[6]
Sniper Localization Using
Acoustic Sensors
Allison Doren
Anne Kitzmiller
Allie Lockhart
Under the Direction of Dr. Arye Nehorai
December 11, 2013
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
= sensor
= shooter
Example of a sensor network[2]
Types of Models
1. 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:
 𝑇𝑂𝐴 = 𝜏𝑖𝑚 =
𝑟𝑖
𝑐
+ 𝜏0
 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)
 𝜏0 ~ 𝑁(0, 𝜎 2 ) = time origin of the muzzle blast (normal
distribution)
 𝑟𝑖 =
𝑥 − 𝑥𝑖 2 + 𝑦 − 𝑦𝑖 2 = distance from the ith sensor
at (𝑥𝑖 , 𝑦𝑖 ) to the sniper position at (𝑥, 𝑦)
Muzzle Blast Model: Second Step
 Muzzle Blast Time Difference of Arrival (TDOA):
 Uses sensor 1 as a reference, for time synchronization purposes
 𝜏𝑖𝑚 = 𝑇𝑂𝐴, 𝜏0𝑖 = time origin of muzzle blast for ith sensor
 𝑇𝐷𝑂𝐴 = 𝑧 = 𝜃 + 𝑒, as defined below, where 𝜃 and 𝑒 are assumed to
be independent, 𝑒 ~ 𝑁 0, 2𝜎 2 , and 𝑧 ~ 𝑁(𝜃, 2𝜎 2 )
𝑇𝐷𝑂𝐴 = 𝑧 =
𝑇𝐷𝑂𝐴 =
𝑟𝑖
𝑐
𝜏𝑖𝑚
𝑟1
−
𝑐
𝜃
−
𝜏1𝑚
𝑟𝑖
𝑟1
𝑖
=
+ 𝜏0 −
+ 𝜏01
𝑐
𝑐
+ (𝜏0𝑖 − 𝜏01 ), for i = 2, 3, …, N
e
Muzzle Blast Model: Second Step
 Maximum Likelihood Estimation, using the conditional probability
distribution p:
𝑚
𝑁
𝑝 𝑧𝜃 =
𝑧𝑖,𝑗 − 𝜃𝑖
−
2𝜎 2
𝑒
2
𝑗=1 𝑖=2
 Maximum Likelihood (ML) and Least Squares (LS) equivalent in this
simulation, because using deterministic ML method, where 𝜃 is the
unknown parameter
 Therefore, maximizing 𝑝 𝑧 𝜃 for the ML method was
2
equivalent to minimizing the error 𝑧𝑖,𝑗 − 𝜃𝑖 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
𝜇 (𝜃) = [𝜇1 (𝜃), 𝜇2 (𝜃), … , 𝜇𝑁 (𝜃)]𝑇
𝐿𝑒𝑡 ∑(𝜃) 𝑏𝑒 𝑡ℎ𝑒 𝑐𝑜𝑣𝑎𝑟𝑖𝑎𝑛𝑐𝑒 𝑚𝑎𝑡𝑟𝑖𝑥
𝑇ℎ𝑒 𝑡𝑦𝑝𝑖𝑐𝑎𝑙 𝑒𝑙𝑒𝑚𝑒𝑛𝑡 𝐽𝑚 ,𝑛 , 𝑜𝑓 𝑡ℎ 𝐹𝐼𝑀 𝑓𝑜𝑟 𝑋 ~ 𝑁 𝜇 (𝜃), ∑(𝜃) 𝑖𝑠:
𝐽𝑚 ,𝑛
𝜕𝜇 𝑇 −1 𝜕𝜇 1
𝜕∑ −1 𝜕∑
=
∑
+ 𝑡𝑟 ∑−1
∑
𝜕𝜃𝑚
𝜕𝜃𝑛 2
𝜃𝑚
𝜃𝑛
𝐹𝑜𝑟 𝑡ℎ𝑒 𝑠𝑝𝑒𝑐𝑖𝑎𝑙 𝑐𝑎𝑠𝑒 𝑤ℎ𝑒𝑟𝑒 ∑(𝜃) = ∑, 𝑎 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡
𝐽𝑚 ,𝑛
𝜕𝜇 𝑇 −1 𝜕𝜇
=
∑
𝜕𝜃𝑚
𝜕𝜃𝑛
Cramér-Rao Bound
 In our case,
𝜇𝑚 𝜃 = 𝜃𝑚
0
0
⋮
𝜕𝜇
= 1
𝜕𝜃𝑚
⋮
0
0 (𝑁−1)×1
𝜎2 ⋯ 0
∑= ⋮
⋱
⋮ = 𝜎 2𝐼
0 ⋯ 𝜎2
Cramér-Rao Bound
 Fisher Information Matrix
1
𝜎2
𝐽= ⋮
0
⋯
0
1
⋮ = 2𝐼
𝜎
1
⋯
𝜎2
⋱
 For T independent measurements,
𝑁−1 ×(𝑁−1)
𝑇
𝐼.
