Geo-locating of wireless devices using a “pass filter” function
A method, wireless device and measuring station are disclosed that determine the best fit geo-location of a target station. According to one aspect, a method includes, using a “Pass Filter” for minimization of the summation of squared miss probabilities SSMP that improves the fitting process of the measured data over the method of minimization of the summation of the squared residuals (SSR) in the presence of spurious measurements. A “Pass Filter” approach is disclosed that reduces the corruption of the fitting process by outlier data and still yields the same result in the limit of clean data as the classic summation of the squared residuals (SSR) method.
1. A method for a measuring station for determining a best fit geo-location of a target station, the target station responding to ranging packets transmitted by the measuring station, the method comprising:
measuring a plurality of round-trip times, RTTs, y i , each RTT being a time elapsed between a transmission of a ranging packet and a reception of a response packet;
for each of a plurality measuring station location variables, x i , determining a first difference, Δ i , between a RTT measurement, y i , and a first function, f(x i , α) of measuring station location variables, x i , and target station location parameters, α; and
determining values of the plurality of target station location parameters, α, that minimize a sum of squared missed probabilities, SSMPs, a missed probability, MP, being a second function of the first difference Δ i .
2. The method of claim 1 , wherein the second function includes a second difference between 1 and a decaying exponential function of the first difference, Δ i .
3. The method of claim 2 , wherein the decaying exponential function is approximated by a first two terms of a Taylor series approximation to the decaying exponential function.
4. The method of claim 3 , wherein minimizing the sum of SSMPs includes multiplying a pass filter weight factor by a gradient of the first function, f(x i , α) with respect to the target location parameters α, the pass filter weight factor being a reciprocal of a variance, σ 2 , multiplied by the decaying exponential function.
5. The method of claim 2 , wherein the decaying exponential function is a function of difference, Δ i /σ, where a is a standard deviation, such that a smaller weight is given to a first RTT measurement having the first difference that is larger than a second difference, Δ i /σ, of a second RTT measurement.
6. The method of claim 5 , wherein the standard deviation Δ is chosen to selectively suppress spurious RTT measurements.
7. The method of claim 5 , wherein a first value of the difference, Δ i /σ, is selected at a start of minimizing the sum of SSMPs and a second value of the difference, Δ i /σ, smaller than the first value is selected after a start of minimizing the sum of SSMPs.
8. The method of claim 7 , wherein a first set of target location parameters a is determined based on the first value of the difference, Δ i /σ, and using the first set of target location parameters a in a subsequent iteration of minimizing the sum of SSMPs.
9. The method of claim 5 , wherein the standard deviation a is selected based at least in part on a standard deviation of the measured RTTs.
10. The method of claim 1 , wherein minimizing the sum of SSMPs includes performing successive iterations of the minimizing until a second difference between successive SSMPs is less than a threshold.
11. A wireless device for a measuring station for determining a best fit geo-location of a target station, the target station responding to ranging packets transmitted by the measuring station, the wireless device comprising:
a transmitter receiver configured to:
measure a plurality of round-trip times, RTTs, y i , each RTT being a time elapsed between a transmission of a ranging packet and a reception of a response packet;
processing circuitry in communication with the transmitter receiver, the processing circuitry being configured to:
for each of a plurality measuring station location variables, x i , determine a first difference, Δ i /σ, between a RTT measurement, y i , and a first function, f(x i , α) of measuring station location variables, x i , and target station location parameters, α; and
determine values of the measuring station location variables, x i , and target station location parameters, α, that minimize a sum of squared missed probabilities, SSMPs, a missed probability, MP i , being a second function of the first difference Δ i .
12. The wireless device of claim 11 , wherein the second function includes a second difference between 1 and a decaying exponential function of the first difference, Δ i .
13. The wireless device of claim 12 , wherein the decaying exponential function is approximated by a first two terms of a Taylor series approximation to the decaying exponential function.
14. The wireless device of claim 13 , wherein the decaying exponential function is a function of a difference, Δ i /σ, where σ is a standard deviation, such that a smaller weight is given to a first RTT measurement having the first difference, Δ i /σ, that is larger than a second difference, Δ i /σ, of a second RTT measurement.
15. The wireless device of claim 12 , wherein the decaying exponential function is a function of a difference, Δ i /σ, such that the difference, Δ i /σ, results in smaller weight given to a first RTT measurement having a difference, Δ i /σ, that is larger than a standard deviation of a second RTT measurement.
16. The wireless device of claim 15 , wherein the standard deviation a is chosen to selectively suppress spurious RTT measurements.
17. The wireless device of claim 15 , wherein a first value of the difference, Δ i /σ, is selected at a start of minimizing the sum of SSMPs and a second value of the difference, Δ i /σ, smaller than the first value is selected after a start of minimizing the sum of SSMPs.
18. The wireless device of claim 17 , wherein a first set of target location parameters α is determined based on the first value of the difference, Δ i /σ, and using the first set of target location parameters α in a subsequent iteration of minimizing the sum of SSMPs.
19. The wireless device of claim 15 , wherein the standard deviation σ is selected based at least in part on a standard deviation of the measured RTTs.
20. The wireless device of claim 11 , wherein minimizing the sum of SSMPs includes performing successive iterations of the minimizing until a second difference between successive SSMPs is less than a threshold.