -       v     z   ⁢           ⁢   0       ⁢     t     k   -   L                 d   ^         n   1     ,   e       ⁡     (     t     k   -   L       )           +       t     k   -   L       ⁢           z     n   2       ⁡     (     t     k   -   L       )       -     z   0     -       v     z   ⁢           ⁢   0       ⁢     t     k   -   L                 d   ^         n   2     ,   e       ⁡     (     t     k   -   L       )             ]       ⁢         η   ^         n   1     ,     n   2     ,   e     C     ⁡     (     w   0     )                 (   48   )               
These values are substituted into EQ. 37 and EQ. 42 to solve for the target emitter  30  parameters.
 
     The mean value of the solution computed using EQ. 42 is: 
                           E   ⁡     [     w   ^     ]       =       ⁢     E   ⁡     [       W   0     +         (       G   H     ⁢   G     )       -   1       ⁢       G   H     ⁡     (     h   -     f   ⁡     (     w   0     )         )           ]                   =       ⁢     E   ⁡     [       w   0     +         (       G   H     ⁢   G     )       -   1       ⁢       G   H     ⁡     (       G   ⁡     (     w   -     w   0       )       +   n     )           ]                   =       ⁢     E   ⁡     [     w   +         (       G   H     ⁢   G     )       -   1       ⁢     G   H     ⁢   n       ]                   =       ⁢     w   +         (       G   H     ⁢   G     )       -   1       ⁢     G   H     ⁢     E   ⁡     [   n   ]                       =       ⁢   w                 (   49   )               
The error covariance can be computed using:
 
                         P   =       ⁢     E   ⁡     [       (       w   ^     -   w     )     ⁢       (       w   ^     -   w     )     H       ]                   =       ⁢     E   ⁡     [         (       G   H     ⁢   G     )       -   1       ⁢     G   H     ⁢     nn   H     ⁢       G   ⁡     (       G   H     ⁢   G     )         -   1         ]                   =       ⁢         (       G   H     ⁢   G     )       -   1       ⁢     G   H     ⁢       NG   ⁡     (       G   H     ⁢   G     )         -   1                       (   50   )               
where N=E[nn H ].
 
     Depending on the set of variables selected for the w, the solution can be found in two or three dimensions, and for stationary or moving target emitters  30  as discussed above. If w=[x y] T , then the covariance matrix can be interpreted as providing information about the error ellipse. If the eigenvalues of P are {λ n }, then the major and minor axes of the error ellipse are 
                           m   0     =       )     -     ⁢   2   ⁢     ln   ⁡     (     1   -     P   e       )       ⁢     λ   max       _                   m   1     =       )     -     ⁢   2   ⁢     ln   ⁡     (     1   -     P   e       )       ⁢     λ   min       _                   (   51   )               
And the rotation of the error ellipse is:
 
     
       
         
           
             
               
                 
                   θ 
                   = 
                   
                     { 
                     
                       
                         
                           
                             
                               1 
                               2 
                             
                             ⁢ 
                             
                               
                                 tan 
                                 
                                   - 
                                   1 
                                 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   
                                     2 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       P 
                                       
                                         1 
                                         , 
                                         2 
                                       
                                     
                                   
                                   
                                     
                                       P 
                                       
                                         1 
                                         , 
                                         1 
                                       
                                     
                                     - 
                                     
                                       P 
                                       
                                         2 
                                         , 
                                         2 
                                       
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                         
                         
                           
                             
                               P 
                               
                                 2 
                                 , 
                                 2 
                               
                             
                             ≤ 
                             
                               P 
                               
                                 1 
                                 , 
                                 1 
                               
                             
                           
                         
                       
                       
                         
                           
                             
                               
                                 1 
                                 2 
                               
                               ⁢ 
                               
                                 
                                   tan 
                                   
                                     - 
                                     1 
                                   
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     
                                       2 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       
                                         P 
                                         
                                           1 
                                           , 
                                           2 
                                         
                                       
                                     
                                     
                                       
                                         P 
                                         
                                           1 
                                           , 
                                           1 
                                         
                                       
                                       - 
                                       
                                         P 
                                         
                                           2 
                                           , 
                                           2 
                                         
                                       
                                     