𝜎2
𝜎2
𝜃 =
𝐼
𝑇
𝐽𝑇 𝜃 = 𝑇 × 𝐽 𝜃 =
𝐶𝑅𝐵 𝜃 = 𝐽𝑇−1
𝜎2
𝑇
𝐶𝑅𝐵 𝜃 = ⋮
0
⋯
0
⋱
⋮
𝜎2
⋯
𝑇
𝑁−1 ×(𝑁−1)
Mean Square Error
 Compare MSE with CRB
𝑀𝑆𝐸 =
1
1
∗
∗
𝑁 − 1 𝑖𝑡𝑒𝑟
𝑁 𝑖𝑡𝑒𝑟
𝜃𝑗,𝑘 − 𝜃𝑗,𝑘
𝑗=2 𝑘=1
 N = number of sensors
 iter = number of iterations
 𝜃 = our parameter
 𝜃= the estimate of our parameter
 Also find the MSE of our sniper position (x, y)
2
Signal-to-Noise Ratio (SNR)
 Compare signal power to noise power
 Signal Power:
1 𝑁
𝑟𝑖 2
∑
,
𝑁 𝑖=1 𝑐
where
𝑟𝑖 2
is
𝑐
as defined previously
 Noise Power: 𝜎 2
𝑆𝑖𝑔𝑛𝑎𝑙 𝑃𝑜𝑤𝑒𝑟
𝑆𝑁𝑅 = 10 log10 (
)
𝑁𝑜𝑖𝑠𝑒 𝑃𝑜𝑤𝑒𝑟
𝑟𝑖 2
𝑆𝑁𝑅 = 10 log10 ( 𝑐2 )
𝜎
Results
 Iterations, iter = 100
 Number of measurements (shots), m = 500
 Number of sensors, N = 6
 𝜎 = 0:0.04:0.36, standard deviation of noise
Placement of sensors in Matlab model and
localization error
160
140
12
120
11
Localization Error
100
Y
80
60
40
10
9
8
20
50
7
50
0
0
0
-20
-100
-50
-80
-60
-40
-20
0
X
20
40
60
(a) Sensor network and shooter position
80
100
-50
Y
(b) Localization error of position
 Variance = 0.01
 Minimum values of error at (0,0), our true sniper location
X
Sensor Network Geometry
100
100
50
50
100
80
60
-50
Y
0
Y
Y
0
40
20
-50
0
-50
0
X
50
100
-100
-100
-50
0
X
50
100
-50
10
5
50
50
0
Y
0
-50
-50
X
Localization Error
15
12
10
8
6
50
0
X
50
14
14
20
Localization Error
Localization Error
-100
-100
50
0
Y
0
-50
-50
12
10
8
6
50
X
0
Y
Comparison of localization performance on various six sensor geometries
 Shooter surrounded by sensors is ideal, but not practical
 Line of sensors does not provide sufficient information
-50
-50
0
X
50
100
100
50
50
50
0
0
0
-50
-50
-100
-100
-50
0
X
50
-100
-100
100
-50
-50
0
X
50
-100
-100
100
8
7
50
50
0
Y
0
-50
-50
X
Localization Error
9
-50
0
X
50
100
30
30
10
Localization Error
Localization Error
Y
100
Y
Y
Sensor Network Geometry
25
20
15
10
50
0
Y
-50
-50
0
X
50
25
20
15
10
50
0
Y
-50
-50
0
X
Comparison of localization performance on various random sensor geometries
