Abstract:
Apparatus having corresponding methods and computer programs comprise a wireless receiver to receive a Very High Frequency (VHF) Omni-directional Radio Range (VOR) signal comprising a frequency-modulated (FM) component; an analog-to-digital converter to generate a digital signal based on the VOR signal, the digital signal comprising data representing the FM component; and a FM phase circuit comprising a correlator to generate a correlation peak based on the data representing the FM component and an ideal representation of the FM component, and a peak detector to determine a phase of the FM component based on the correlation peak.

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     This application is a divisional of U.S. patent application Ser. No. 11/535,485 filed Sep. 27, 2006, which claims benefit of U.S. Provisional Patent Application Ser. No. 60/721,562 filed Sep. 28, 2005, and which claims benefit of U.S. Provisional Patent Application Ser. No. 60/726,510 filed Oct. 13, 2005, and which claims benefit of U.S. Provisional Patent Application Ser. No. 60/748,331 filed Dec. 7, 2005, the disclosures thereof incorporated by reference herein in their entirety. 
    
    
     BACKGROUND 
     The present invention relates generally to position determination. More particularly, the present invention relates to precise position determination using Very High Frequency (VHF) Omni-directional Radio Range (VOR) signals. 
     The VOR signal is a broadcast signal currently used for radionavigation. The VOR signal is a VHF radio signal that encodes both the identity of the VOR transmitter and the azimuth defined by a line extending from the VOR transmitter to a receiver relative to magnetic north. 
     SUMMARY 
     In general, in one aspect, the invention features an apparatus comprising: a wireless receiver to receive a Very High Frequency (VHF) Omni-directional Radio Range (VOR) signal comprising a frequency-modulated (FM) component; an analog-to-digital converter to generate a digital signal based on the VOR signal, the digital signal comprising data representing the FM component; and a FM phase circuit comprising a correlator to generate a correlation peak based on the data representing the FM component and an ideal representation of the FM component, and a peak detector to determine a phase of the FM component based on the correlation peak. 
     In some embodiments, the FM phase circuit comprises: a digital signal processor. Some embodiments comprise an integrator to coherently integrate the digital signal. In some embodiments, an azimuth of the apparatus with respect to a transmitter of the VOR signal is determined based on the phase of the FM component. Some embodiments comprise an azimuth circuit to determine the azimuth of the apparatus based on the phase of the FM component. In some embodiments, a position of the apparatus is determined based on the azimuth of the apparatus. Some embodiments comprise a position circuit to determine the position of the apparatus based on the azimuth of the apparatus. In some embodiments, the VOR signal further comprises an amplitude-modulated (AM) component: wherein the azimuth of the apparatus is determined based on the phase of the FM component and a phase of the AM component. Some embodiments comprise an AM phase circuit to determine the phase of the AM component. In some embodiments, the AM phase circuit determines the phase of the AM component according to an acausal technique. Some embodiments comprise an azimuth circuit to determine the azimuth of the apparatus based on the phase of the FM component and the phase of the AM component. In some embodiments, a position of the apparatus is determined based on the azimuth of the apparatus. Some embodiments comprise a position circuit to determine the position of the apparatus based on the azimuth of the apparatus. 
     In general, in one aspect, the invention features an apparatus comprising: wireless receiver means for receiving a Very High Frequency (VHF) Omni-directional Radio Range (VOR) signal comprising a frequency-modulated (FM) component; analog-to-digital converter means for generating a digital signal based on the VOR signal, the digital signal comprising data representing the FM component; and correlator means for generating a correlation peak based on the data representing the FM component and an ideal representation of the FM component, and peak detector means for determining a phase of the FM component based on the correlation peak. 
     Some embodiments comprise integrator means for coherently integrating the digital signal. In some embodiments, an azimuth of the apparatus with respect to a transmitter of the VOR signal is determined based on the phase of the FM component. Some embodiments comprise azimuth means for determining the azimuth of the apparatus based on the phase of the FM component. In some embodiments, a position of the apparatus is determined based on the azimuth of the apparatus. Some embodiments comprise position means for determining the position of the apparatus based on the azimuth of the apparatus. In some embodiments, the VOR signal further comprises an amplitude-modulated (AM) component: wherein the azimuth of the apparatus is determined based on the phase of the FM component and a phase of the AM component. Some embodiments comprise AM phase means for determining the phase of the AM component. In some embodiments, the AM phase means determines the phase of the AM component according to an acausal technique. Some embodiments comprise azimuth means for determining the azimuth of the apparatus based on the phase of the FM component and the phase of the AM component. In some embodiments, a position of the apparatus is determined based on the azimuth of the apparatus. Some embodiments comprise position means for determining the position of the apparatus based on the azimuth of the apparatus. 