                                   
                                   ) 
                                 
                               
                             
                             - 
                             
                               π 
                               / 
                               2 
                             
                           
                         
                         
                           
                             
                               P 
                               
                                 2 
                                 , 
                                 2 
                               
                             
                             &gt; 
                             
                               P 
                               
                                 1 
                                 , 
                                 1 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   52 
                   ) 
                 
               
             
           
         
       
     
     Referring again to  FIG. 1B  and  FIGS. 2A-2B , in embodiments in which a receiver node  10 ,  10 -N includes at least a first tuner  250  a second tuner  255 , and where a target emitter  30  and a reference emitter  20  are transmitting signals in widely separated bandwidths, such as when the reference emitter is a wide-band signal transmitter and the target emitter is a narrow-band signal transmitter. As previously described, it may not be possible to use a single tuner to receive and process both signals, and separate tuners may be used to pick up signals in each band. For any one tuner, with respect to measurements of reference emitter  20 , EQ. 26 may be written as: 
                         δ   ^         n   1     ,     n   2     ,     e   2     ,   A       ⁡     (     t   k     )       =     K   ⁢           ⁢     exp   ⁡     [     j   ⁡     (         -       2   ⁢   π       λ     e   2           ⁢     (         d       n   1     ,     e   2         ⁡     (     t   k     )       -       d       n   2     ,     e   2         ⁡     (     t   k     )         )       +       α       n   1     ,   A       ⁡     (     t   k     )       -       α       n   2     ,   A       ⁡     (     t   k     )         )       ]                 (   53   )               
where the notation is extended to indicate that the measurements were made on a particular tuner, such as first tuner  250 . Since the reference emitter is at a known location, the following can be formed:
 
                               δ   ^         n   1     ,     n   2     ,     e   2     ,   A     C     ⁡     (     t   k     )       =       ⁢           δ   ^         n   1     ,     n   2     ,     e   2     ,   A       ⁡     (     t   k     )       ·                     ⁢     exp   ⁡     [     j   ⁢       2   ⁢   π       λ     e   1         ⁢     (           d   ^         n   1     ,     e   2         ⁡     (     t   k     )       -         d   ^         n   2     ,     e   2         ⁡     (     t   k     )         )       ]                   ≈       ⁢     K   ⁢           ⁢     exp   ⁢           [     j   ⁢     (         α       n   1     ,   A       ⁡     (     t   k     )       -       α       n   2     ,   A       ⁡     (     t   k     )         )       ]                     (   54   )               
The measurement of the target emitter  30  on the tuner is:
 
                         δ   ^         n   1     ,     n   2     ,     e   2     ,   B       ⁡     (     t   k     )       =     K   ⁢           ⁢     exp   ⁡     [     j   ⁡     (         -       2   ⁢   π       λ     e   1           ⁢     (         d       n   1     ,     e   1         ⁡     (     t   k     )       -       d       n   2     ,     e   1         ⁡     (     t   k     )         )       +       α       n   1     ,   B       ⁡     (     t   k     )       -       α       n   2     ,   B       ⁡     (     t   k     )         )       ]                 (   55   )               
It may be assumed that each tuner is phase-locked to the reference oscillator  260 . Then the instantaneous RF phase of the local oscillator for tuner s is modeled as:
 
                       ψ   s     ⁡     (   t   )       =         f     c   ,   s         f   ref       ×       ψ   ref     ⁡     (   t   )                 (   56   )               
where f c,s  is the center frequency for tuner s.
 
     Based on the model in EQ. 56 the measured residual phase on first tuner  250  can be measured to estimate the residual phase on second tuner  255  due to the reference oscillator and remove its effect. After performing this correction, the remaining phase variation on second tuner  255  is 
                     Δ   ⁢           ⁢       ψ   B     ⁡     (   t   )         =         ψ   B     ⁡     (   t   )       -         f     c   ,   B         f     c   ,   A         ×       ψ   A     ⁡     (   t   )                   (   57   )               
For the measurements above, after computing this difference, the mean residual phase slope of Δψ B (t) is less than 10 −4  Hz for both of the radios. This suggests that a major contributor to the residual phase variation on each slice is due to the reference oscillator  260 , and that the phase measurements from first tuner  250  can be used to characterize and remove the residual phase variation on second tuner  255 .
 
