Sniper Localization Using Acoustic Sensors
Allison Doren
Anne Kitzmiller
Allie Lockhart
Under the Direction of Dr. Arye NehoraiDecember 11, 2013
[6]
Outline
Background
Muzzle Blast Model
Sniper Localization Maximum Likelihood Cramér-Rao Bound Mean Square Error
Results
Detection
Conclusions
Background
Existing Work: “Shooter Localization in Wireless Microphone Networks,” comparing muzzle blast and shock wave
models and using Cramér-Rao lower bound analysis[1]
“Analysis of Sniper Localization for Mobile, Asynchronous Sensors”, relying on time difference of arrival measurements, and providing a Cramér-Rao bound for the models[2]
“ShotSpotter” uses acoustic sensors to detect outside gunshot incidents in the D.C. area[5]
Applications: Military Operations: can be worn by soldiers or placed in vehicles Civilian Environments: can detect gunfire to alert local authorities
Example of a sensor network[2]
= sensor= shooter
Types of Models1. Shockwave Model (SW)
Exploits the shockwave of a gun shot, which comes about as a result of the supersonic bullets
2. Muzzle Blast Model (MB) Exploits the “bang” of a gun shot
3. Combined Model (Shockwave and Muzzle Blast)
The shockwave from the supersonic bullet reaches the microphone before the muzzle blast [1]
Muzzle Blast Model: First Step
Time of Arrival (TOA), for the ith sensor and the mth measurement:
Define Parameters: N = total number of sensors (N = 6) iter = number of iterations (iter = 100) m = total number of measurements (m = 500) i = ith sensor (i = 1, 2, …, N) c = speed of sound (330 m/s) = time origin of the muzzle blast (normal distribution) = distance from the ith sensor at to the sniper position
at
Muzzle Blast Model: Second Step
Muzzle Blast Time Difference of Arrival (TDOA): Uses sensor 1 as a reference, for time synchronization purposes = time origin of muzzle blast for ith sensor , as defined below, where and are assumed to be independent, , and
, for i = 2, 3, …, N
𝜃 e
Muzzle Blast Model: Second Step
Maximum Likelihood Estimation, using the conditional probability distribution p:
Maximum Likelihood (ML) and Least Squares (LS) equivalent in this simulation, because using deterministic ML method, where is the unknown parameter
Therefore, maximizing for the ML method was equivalent to minimizing the error for the LS method.
Cramér-Rao Bound
The Cramér-Rao Bound (CRB) is a lower bound on the variance of an unbiased estimator
We use a Multivariate Normal Distribution, because TDOA vector has a length equal to N-1
Cramér-Rao Bound
CRB for Multivariate Case The Fisher Information Matrix (FIM) for N-variate multivariate normal
distribution 𝜇ሺ𝜃ሻ= [𝜇1ሺ𝜃ሻ,𝜇2ሺ𝜃ሻ,…,𝜇𝑁ሺ𝜃ሻ]𝑇
𝐿𝑒𝑡 ∑ሺ𝜃ሻ 𝑏𝑒 𝑡ℎ𝑒 𝑐𝑜𝑣𝑎𝑟𝑖𝑎𝑛𝑐𝑒 𝑚𝑎𝑡𝑟𝑖𝑥 𝑇ℎ𝑒 𝑡𝑦𝑝𝑖𝑐𝑎𝑙 𝑒𝑙𝑒𝑚𝑒𝑛𝑡 𝐽𝑚,𝑛,𝑜𝑓 𝑡ℎ 𝐹𝐼𝑀 𝑓𝑜𝑟 𝑋 ~ 𝑁൫𝜇ሺ𝜃ሻ,∑ሺ𝜃ሻ൯𝑖𝑠:
𝐽𝑚,𝑛 = 𝜕𝜇𝑇𝜕𝜃𝑚 ∑−1 𝜕𝜇𝜕𝜃𝑛 + 12𝑡𝑟൬∑−1 𝜕∑𝜃𝑚 ∑−1 𝜕∑𝜃𝑛൰
𝐹𝑜𝑟 𝑡ℎ𝑒 𝑠𝑝𝑒𝑐𝑖𝑎𝑙 𝑐𝑎𝑠𝑒 𝑤ℎ𝑒𝑟𝑒 ∑ሺ𝜃ሻ= ∑,𝑎 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡
𝐽𝑚,𝑛 = 𝜕𝜇𝑇𝜕𝜃𝑚 ∑−1 𝜕𝜇𝜕𝜃𝑛
Cramér-Rao Bound
In our case,
Cramér-Rao Bound
Fisher Information Matrix
For T independent measurements,
Compare MSE with CRB
N = number of sensors
iter = number of iterations
= our parameter
= the estimate of our parameter
Also find the MSE of our sniper position (x, y)
Mean Square Error
Signal-to-Noise Ratio (SNR)
Compare signal power to noise power
Signal Power: , where is as defined previously
Noise Power:
Results
Iterations, iter = 100
Number of measurements (shots), m = 500
Number of sensors, N = 6
= 0:0.04:0.36, standard deviation of noise
-100 -80 -60 -40 -20 0 20 40 60 80 100-20
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507
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Loca
lizat
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Err
or(a) Sensor network and shooter position (b) Localization error of position
Placement of sensors in Matlab model and localization error
Variance = 0.01
Minimum values of error at (0,0), our true sniper location
-100 -50 0 50 100-100