     In general, in one aspect, the invention features a method comprising: receiving a Very High Frequency (VHF) Omni-directional Radio Range (VOR) signal comprising a frequency-modulated (FM) component; generating a digital signal based on the VOR signal, the digital signal comprising data representing the FM component; and determining a phase of the FM component, comprising generating a correlation peak based on the data representing the FM component and reference data, and determining the phase of the FM component based on the correlation peak. 
     Some embodiments comprise coherently integrating the digital signal. In some embodiments, an azimuth of a receiver of the VOR signal with respect to a transmitter of the VOR signal is determined based on the phase of the FM component. Some embodiments comprise determining the azimuth of the receiver based on the phase of the FM component. In some embodiments, a position of the receiver is determined based on the azimuth of the receiver. Some embodiments comprise determining the position of the receiver based on the azimuth of the receiver. In some embodiments, the VOR signal further comprises an amplitude-modulated (AM) component: wherein the azimuth of the receiver is determined based on the phase of the FM component and a phase of the AM component. Some embodiments comprise determining the phase of the AM component. In some embodiments, the phase of the AM component is determined according to an acausal technique. Some embodiments comprise determining the azimuth of the receiver based on the phase of the FM component and the phase of the AM component. In some embodiments, a position of the receiver is determined based on the azimuth of the receiver. Some embodiments comprise determining the position of the receiver based on the azimuth of the receiver. 
     In general, in one aspect, the invention features computer-readable media embodying instructions executable by a computer to perform a method comprising: generating a digital signal based on a Very High Frequency (VHF) Omni-directional Radio Range (VOR) signal received by an apparatus, the VOR signal comprising a frequency-modulated (FM) component, the digital signal comprising data representing the FM component; and determining a phase of the FM component, comprising generating a correlation peak based on the data representing the FM component and reference data, and determining the phase of the FM component based on the correlation peak. 
     In some embodiments, the method further comprises: coherently integrating the digital signal. In some embodiments, an azimuth of a receiver of the VOR signal with respect to a transmitter of the VOR signal is determined based on the phase of the FM component. In some embodiments, the method further comprises: determining the azimuth of the receiver based on the phase of the FM component. In some embodiments, a position of the receiver is determined based on the azimuth of the receiver. In some embodiments, the method further comprises: determining the position of the receiver based on the azimuth of the receiver. In some embodiments, the VOR signal further comprises an amplitude-modulated (AM) component: wherein the azimuth of the receiver is determined based on the phase of the FM component and a phase of the AM component. In some embodiments, the method further comprises: determining the phase of the AM component. In some embodiments, the phase of the AM component is determined according to an acausal technique. In some embodiments, the method further comprises: determining the azimuth of the receiver based on the phase of the FM component and the phase of the AM component. In some embodiments, a position of the receiver is determined based on the azimuth of the receiver. In some embodiments, the method further comprises: determining the position of the receiver based on the azimuth of the receiver. 
     In general, in one aspect, the invention features an apparatus comprising: a wireless receiver to receive a Very High Frequency (VHF) Omni-directional Radio Range (VOR) signal comprising an amplitude-modulated (AM) component; an analog-to-digital converter to generate a digital signal based on the VOR signal, the digital signal comprising data representing the AM component; and an AM phase circuit to determine a phase of the AM component of the VOR signal as received at a station remote from the apparatus based on the data representing the AM component. 
     In some embodiments, the AM phase circuit determines the phase of the AM component according to an acausal technique. In some embodiments, the AM phase circuit comprises: a digital signal processor. In some embodiments, an azimuth of the station with respect to a transmitter of the VOR signal is determined based on the phase of the AM component. Some embodiments comprise an azimuth circuit to determine the azimuth of the station. In some embodiments, the azimuth circuit corrects the azimuth of the station based on at least one correction signal, wherein each correction signal represents a difference between a first azimuth determined by a respective further station based on the VOR signal and a second azimuth determined by the respective further station based on one or more other signals. In some embodiments, the VOR signal further comprises a frequency-modulated (FM) component, the apparatus further comprising: a further wireless receiver to receive a signal representing a phase of the FM component of the VOR signal as received at the station; and an azimuth circuit to determine the azimuth of the station based on the phase of the AM component and the phase of the FM component. In some embodiments, a position of the station is determined based on the azimuth of the apparatus. Some embodiments comprise a position circuit to determine the position of the station based on the azimuth of the apparatus. 
