Determination and compensation for systematic errors in passive geolocation of wireless local area devices

1 . A method for operating a measuring station to generate a circular error probability (CEP) ellipse that encompasses the location of a target wireless station, the method comprising:

receiving, at a measuring station, a plurality of beacons transmitted by a target wireless station;

identifying a plurality of Time of Departures (TODs) and a plurality of Time of Arrivals (TOAs) corresponding to the plurality of beacons;

determining a plurality of Times of Flight (TOFs) corresponding to the plurality of beacons based on the plurality of TODs and the plurality of TOAs;

calculating a plurality of CEP ellipses for the target wireless station based, at least in part, on the plurality of TOFs;

determining whether the plurality of CEP ellipses exhibit an orbital radius;

in response to determining that the plurality of CEP ellipses exhibit an orbital radius, calculating a curvature radius of the plurality of CEP ellipses; and

generating, based on the curvature radius of the plurality of CEP ellipses, an adjusted CEP ellipse encompassing a location of the target wireless station.

2 . The method of claim 1 , wherein identifying the plurality of TODs comprises retrieving, from each of the plurality of beacons, a time of transmission determined by a timer associated with the target wireless station, and identifying the plurality of TOAs comprises determining, for each of the plurality of beacons, an arrival time using a timer associated with the measuring station.

3 . The method of claim 2 , further comprising:

in response to determining that the target wireless station has completed at least one orbit, synchronizing the timer associated with the target wireless station with the timer associated with the measuring station,

wherein determining the plurality of TOFs corresponding to the plurality of beacons based on the plurality of TODs and the plurality of TOAs further comprises determining the plurality of TOFs based on the synchronization of the timer associated with the target wireless station and the timer associated with the measuring station.

4 . The method of claim 1 , wherein determining whether the plurality of CEP ellipses exhibit an orbital radius comprises:

recording a time t for each CEP ellipse; and

for each consecutive pair of CEP ellipses, calculating an angle γ of a reference line connecting the pair of CEP ellipses.

5 . The method of claim 4 , further comprising:

calculating a time difference value between times t for a consecutive pair of CEP ellipses;

comparing the time difference value to a time difference threshold;

in response to the time difference value being less than the time difference threshold, proceeding with the orbital radius determination; and

in response to the time difference value being greater than the time difference threshold, forgoing the orbital radius determination.

6 . The method of claim 5 , further comprising:

calculating an angle difference value between a first angle γ 1 corresponding to a first and second CEP ellipse and a second angle γ 2 corresponding to the second CEP ellipse and a third CEP ellipse;

comparing the angle difference value to an angle difference threshold;

in response to the angle difference value being less than the angle difference threshold, proceeding with the orbital radius determination; and

in response to the angle difference value being greater than the angle difference threshold, forgoing the orbital radius determination.

7 . The method of claim 6 , further comprising:

determining an angular direction of the plurality of CEP ellipses;

comparing a sign of the angle difference value to the angular direction of the plurality of CEP ellipses;

in response to the sign of the angle difference value matching the angular direction of the plurality of CEP ellipses, proceeding with the orbital radius determination; and

in response to the sign of the angle difference value not matching the angular direction of the plurality of CEP ellipses, foregoing the orbital radius determination.

8 . The method of claim 6 , further comprising:

calculating an average distance value using a first distance k 1 between the first and second CEP ellipses and a second distance k 2 between the second CEP and third CEP ellipses; and

calculating the curvature radius of the plurality of CEP ellipses based on the angle difference value and the average distance value.

9 . The method of claim 8 , further comprising:

calculating an angle minimum value based on an orbit period of the target wireless station and the time difference so that (i) the angle difference threshold corresponds to a multiple of the angle minimum value, and (ii) calculating the curvature radius of the plurality of CEP ellipses based on the angle difference value and the average distance value comprises calculating the curvature radius based on the angle minimum value.

10 . The method of claim 1 , wherein generating the adjusted CEP ellipse comprises generating the adjusted CEP ellipse with a radius corresponding to a multiple of the curvature radius.