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Comparison of localization performance on various six sensor geometries
Sensor Network Geometry
Shooter surrounded by sensors is ideal, but not practical
Line of sensors does not provide sufficient information
-500
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Comparison of localization performance on various random sensor geometries
Sensor Network Geometry
Increased number of sensors increases accuracy, but not realistic to have this many sensors in close range
5 10 15 20 250
5
10
15
20
25
SNR
MS
E (
met
ers)
SNR vs MSEMSE of sniper position (x, y) vs. SNR
As the signal-to-noise ratio increases, error decreases
Thus as noise increases, error increases
MSE of position vs. SNR
MSE of vs. SNR, with CRB
5 10 15 20 250
0.5
1
1.5
2
2.5
3
3.5x 10
-4
SNR
MS
E (
seco
nds)
SNR vs MSE with CRB
MSE
CRB
r
MSE converges to the CRB as SNR increases
MSE of , the TDOA, vs. SNR with CRB
Detection - general
The Neyman-Pearson Lemma [7] uses a likelihood-ratio test to choose a critical region that maximizes the power of a hypothesis test
=, false alarm
If are independent and identically distributed random samples of , and the following hypothesis test is given
.
It follows that the critical region is
where k is calculated from
Detection of a shot
For this simulation,
, where
, where .
If
then the critical region is of the form
Detection of a shot is rejected if and a is calculated from where . Then,
and
Therefore,
will be rejected if , and will be accepted if
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
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0.8
0.9
1
Alpha (False Positive Rate)
Pow
er (
Tru
e P
ositi
ve R
ate)
ROC Curve
ROC Curve
ROC Curve generated from detection applied in the scalar case (2 sensors)
Power, PD =
As increases, the critical region also increases, and thus power increases.
PD
𝛼
Conclusions
We used the Maximum Likelihood Method, Cramér-Rao Bound, and Mean Square Error in the Muzzle Blast Model to analyze our simulated shooter data, with different values of variance (noise) As predicted, MSE increases as noise increases MSE converges to the CRB as SNR increases
We studied the concept of detection and applied it to the scalar case of detecting a sniper with two sensors
We would have liked to compare our results to actual data obtained from sensors
Further Research Adding walls or other obstacles to sensor model Using different types of sensors, ie. optical, infrared Explore shockwave or combined MB-SW model Compare results to real data
References
1. D. Lindgren, O. Wilsson, F. Gustafsson, and H. Habberstad, “Shooter localization in wireless sensor networks,” Information Fusion, 2009, FUSION ’09, 12th International Conference on, pp. 404-411, 2009.
2. G. T. Whipps, L. M. Kaplan, and R. Damarla, “Analysis of sniper localization for mobile, asynchronous sensors,” Signal Processing, Sensor Fusion, and Target Recognition XVIII, vol. 7336, 2009.
3. P. Bestagini, M. Compagnoni, F. Antonacci, A. Sarti, and S. Tubaro, “TDOA-based acoustic source localization in the space-range reference frame,” Multidimensional Systems and Signal Processing, Vol. March, 2013.
4. Stephen, Tan Kok Sin. (2006). Source localization using wireless sensor networks (Master’s thesis). Naval Postgraduate School, 2006. Web. Sept 2013.
5. Berkowitz, Bonnie, Emily Chow, Dan Keating and James Smallwood. “Shots heard around the District.” The Washington Post 2 Nov. 2013. Investigations Web. Nov. 2013.
6. Photograph of Sniper. Photograph. n.d. Shooter Localization Mobile App Pinpoints Enemy Snipers. Vanderbilt School of Engineering. Web. 11 Nov 2013.
7. Hogg, Robert V., and Allen T. Craig. Introduction to Mathematical Statistics. New York: Macmillan, 1978. 90-98. Print.
Thank You!
Thank you to Keyong Han, the PhD student who has been guiding us throughout this project.
Thank you to Dr. Arye Nehorai for all of his help in overseeing our work and our progress.
Questions?
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