     In general, in one aspect, the invention features an apparatus comprising: wireless receiver means for receiving a Very High Frequency (VHF) Omni-directional Radio Range (VOR) signal comprising an amplitude-modulated (AM) component; analog-to-digital converter means for generating a digital signal based on the VOR signal, the digital signal comprising data representing the AM component; and AM phase means for determining a phase of the AM component of the VOR signal as received at a station remote from the apparatus based on the data representing the AM component. 
     In some embodiments, the AM phase means determines the phase of the AM component according to an acausal technique. In some embodiments, an azimuth of the station with respect to a transmitter of the VOR signal is determined based on the phase of the AM component. Some embodiments comprise azimuth means for determining the azimuth of the station. In some embodiments, the azimuth means corrects the azimuth of the station based on at least one correction signal, wherein each correction signal represents a difference between a first azimuth determined by a respective further station based on the VOR signal and a second azimuth determined by the respective further station based on one or more other signals. In some embodiments, the VOR signal further comprises a frequency-modulated (FM) component, the apparatus further comprising: further wireless receiver means for receiving a signal representing a phase of the FM component of the VOR signal as received at the station; and azimuth means for determining the azimuth of the station based on the phase of the AM component and the phase of the FM component. In some embodiments, a position of the station is determined based on the azimuth of the apparatus. Some embodiments comprise position means for determining the position of the station based on the azimuth of the apparatus. 
     In general, in one aspect, the invention features a method comprising: receiving, at a first station, a Very High Frequency (VHF) Omni-directional Radio Range (VOR) signal comprising an amplitude-modulated (AM) component; generating a digital signal based on the VOR signal, the digital signal comprising data representing the AM component; and determining a phase of the AM component of the VOR signal as received at a second station remote from the first station based on the data representing the AM component. 
     In some embodiments, the phase of the AM component is determined according to an acausal technique. In some embodiments, an azimuth of the second station with respect to a transmitter of the VOR signal is determined based on the phase of the AM component. Some embodiments comprise determining the azimuth of the second station. Some embodiments comprise correcting the azimuth of the second station based on at least one correction signal, wherein each correction signal represents a difference between a first azimuth determined by a respective remote station based on the VOR signal and a second azimuth determined by the respective remote station based on one or more other signals. In some embodiments, the VOR signal further comprises a frequency-modulated (FM) component, the method further comprising: receiving a signal representing a phase of the FM component of the VOR signal as received at the second station; and determining the azimuth of the second station based on the phase of the AM component and the phase of the FM component. In some embodiments, a position of the second station is determined based on the azimuth of the second station. Some embodiments comprise determining the position of the second station based on the azimuth of the second station. 
     In general, in one aspect, the invention features computer-readable media embodying instructions executable by a computer to perform a method comprising: generating a digital signal based on a Very High Frequency (VHF) Omni-directional Radio Range (VOR) signal received by a first station, the VOR signal comprising an amplitude-modulated (AM) component, the digital signal comprising data representing the AM component; and determining a phase of the AM component of the VOR signal as received at a second station remote from the first station based on the data representing the AM component. 
     In some embodiments, the phase of the AM component is determined according to an acausal technique. In some embodiments, an azimuth of the second station with respect to a transmitter of the VOR signal is determined based on the phase of the AM component. In some embodiments, the method further comprises: determining the azimuth of the second station. Some embodiments comprise correcting the azimuth of the second station based on at least one correction signal, wherein each correction signal represents a difference between a first azimuth determined by a respective further second station based on the VOR signal and a second azimuth determined by the respective further second station based on one or more other signals. In some embodiments, the VOR signal further comprises a frequency-modulated (FM) component, the method further comprising: determining a phase of the FM component of the VOR signal as received at the second station based on a signal received by the first station from the second station; and determining the azimuth of the second station based on the phase of the AM component and the phase of the FM component. In some embodiments, a position of the second station is determined based on the azimuth of the second station. In some embodiments, the method further comprises: determining the position of the second station based on the azimuth of the second station. 
     The details of one or more implementations are set forth in the accompanying drawings and the description below. Other features will be apparent from the description and drawings, and from the claims. 
    