     A modified version of the metric given by EQ. 27 may be formed: 
                               ρ   ^         n   1     ,     n   2     ,     e   1       C     ⁡     (     t   k     )       =       ⁢           δ   ^         n   1     ,     n   2     ,     e   1     ,   B       ⁡     (     t   k     )       ·       (         δ   ^         n   1     ,     n   2     ,       e   2     ⁢   A         C   *       ⁡     (     t   k     )       )         f   B     /     f   A                       ≈       ⁢     K   ⁢           ⁢     exp   [     j   (         -       2   ⁢   π       λ     e   1           ⁢     (         d       n   1     ,     e   1         ⁡     (     t   k     )       -       d       n   2     ,     e   1         ⁡     (     t   k     )         )       +                               ⁢         α       n   1     ,   B       ⁡     (     t   k     )       -       α       n   2     ,   B       ⁡     (     t   k     )         )     ]     ×                   ⁢     exp   ⁡     [       -   j     ⁢       f     c   ,   B         f     c   ,   A         ⁢     (         α       n   1     ,   A       ⁡     (     t   k     )       -       α       n   2     ,   A       ⁡     (     t   k     )         )       ]                   ≈       ⁢     K   ⁢           ⁢     exp   [     -     j   (           2   ⁢   π       λ     e   1         ⁢     (         d       n   1     ,     e   1         ⁡     (     t   k     )       -       d       n   2     ,     e   1         ⁡     (     t   k     )         )       +                             ⁢         α       n   1     ,   B       ⁡     (     t   k     )       -       α       n   2     ,   B       ⁡     (     t   k     )       -         f     c   ,   B         f     c   ,   A         ⁢     (         α       n   1     ,   A       ⁡     (     t   k     )       -                               ⁢       α       n   2     ,   A       ⁡     (     t   k     )       )     )     ]                 (   58   )               
Using EQ. 57,
 
                       α     n   ,   s       ⁡     (     t   k     )       =         f     c   ,   n         f   Ref       ⁢       α     n   ,   Ref       ⁡     (     t   k     )                 (   59   )               
and
 
                         α       n   1     ,   B       ⁡     (     t   k     )       -       α       n   2     ,   B       ⁡     (     t   k     )       -         f     c   ,   B         f     c   ,   A         ⁢     (         α       n   1     ,   A       ⁡     (     t   k     )       -       α       n   2     ,   A       ⁢     (     t   k     )         )         =             f     c   ,   B         f   Ref       ⁢       α       n   1     ,   Ref       ⁡     (     t   k     )         -         f     c   ,   B         f   Ref       ⁢       α       n   2     ,   Ref       ⁡     (     t   k     )         -         f     c   ,   B         f     c   ,   A         ⁢     (           f     c   ,   A         f   Ref       ⁢       α       n   1     ,   Ref       ⁡     (     t   k     )         -         f     c   ,   A         f   Ref       ⁢       α       n   2     ,   Ref       ⁡     (     t   k     )           )         =             f     c   ,   B         f   Ref       ⁢       α       n   1     ,   Ref       ⁡     (     t   k     )         -         f     c   ,   B         f   Ref       ⁢       α       n   2     ,   Ref       ⁡     (     t   k     )         -         f     c   ,   B         f   Ref       ⁢       α       n   1     ,   Ref       ⁡     (     t   k     )         +         f     c   ,   B         f   Ref       ⁢       α       n   2     ,   Ref       ⁡     (     t   k     )           =   0               (   60   )               
Substituting above,
 
                         ρ   ^         n   1     ,     n   2     ,     e   1     ,     C     ⁡     (     t   k     )       ≈     K   ⁢           ⁢     exp   ⁡     [     -     j   ⁡     (         2   ⁢   π       λ     e   1         ⁢     (         d       n   1     ,     e   1         ⁡     (     t   k     )       -       d       n   2     ,     e   1         ⁡     (     t   k     )         )       )         ]                 (   61   )               
Then EQ. 58 can be used in EQ. 32 to find the triple difference metric.
 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             
                               η 
                               ^ 
                             
                             
                               
