Transcript ChinaSIP Poster_finalx

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An improved wavelet based shockwave detector
Wei Xie, Xiao-Ping Zhang, Ming Bao, and Xiaodong Li
Key Laboratory of Noise and Vibration Research, Institute of Acoustics
Chinese Academy of Sciences, Beijing, China
ID:3204
Abstract
The probability density function of lognormal distribution is
In this paper, the detection of shock wave that generated by
supersonic bullet is considered. We present a new framework based on wavelet
multi-scale products method. We analyze the method under the standard
likelihood ratio test. It is found that the multi-scale product method is made in an
assumption that is extremely restricted, just hold for a special noise condition.
Based on the analysis, a general condition is considered for the detection. An
optimal detector under the standard likelihood ratio test is proposed. Monte
Carlo simulations is conducted with simulated shock waves under additive white
Gaussian noise. The result shows that this new detection algorithm outperforms
the conventional detection algorithm.
f Y ( y;  ,  ) 
1
e
2 2
,y0
y 2
Suppose wavelet coefficients from scale 2 j1 to 2 jm are considered, we have
p( X j1 (k ), X j2 (k ),..., X jm (k ) | H1 )
max p( X
{k }
sm
j1
(k ), X j2 (k ),..., X jm (k ) | H 0 )
    
 0
 j

jm
 max   X j (k ) 
{k }

1
0
j  j
 /
2
j

j  j1
1 2
 j  /2 2j
2

where   e
.
Based on the analysis, it is straightforward to propose the new detector
j  s1
Introduction
jm
The gunshots detection is of topical interest in areas related to
anti-terrorism, public security, and military actions. As a distinguishable
signature of acoustical gunshot signal, the shock wave has been widely used for
shooter localization and weapon classification. This paper focuses on the
detection of shock wave. Detection of shock wave is the key part of an initial
low-power processing stage. If a gunshot exists, this detector would activate an
advanced algorithm including localization and classification.
2
ln y   



L( X )  max  X j (k ) 
{k }
We
found
that
jm
max  X
{k }
j  j1
j
if

1
j
1
0
j  j
 /
j  j1
   /   1, j  j1 , j2 ,... jm ,
0
j
2
j
2
j
.
test
statistics
L( X )
is
(k )      . This detector was proposed by Salder [1][2] and has been widely
used for shock wave detection.
Methods
Simulation
The detection problem is formed in wavelet domain,
H 0 : X j [k ] p( X j [k ] | H 0 ) k  0,1,..., N  1,
H1 : X j [k ] p( X j [k ] | H1 ) k  0,1,..., N  1
where x[ n ] represents a noisy observation at a discrete time n .
The likelihood ratio detector is
N
p ( X | H1 )
L( X ) 

p( X | H 0 )
 p( X (k ) | H )
1
k 1
N
 p( X (k ) | H
k 1
0
)
(a) SNR = 7dB
where X (k )  { X j1 (k ), X j2 (k )..., X jm (k )} .
(b) SNR = 8dB
Fig. 1. Simulated ideal shock wave.
Here we assumed that the coefficients are independent with each other.
Consider detecting the edges of shock wave by tracking maxima across scales. For
simplicity, first we consider the maxima across scales correspond to one edge of
shock waves. We focus on the maxima across scales. Under the two hypothesizes, we
have
H 0 : X j [k ] p  X j [k ] H 0  k  0,1,...N  1,
H1 : X j [ k ]
X j [k ]
p  X j [ k ] H 1  k  iE ,
p  X j [ k ] H 0  k  iE , ,
(c) SNR = 9dB
where iE denotes the time location of the edge.
Then the detector becomes:
p( X (k ) | H1 ) k i   p( X (k ) | H 0 )
E
p( X | H1 )
p( X (k ) | H1 )
k 1,... N , k iE
L( X ) 


p( X | H 0 )
p ( X ( k ) | H 0 ) k i
 p( X (k ) | H 0 )
Since iE is time location of the only edge under H1 , it is straightforward to have
p ( X ( k ) | H1 )
p ( X ( k ) | H1 )

. In other words, at the time location iE , the
p ( X ( k ) | H 0 ) k i
p( X (k ) | H 0 ) k i
E
likelihood ratio has maximal value.
It is reasonable to argue that the time location
p ( X ( k ) | H1 )
iE  arg max
.
p( X (k ) | H 0 )
k
Thus the detector becomes:
p ( X ( k ) | H1 )
p ( X ( k ) | H1 )
L( X ) 
 max
p ( X ( k ) | H 0 ) k i
p( X (k ) | H 0 )
{k }
.
In order to achieve the form of the Multi-scale product detector, we assume that
under H1 and H 0 , the absolute wavelet coefficients follow Log normal distribution.
The distributions are independent across scales. For simplicity, we denote the
1
2
X
(
k
)
p
X
(
k
)
|
H
LN
(

(
k
),

(
k
))
absolute coefficients j
as X j (k ) .We has  j
and
1
j
j
E
p  X j (k ) | H 0 
LN (  (k ),  j (k )) , where  (k ) and  (k ) represent the mean
0
j
2
1
j
0
j
value of scale j under H1 and H 0 respectively. Under both of the two hypotheses,
the variance is  j (k ) .
2
Fig. 3. ROC curves for the MSP detector(blue line)
and the proposed detector(red line).
Conclusions
E
k 1,... N
E
Fig. 2. Performance of detectors for method MSP (blue
line) and the proposed method (red line): probability of
detection versus SNR for Pf a = 10−1, 10−2, 10−3.
(d) SNR = 10dB
In this paper, we have analyzed the Multi-scale product detector. We
derived the detector under log normal distribution and found that the detector
can be derived with limitations of the parameters of the distribution. It is a suboptimal detector under that distribution. We proposed the optimal detector
under the distribution without the limitations of the parameters. The
parameters of distribution can be estimated by tracking the maxima of the
wavelet coefficients across scales. The simulation shows that the performance
of proposed detector is better than the MSP detector.
References
1 Brian M Sadler, Tien Pham, and Laurel C Sadler, “Optimal and wavelet-based shock
wave detection and estimation,” The Journal of the Acoustical Society of America, vol.
104, no. 2, pp. 955–963, 1998.
2 Brian M Sadler and Ananthram Swami, “Analysis of multiscale products for step
detection and estimation,” IEEE Trans. Inf. Theory, vol. 45, no. 3, pp. 1043–1051, 1999.