    
     
       DESCRIPTION OF DRAWINGS 
         FIG. 1  shows the spectrum of a conventional VOR signal. 
         FIG. 2  shows a VOR receiver according to an embodiment of the present invention. 
         FIG. 3  shows a process for the VOR receiver of  FIG. 2  according to an embodiment of the present invention. 
         FIG. 4  shows a position determination system according to an embodiment of the present invention. 
         FIG. 5  shows a process for the position determination system of  FIG. 4  according to an embodiment of the present invention. 
     
    
    
     The leading digit(s) of each reference numeral used in this specification indicates the number of the drawing in which the reference numeral first appears. 
     DETAILED DESCRIPTION 
     Embodiments of the present invention provide ways to maximize the azimuth estimation accuracy for a given received VOR signal power, and to evaluate the accuracy of this estimate, using digital signal processing techniques. Equivalently, embodiments of the present invention minimize the received power necessary to achieve a desired azimuth estimation accuracy, thus extending the useful footprint of a VOR transmitter. 
     Though VOR transmitters were originally deployed for the purpose of aircraft navigation, they can also be used by ground-based receivers. A two-dimensional position fix can be achieved by combining azimuth measurements from multiple VOR transmitters, or by combining azimuth measurements with range measurements or time differences of arrival (TDOA) derived from other types of signals. 
     VOR systems use the phase relationship between two transmitted 30 Hz signals to encode azimuth.  FIG. 1  shows the spectrum  100  of a conventional VOR signal. Spectrum  100  comprises a VHF carrier  102 , an AM component  104  comprising a simple AM tone offset 30 Hz from carrier  102 , and an FM component  106  comprising a 30 Hz signal that is FM modulated on a 9960 Hz subcarrier. By means of an electrically rotating phased-array antenna, the phase difference between the two 30 Hz signals encodes the azimuth from the VOR transmitter. 
     The baseband modulating waveform for a VOR transmitter is given by
 
 m ( t )=1+0.3 cos(2 πf   m   t+θ   AM )+0.3 cos(2π f   sc   t+ 16 cos(2π f   m   t+θ   FM ))  (1)
 
where
 
     f m =30 Hz 
     f sc =9960 Hz 
     φ=θ AM −θ FM    
     Taken left-to-right, the terms in equation (1) represent carrier  102 , AM component  104 , and FM component  106 . The value of φ encodes the azimuth from the transmitter with respect to magnetic north. 
       FIG. 2  shows a VOR receiver  200  according to an embodiment of the present invention. VOR receiver  200  comprises an antenna  202 , a mixer  204 , an analog-to-digital converter (ADC)  206 , an integrator  208 , an AM bandpass filter (BPF)  210 , an FM BPF  212 , an AM phase circuit  214 , an FM phase circuit  216 , an azimuth circuit  218 , and a position circuit  220 . FM phase circuit  216  preferably comprises a correlator  222  and a peak detector  224 . AM BPF  210 , FM BPF  212 , AM phase circuit  214 , FM phase circuit  216 , azimuth circuit  218 , and position circuit  220  can be implemented as one or more digital signal processors. 
       FIG. 3  shows a process  300  for VOR receiver  200  of  FIG. 2  according to an embodiment of the present invention. Although in the described embodiments, the elements of process  300  are presented in one arrangement, other embodiments may feature other arrangements, as will be apparent to one skilled in the relevant arts based on the disclosure provided herein. 
     Antenna  202  receives a VOR signal (step  302 ). Embodiments of the present invention employ digital signal processing techniques. Hence, VOR receiver  200  digitizes the received VOR signal (step  304 ). In particular, mixer  204  mixes the received VOR signal down to baseband, and ADC  206  digitizes the signal at a sampling frequency of f s , though not necessarily in that order. 
     The VOR signal is periodic with a frequency of 30 Hz, so in some embodiments coherent integration is employed, for example using a phase-locked loop (PLL) locked to carrier  102 . This integration serves not only to lower the White Gaussian noise level, but also acts as a comb filter to attenuate tone-like signals and noise except at the frequencies of interest, which are all non-zero multiples of 30 Hz. In these embodiments, integrator  208  integrates the digitized VOR signal (step  306 ). The recovered discrete-time signal r(n) consists of the transmitted signal m(n/f s ) plus noise ε(n/f s ), as shown in equation (2).
 