                                 n 
                                 1 
                               
                               , 
                               
                                 n 
                                 2 
                               
                               , 
                               
                                 e 
                                 1 
                               
                             
                             C 
                           
                           ⁡ 
                           
                             ( 
                             
                               
                                 t 
                                 k 
                               
                               , 
                               
                                 t 
                                 
                                   k 
                                   - 
                                   L 
                                 
                               
                             
                             ) 
                           
                         
                         = 
                           
                         ⁢ 
                         
                           
                             
                               
                                 ρ 
                                 ^ 
                               
                               
                                 
                                   n 
                                   1 
                                 
                                 , 
                                 
                                   n 
                                   2 
                                 
                                 , 
                                 
                                   e 
                                   1 
                                 
                               
                               c 
                             
                             ⁡ 
                             
                               ( 
                               
                                 t 
                                 k 
                               
                               ) 
                             
                           
                           × 
                           
                             
                               
                                 ρ 
                                 ^ 
                               
                               
                                 
                                   n 
                                   1 
                                 
                                 , 
                                 
                                   n 
                                   2 
                                 
                                 , 
                                 
                                   e 
                                   1 
                                 
                               
                               
                                 c 
                                 * 
                               
                             
                             ⁡ 
                             
                               ( 
                               
                                 t 
                                 
                                   k 
                                   - 
                                   L 
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         ≈ 
                           
                         ⁢ 
                         
                           K 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             exp 
                             [ 
                             
                               
                                 - 
                                 j 
                               
                               ⁢ 
                               
                                 
                                   2 
                                   ⁢ 
                                   π 
                                 
                                 
                                   λ 
                                   
                                     e 
                                     1 
                                   
                                 
                               
                               ⁢ 
                               
                                 ( 
                                 
                                   
                                     
                                       d 
                                       
                                         
                                           n 
                                           1 
                                         
                                         , 
                                         
                                           e 
                                           1 
                                         
                                       
                                     
                                     ⁡ 
                                     
                                       ( 
                                       
                                         t 
                                         k 
                                       
                                       ) 
                                     
                                   
                                   - 
                                   
                                     
                                       d 
                                       
                                         
                                           n 
                                           2 
                                         
                                         , 
                                         
                                           e 
                                           1 
                                         
                                       
                                     
                                     ⁢ 
                                     
                                       ( 
                                       
                                         t 
                                         k 
                                       
                                       ) 
                                     
                                   
                                   - 
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         
                             
                           ⁢ 
                           
                             
                               
                                 d 
                                 
                                   
                                     n 
                                     1 
                                   
                                   , 
                                   
                                     e 
                                     1 
                                   
                                 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   t 
                                   
                                     k 
                                     - 
                                     L 
                                   
                                 
                                 ) 
                               
                             
                             + 
                             
                               
                                 d 
                                 
                                   
                                     n 
                                     2 
                                   
                                   , 
                                   
                                     e 
                                     1 
                                   
                                 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   t 
                                   
                                     k 
                                     - 
                                     L 
                                   
                                 
                                 ) 
                               
                             
                           
                           ) 
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   62 
                   ) 
                 
               
             
           
         
       
     