 Increased number of sensors increases accuracy, but not realistic to have
this many sensors in close range
50
MSE of sniper position
(x, y) vs. SNR
SNR vs MSE
25
MSE (meters)
20
15
10
5
0
5
10
15
SNR
20
25
MSE of position vs. SNR
 As the signal-to-noise ratio increases, error decreases
 Thus as noise increases, error increases
MSE of 𝜽 vs. SNR, with CRB
r
-4
3.5
SNR vs MSE with CRB
x 10
MSE
CRB
3
MSE (seconds)
2.5
2
1.5
1
0.5
0
5
10
15
SNR
20
MSE of 𝜽, the TDOA, vs. SNR with CRB
 MSE converges to the CRB as SNR increases
25
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
 𝛼 = 𝑃(critical region|𝐻0 ), false alarm
 If 𝑋~𝑓 𝑥; 𝜃 , 𝑋1 , 𝑋2 , … , 𝑋𝑛 are independent and identically distributed random
samples of 𝑋, and the following hypothesis test is given
𝐻0 : 𝜃 = 𝜃 ′
𝐻1 : 𝜃 = 𝜃′′.
 It follows that the critical region is
𝐿(𝜃 ′ ; 𝑿)
𝑿 = 𝑋1 , 𝑋2 , … , 𝑋𝑛 :
≤𝑘
𝐿(𝜃 ′′ ; 𝑿)
 where k is calculated from
𝐿(𝜃 ′ ; 𝑿)
𝑃
≤ 𝑘 𝐻0 = 𝛼.
𝐿(𝜃 ′′ ; 𝑿)
Detection of a shot
 For this simulation,
𝜏=(
𝑟2 𝑟1
− ) + (𝜏0,2 − 𝜏0,1 )
𝑐
𝑐
𝐻0 : 𝜏 = 𝜏0,2 − 𝜏0,1 , where 𝜏 ~ 𝑁(0, 2𝜎 2 )
𝑟
𝑐
𝑟
𝑐
𝑟
𝑐
𝐻1 : 𝜏 = ( 2 − 1 ) + (𝜏0,2 − 𝜏0,1 ), where 𝜏 ~ 𝑁( 2 −
 If
𝜃′ = 0
𝑟2 𝑟1
′′
𝜃 = −
𝑐
𝑐
 then the critical region is of the form
𝒙≥−
𝑛 𝑟 2
− 2 𝑐𝑖
ln(𝑘𝑒
𝑟
𝑛 𝑐𝑖
≡ 𝑎.
𝑟1
, 2𝜎 2 ).
𝑐
Detection of a shot
 𝐻0 is rejected if 𝒙 ≥ 𝑎, and a is calculated from 𝑃 𝒙 ≥ 𝑎 𝐻0 = 𝛼, where
𝐻0 : 𝑋~𝑁 𝜃 ′ , 1 . Then,
1
′
𝑿~𝑁(𝜃 , )
𝑛
𝑃
 Therefore,
and
𝑎 − 𝜃′
≥
= 𝛼.
1
1
𝑛
𝑛
𝑿 − 𝜃′
𝑎 − 𝜃′
1
 𝐻0 will be rejected if 𝒙 ≥ 𝜃 ′ +
1
𝑧 ,
𝑛 𝑎
= 𝑧𝛼 .
𝑛
and 𝐻0 will be accepted if 𝒙 < 𝜃 ′ +
1
𝑧 .
𝑛 𝑎
ROC Curve
ROC Curve
1
0.9
Power (True Positive Rate)
0.8
PD
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Alpha (False Positive Rate)
0.8
0.9
1
𝛼
ROC Curve generated from detection applied in the
scalar case (2 sensors)
 Power, PD = 𝑃(critical region|𝐻1 )
 As 𝛼 increases, the critical region also increases, and thus power increases.
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?