 r ( n )= m ( n/f   s )+ε( n/f   s )  (2)
 
     After integration, the signal is split into two components: one component r AM (n) for analysis of AM component  104 , and one component r FM (n) for analysis of FM component  106 . AM BPF  210  extracts the AM component, r AM (n) (step  308 ), by bandpass filtering around 30 Hz (in addition to any comb-filtering that took place as a result of coherent integration at integrator  208 ). 
     AM phase circuit  114  determines the phase θ AM  of AM component  104  (step  310 ), for example using an acausal technique such as that shown in equation (3). 
                     θ   AM     =     -       tan     -   1       (         ∑     n   =   0       N   -   1       ⁢           ⁢       r   ⁡     (   n   )       ⁢     sin   ⁡     (       ω   m     ⁢   n     )               ∑     n   =   0       N   -   1       ⁢           ⁢       r   ⁡     (   n   )       ⁢     cos   ⁡     (       ω   m     ⁢   n     )             )               (   3   )               
where
 
     ω m =2πf m /f s    
     f s =sample rate 
     N=f s /30 
     Equation (3) represents an approximate Maximum Likelihood estimate of the phase angle of a sinusoid in Gaussian white noise. For typical desired azimuth accuracies, the estimator of equation (3) is nearly unbiased and exhibits performance close to the Cramer-Rao lower bound, which is given by equation (4). 
     
       
         
           
             
               
                 
                   
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     Hence, no better estimate of AM phase angle can be obtained. In equation 4, CNR refers to the carrier-to-noise ratio of the 30 Hz sideband, not the RF carrier. 
     Equation (4) indicates that the phase estimate error decreases roughly inverse-proportionally to the square root of the number of samples. Hence, the phase accuracy estimate can be improved by oversampling beyond the Nyquist limit. 
     FM BPF  212  extracts FM component  106 , r FM (n) (step  312 ), by bandpass filtering to approximately 9960 Hz+/−480 Hz. FM phase circuit  116  determines the phase θ FM  of FM component  106  (step  314 ). In particular, Correlator  222  performs a circular cross-correlation between FM component  106 , r FM (n) and an ideal representation of the FM component, for example according to equation (5), thereby generating a correlation peak. 
     
       
         
           
             
               
                 
                   
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     Peak detector  224  determines the phase θ FM  of FM component  106  based on the position of the correlation peak, for example according to equation (6). 
     
       
         
           
             
               
                 
                   
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     Azimuth circuit  218  determines the azimuth φ of VOR receiver  200  (step  316 ) by taking the difference between the phase θ AM  of AM component  104  and the phase θ FM  of FM component  106 , as shown in equation (7).
 