11 . A measuring station configured to generate a circular error probability (CEP) ellipse that encompasses the location of a target wireless station, the measuring station comprising:

at least one memory device with computer-executable instructions stored thereon; and

at least one processor for executing the computer-executable instructions stored on the at least one memory device, wherein execution of the computer-executable instructions by the at least one processor causes the at least one processor to perform operations comprising:

receiving a plurality of beacons transmitted by a target wireless station;

identifying a plurality of Time of Departures (TODs) and a plurality of Time of Arrivals (TOAs) corresponding to the plurality of beacons;

determining a plurality of Times of Flight (TOFs) corresponding to the plurality of beacons based on the plurality of TODs and the plurality of TOAs;

calculating a plurality of CEP ellipses for the target wireless station based, at least in part, on the plurality of TOFs;

determining whether the plurality of CEP ellipses exhibit an orbital radius;

in response to determining that the plurality of CEP ellipses do exhibit an orbital radius, calculating a curvature radius of the plurality of CEP ellipses; and

generating, based on the curvature radius of the plurality of CEP ellipses, an adjusted CEP ellipse encompassing a location of the target wireless station.

12 . The measuring station of claim 11 , wherein identifying the plurality of TODs comprises retrieving, from each of the plurality of beacons, a time of transmission determined by a timer associated with the target wireless station and identifying the plurality of TOAs comprises determining, for each of the plurality of beacons, an arrival time using a timer associated with the measuring station.

13 . The measuring station of claim 12 , wherein execution of the computer-executable instructions by the at least one processor causes the at least one processor to perform further operations comprising:

in response to determining that the target wireless station has completed at least one orbit, synchronizing the timer associated with the target wireless station with the timer associated with the measuring station,

wherein determining the plurality of TOFs corresponding to the plurality of beacons based on the plurality of TODs and the plurality of TOAs further comprises determining the plurality of TOFs based on the synchronization of the timer associated with the target wireless station and the timer associated with the measuring station.

14 . The measuring station of claim 11 , wherein determining whether the plurality of CEP ellipses exhibit an orbital radius comprises:

recording a time t for each CEP ellipse; and

for each consecutive pair of CEP ellipses, calculating an angle γ of a reference line connecting the pair of CEP ellipses.

15 . The measuring station of claim 14 , wherein execution of the computer-executable instructions by the at least one processor causes the at least one processor to perform operations comprising:

calculating a time difference value between times t for a consecutive pair of CEP ellipses;

comparing the time difference value to a time difference threshold;

in response to the time difference value being less than the time difference threshold, proceeding with the orbital radius determination; and

in response to the time difference value being greater than the time difference threshold, forgoing the orbital radius determination.

16 . The measuring station of claim 15 , wherein execution of the computer-executable instructions by the at least one processor causes the at least one processor to perform operations comprising:

calculating an angle difference value between a first angle γ 1 corresponding to a first and second CEP ellipse and a second angle γ 2 corresponding to the second CEP ellipse and a third CEP ellipse;

comparing the angle difference value to an angle difference threshold;

in response to the angle difference value being less than the angle difference threshold, proceeding with the orbital radius determination; and

in response to the angle difference value being greater than the angle difference threshold, forgoing the orbital radius determination.

17 . The measuring station of claim 16 , wherein execution of the computer-executable instructions by the at least one processor causes the at least one processor to perform operations comprising:

determining an angular direction of the plurality of CEP ellipses;

comparing a sign of the angle difference value to the angular direction of the plurality of CEP ellipses;

in response to the sign of the angle difference value matching the angular direction of the plurality of CEP ellipses, proceeding with the orbital radius determination; and

in response to the sign of the angle difference value not matching the angular direction of the plurality of CEP ellipses, foregoing the orbital radius determination.

18 . The measuring station of claim 16 , wherein execution of the computer-executable instructions by the at least one processor causes the at least one processor to perform operations comprising:

calculating an average distance value using a first distance k 1 between the first and second CEP ellipses and a second distance k 2 between the second CEP and third CEP ellipses; and

calculating the curvature radius of the plurality of CEP ellipses based on the angle difference value and the average distance value.

19 . The measuring station of claim 18 , wherein execution of the computer-executable instructions by the at least one processor causes the at least one processor to perform operations comprising:

calculating an angle minimum value based on an orbit period of the target wireless station and the time difference so that (i) the angle difference threshold corresponds to a multiple of the angle minimum value and (ii) calculating the curvature radius of the plurality of CEP ellipses based on the angle difference value and the average distance value comprises calculating the curvature radius based on the angle minimum value.

20 . The measuring station of claim 11 , wherein generating the adjusted CEP ellipse comprises generating the adjusted CEP ellipse with a radius corresponding to a multiple of the curvature radius.