     As will be appreciated by one skilled in the art, one or more aspects as described herein may be embodied as a computer program product. Accordingly, one or more aspects described herein may take the form of an entirely hardware embodiment, an entirely software embodiment (including firmware, resident software, micro-code, etc.) or an embodiment combining software and hardware aspects that may all generally be referred to herein as a “circuit,” such as a signal processing circuit or a digital signal processing circuit. Furthermore, aspects of the disclosed method may take the form of a computer program product embodied in one or more computer readable medium(s) having computer readable program code embodied thereon. A computer readable storage medium may be, for example, but not limited to, an electronic, magnetic, optical, electromagnetic, infrared or semiconductor system, apparatus, or device, or any suitable combination of the foregoing. More specific examples (a non-exhaustive list) of the computer readable storage medium include the following: an electrical connection having one or more wires, a portable computer diskette, a hard disk, a random access memory (RAM), a read-only memory (ROM), an erasable programmable read-only memory (EPROM or Flash memory), an optical fiber, a portable compact disc read-only memory (CD-ROM), an optical storage device, a magnetic storage device, or any suitable combination of the foregoing. In the context of this document, a computer readable storage medium may be any tangible medium that can contain or store a program for use by or in connection with one or more processors. One or more processors as set forth herein can execute computer program code for performance of one or more methods as set forth herein, e.g. method  300 . 
       FIG. 4  depicts one example of a computer program product  400  that may include, for instance, one or more computer readable storage media  402  to store computer readable program code  404  stored on one or more computer readable media  402  thereon to provide and facilitate one or more aspects as described herein. Computer readable program code  404  may be executed by one or more processors for performance of one or more functions of method  300 . Program code herein may be regarded as program logic. 
     Computer readable program code embodied on a computer readable medium, for instance a computer program product of  FIG. 4 , may be transmitted using an appropriate medium, including but not limited to wireless, wireline, optical fiber cable, radio frequency, etc., or any suitable combination of the foregoing. Any of the foregoing may be or be included in an input communications interface  280 . The computer program code for execution by one or more processors for carrying out operations for aspects described herein, e.g. method  300 , may be written in any combination of one or more programming languages. Some of the program code may be stored on computer readable storage media  402  included in one or more memory(ies) of one or more processor systems  223  of a digital signal processing circuit  220 , as described in  FIG. 1C  and  FIG. 1E . In one embodiment, a processor or processor system  223  of node  40  can execute program code stored on a memory of processor system  223  of node  40  for performance of a method, e.g. method  300 . In one embodiment of an exemplary distributive processing system, a first processor of processor system  223  of a first node can execute program code stored on a memory of processor system  223  of the first node for performance of a first subset of functions or methods, e.g. of method  300 , and a second processor or processing system  223  of a second node external to the first node can execute program code stored on a memory of a processing system  223  of the second node for performance of a second subset of functions of the method, e.g. method  300 . In one embodiment the program code executed by processors of external nodes can be stored on a computer readable storage medium  404  external to either of first or second nodes. Program code, prior to execution, can be transmitted from such computer readable medium to a memory of one or more of nodes  40 ,  10 ,  10 -N, and/or  20  for execution by a processor of the one or more node. In one embodiment one or more processors for execution of program code stored on one or more computer readable media can, for example, include parallel processors, e.g., bus connected parallel processors of a first node. 
     Approximating language, as used herein throughout the specification and claims, may be applied to modify any quantitative representation that could permissibly vary without resulting in a change in the basic function to which it is related. Accordingly, a value modified by a term or terms, such as “about,” is not limited to the precise value specified. In some instances, the approximating language may correspond to the precision of an instrument for measuring the value. 
     The terminology used herein is for the purpose of describing particular examples only and is not intended to be limiting of the invention. As used herein, the singular forms “a”, “an” and “the” are intended to include the plural forms as well, unless the context clearly indicates otherwise. It will be further understood that the terms “comprise” (and any form of comprise, such as “comprises” and “comprising”), “have” (and any form of have, such as “has” and “having”), “include (and any form of include, such as “includes” and “including”), and “contain” (and any form of contain, such as “contains” and “containing”) are open-ended linking verbs. As a result, a method or device that “comprises,” “has,” “includes” or “contains” one or more steps or elements possesses those one or more steps or elements, but is not limited to possessing only those one or more steps or elements. Likewise, a step of a method or an element of a device that “comprises,” “has,” “includes” or “contains” one or more features possesses those one or more features, but is not limited to possessing only those one or more features. 
     As used herein, the terms “may” and “may be” indicate a possibility of an occurrence within a set of circumstances; a possession of a specified property, characteristic or function; and/or qualify another verb by expressing one or more of an ability, capability, or possibility associated with the qualified verb. Accordingly, usage of “may” and “may be” indicates that a modified term is apparently appropriate, capable, or suitable for an indicated capacity, function, or usage, while taking into account that in some circumstances the modified term may sometimes not be appropriate, capable or suitable. For example, in some circumstances, an event or capacity can be expected, while in other circumstances the event or capacity cannot occur—this distinction is captured by the terms “may” and “may be.” 
     While several aspects have been described and depicted herein, alternative aspects may be effected by those skilled in the art to accomplish the same objectives. Accordingly, it is intended by the appended claims to cover all such alternative aspects as fall within the true spirit and scope of the invention.