φ=θ AM −θ FM   (7)
 
     Position circuit  220  determines the position of VOR receiver  200  based on the azimuth φ of VOR receiver  200  (step  318 ), in combination with for example the location of the VOR transmitter and other measurements such as measurements of other VOR signals or other signals such as GPS, TV, FM, and the like. 
     In some embodiments, VOR receiver  200  determines the signal-to-noise ratio (SNR) of the received VOR signal (step  320 ). In these embodiments, referring again to  FIG. 2 , VOR receiver  200  also includes a SNR unit  226 . SNR unit  226  reconstructs the VOR signal based on the estimated phase θ AM  of AM component  104  and the estimated phase θ FM  of FM component  106 , for example using least squares fitting or the like. The reconstructed VOR signal is then subtracted from the received VOR signal to estimate the SNR of the received VOR signal. The SNR can be used to compute the variance in the estimate of the azimuth φ. 
     Ideally, an error-free measurement of the value of φ would represent the azimuth of a line extending from the VOR transmitter to VOR receiver  200  with respect to magnetic north. However, VOR transmitters generally do not exhibit the degree of accuracy desired for precision fixes (about 1 milliradian). For one, the initial alignment of the VOR transmitter to magnetic north may not be perfect. Second, the direction of magnetic north with respect to true north evolves over time and most VOR transmitters are never realigned after initial installation. Third, phase shifts may occur as the VOR transmitter electronics age. 
     In some embodiments, these problems are addressed by a monitoring system that accurately measures the transmitted value of φ from a known azimuth. The monitoring system includes monitor stations having clear line-of-sight to the VOR transmitter and sufficiently high SNR to achieve high measurement accuracy of φ. The difference between the true azimuth, φ, and the transmitted value, φ transmitted , represents the VOR transmitter&#39;s azimuth error, φ error , as shown in equation (8).
 
φ transmitted =φ+φ error   (8)
 
     The value of the VOR transmitter&#39;s azimuth error, φ error , can be continuously monitored and used to correct the measured azimuth of roaming VOR receivers such as VOR receiver  200 . 
     Another problem in this category is “scalloping”, a variation in φ error  that is quasi-periodic over azimuth. This error is primarily the result of multipath, particularly from terrain or objects near the VOR transmitter. Hence, the value of φ error  is a function of azimuth and range (though the dependence on range is expected to be weak as it grows larger), as shown in equation (9).
 
φ transmitted =φ+φ error (φ, r )  (9)
 
     It is impractical to scatter thousands of dedicated monitoring stations to derive azimuth correction factors at the level of granularity required for high-precision azimuth measurements. However, the individual mobile users of a hybrid positioning service can instead be aggregated to develop these correction factors, effectively serving as a distributed monitoring network. Each one of these user receivers employs multiple positioning technologies to generate a position, for example using GPS, TV, FM, VOR, and the like. When a VOR measurement is made by a receiver device, but an accurate position of that receiver can be determined without making use of the VOR measurement, then the difference between the actual and measured azimuth can be transmitted to a central server. (If the position of the receiver is known, so is its azimuth and range with respect to any VOR transmitter.) By filtering and interpolation, the server gradually generates an estimate of the VOR correction factors for all azimuths and ranges, as shown in equation (10).
 
φ=φ measured +φ′ error (φ measured   ,r )  (10)
 
     For a given CNR, the phase estimate of the phase θ FM  of FM component  106  is less accurate than the phase estimate of the phase θ AM  of AM component  104 . Experimentally, the AM measurement also appears to be more sensitive to real-world distortion and interference. In some embodiments, a monitoring system similar to the one described above allows VOR receiver  200  to avoid the use of the AM portion of the signal, while increasing both the azimuth measurement&#39;s accuracy and its robustness. 
     According to these embodiments, the phase θ AM  of AM component  104  as it appears at VOR receiver  200  is reconstructed from measurements collected at a monitor station. The monitor station measures both the phase θ AM  of AM component  104  and the frequency of the AM subcarrier, ω AM . (Although ω AM  is nominally 30 Hz, it may deviate from this standard by a small amount.) 
       FIG. 4  shows a position determination system  400  according to an embodiment of the present invention. Position determination system  400  comprises a VOR transmitter  402 , one or more VOR monitor stations  404 , one or more VOR receivers  406 , and a location server  408 . According to these embodiments, VOR receivers  406  can be implemented as described above for VOR receiver  200  of  FIG. 2 , but without the need for AM signal processing elements such as AM BPF  210 , AM phase circuit  214 , and the like. 
     VOR monitor station  404  comprises an antenna  410 , a mixer  412 , an analog-to-digital converter (ADC)  414 , an integrator  415 , an AM BPF  416 , an AM phase circuit  418 , an AM frequency circuit  420 , an AM user phase circuit  422 , and a transmitter  424 . Location server  408  comprises a receiver  426 , an azimuth circuit  428 , and a position circuit  430 . Elements of position determination system  400  can be implemented as one or more digital signal processors. 
       FIG. 5  shows a process for position determination system  400  of  FIG. 4  according to an embodiment of the present invention. VOR transmitter  402  transmits a conventional VOR signal (step  502 ). VOR receiver  406  receives the VOR signal, determines the phase θ FM  of FM component  106  as described above (step  504 ), and communicates phase θ FM  to location server  408 . 
     VOR monitor station  404  also receives the VOR signal. AM phase circuit  418  determines the phase θ AM,monitor  of AM component  104  as received at VOR monitor station  404  (step  506 ), for example using the techniques described above for VOR receiver  200  of  FIG. 2 . AM frequency circuit  420  of VOR monitor station  404  determines the frequency of the AM subcarrier, ω AM  (step  508 ). AM user phase circuit  422  determines the phase ω AM,user  of AM component  104  as received at VOR receiver  406  based on the values of θ AM,monitor  and ω AM  plus knowledge of time t at VOR receiver  406 , time t 0  of the measurement at VOR monitor station  404 , the range r monitor  between VOR monitor station  404  and VOR transmitter  402 , and a coarse estimate of the range r user  between VOR receiver  406  and VOR transmitter  402  (step  510 ), for example according to equation (11). 
     
       
         
           
             
               
                 
                   
                     
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                   11 
                   ) 
                 
               
             
           
         
       
     
     Transmitter  424  communicates the value of phase θ AM,user  to location server  408 . Receiver  426  of location server receives the value of phase θ AM,user . Azimuth circuit  428  determines the azimuth φ of VOR receiver  406  (step  512 ) by taking the difference between the reconstructed phase θ AM,user  received from VOR monitor station  404  and the phase θ FM  of FM component  106  received from VOR receiver  406 , for example as shown in equation (12).
 
φ=θ AM,user −θ FM   (12)
 
     Position circuit  430  of location server  408  determines the position of VOR receiver  406  based on the azimuth φ of VOR receiver  406  (step  514 ), for example in combination with the location of VOR transmitter  402  and other measurements such as measurements of other VOR signals or other signals such as GPS, TV, FM, and the like. 
     Referring again to equation (11), the latter term compensates for the range difference from VOR transmitter  402  to VOR monitor station  404  and VOR receiver  406 . While the range r monitor  between VOR monitor station  404  and VOR transmitter  402  can be surveyed very accurately, for example with GPS, the range r user  between VOR receiver  406  and VOR transmitter  402  need only be known to within about ±10 km in order to ensure that its phase error contribution is less than 1 milliradian. 
     The absolute time t at VOR receiver  406  can be determined by an accurate local clock or through a time-transfer mechanism, for example such as one that relies on the reception of RF signals with a known emission time combined with constraints on the range r user  between VOR receiver  406  and VOR transmitter  402  and the like. For 1 milliradian maximum error, equation (11) implies that range r user  need only be known to within about ±1.6 km. In dense urban areas, this is larger than the size of a cellular site&#39;s service radius. Hence, knowledge of the serving tower in conjunction with an RF synchronizing signal provides sufficiently accurate position to accurately reconstruct the phase θ AM,user  of AM component  104  as received at VOR receiver  406 . 
     If time t at VOR receiver  406  is known with high accuracy, but the range r user  from VOR transmitter  402  is not, the phase θ AM,user  of AM component  104  as received at VOR receiver  406  can still be reconstructed with relatively high precision, thereby allowing an estimate of azimuth φ of VOR receiver  406  to be made which, in turn, may be combined with other known constraints about the position of VOR receiver  406 , for example a line of position estimated from TDOA signals, to iteratively improve the position estimate for VOR receiver  406 . At each step, the position estimate is used to reduce the uncertainty bounds of the range r user  between VOR transmitter  402  and VOR receiver  406 . 
     Even when only a very coarse time transfer to VOR receiver  406  is possible, a line of position can be formed using a pair of VOR transmitters  402 . This line of position represents a constant azimuth difference between the two VOR transmitters  402 , as shown in equation (13).
 
φ 1 −φ 2 =(θ AM1 −θ FM1 )−(θ AM2 −θ FM2 )= k   (13)
 
     The shape of this constraint is a circular arc that joins the pair of VOR transmitters  402  and passes through the position of VOR receiver  406 . The constant k is formed by a combination of user and monitor measurements. According to these embodiments, the AM measurements are made only at VOR monitor stations  404 , where they will have high accuracy due to high SNR and a guaranteed direct path to the VOR, as shown in equation (14).
 
θ AM1,monitor ( t   0 )−θ AM2,monitor ( t   0 )+(ω AM1 −ω AM2 )( t−t   0 )−θ FM1,user ( t )+θ FM2,user ( t )= k   (14)
 
     Note that the term reflecting the range r user  between VOR transmitter  402  and VOR receiver  406  has been discarded since it is negligible. Also, since the ratio of ω AM1  to ω AM2  is very close to one, the value of the (ω AM1 −ω AM2 ) term is small (perhaps 10 −2  to 2·10 −4 ), so the value of k is relatively insensitive to the exact time at which the measurement is made. For example, if the relative rates of two VOR transmitters  402  differ by 1 ppm, then even a time transfer error of 5 seconds at VOR receiver  406  would result in less than 1 milliradian of estimation error in the value of k. 
     Multiple lines of position can be formed from multiple pairs of VOR transmitters  402 , and their intersection yields the two-dimensional position of VOR receiver  406 . Similarly, lines of position from combinations of pairs of VOR transmitters  402  and TDOA ranging pairs from other types of signals such as TV, FM, and the like, can be combined to produce a position fix. 
     While in the described embodiments, the elements and processes of VOR receiver  200  of  FIG. 2  and position determination system  400  of  FIG. 4  are described in particular arrangements, in other embodiments, the elements and processes are distributed in other arrangements. For example, all or part of location server  408  can be implemented within VOR monitor unit  404  or VOR receiver  406 . As another example, all or part of VOR monitor unit  404  can be implemented within VOR receiver  406  or location server  408 . In these arrangements, the required measurements are transferred among the units as needed. For example, AM user phase circuit  422  can be implemented within VOR monitor unit  404  or VOR receiver  406 . In these embodiments, VOR monitor unit transmits the phase θ AM,monitor  of AM component  104  as received at VOR monitor station  404  and the frequency of the AM subcarrier, ω AM , to the unit comprising AM user phase circuit  422 , which determines the phase θ AM,user  of AM component  104  as received at VOR receiver  406 . Of course, other arrangements are contemplated. 
     Embodiments of the invention can be implemented in digital electronic circuitry, or in computer hardware, firmware, software, or in combinations of them. Apparatus of the invention can be implemented in a computer program product tangibly embodied in a machine-readable storage device for execution by a programmable processor; and method steps of the invention can be performed by a programmable processor executing a program of instructions to perform functions of the invention by operating on input data and generating output. The invention can be implemented advantageously in one or more computer programs that are executable on a programmable system including at least one programmable processor coupled to receive data and instructions from, and to transmit data and instructions to, a data storage system, at least one input device, and at least one output device. Each computer program can be implemented in a high-level procedural or object-oriented programming language, or in assembly or machine language if desired; and in any case, the language can be a compiled or interpreted language. Suitable processors include, by way of example, both general and special purpose microprocessors. Generally, a processor will receive instructions and data from a read-only memory and/or a random access memory. Generally, a computer will include one or more mass storage devices for storing data files; such devices include magnetic disks, such as internal hard disks and removable disks; magneto-optical disks; and optical disks. Storage devices suitable for tangibly embodying computer program instructions and data include all forms of non-volatile memory, including by way of example semiconductor memory devices, such as EPROM, EEPROM, and flash memory devices; magnetic disks such as internal hard disks and removable disks; magneto-optical disks; and CD-ROM disks. Any of the foregoing can be supplemented by, or incorporated in, ASICs (application-specific integrated circuits). 
     A number of implementations of the invention have been described. Nevertheless, it will be understood that various modifications may be made without departing from the spirit and scope of the invention. Accordingly, other implementations are within the scope of the following claims.