- Email: [email protected]
An Algorithm for UWB Signals Tracking Based on Extended H∞ Filter
Available online at www.sciencedirect.com
Physics Procedia 33 (2012) 905 – 911
2012 International Conference on Medical Physics and Biomedical Engineering
An Algorithm for UWB Signals Tracking Based on Extended H Filter FuCheng Cao*, MingJing Li College of Electronic Information and Engineering,Changchun University ,Changchun 130022, China Email: [email protected]
Abstract In this paper, we present a positioning and tracking method for UWB signals. By the low complexity array processing, the time of arrival (TOA) and direction of arrival (DOA) is estimated. A robust algorithm based on the extended H -time location and velocity. This method is effective to the no-Gaussian or biased system model and the unknown or not fully known observation errors statistical properties.
©2012 2011Published Published by Elsevier Ltd. Selection and/or peer-review under responsibility of [name Committee. organizer] © by Elsevier B.V. Selection and/or peer review under responsibility of ICMPBE International Keywords:UWB; TOA; DOA; extended H
lter; tracking
1.Introduction Ultra-wideband (UWB) signals promise interesting perspectives for location estimation and object cation in short range environments. The use of location and tracking information is an excellent tool to improve productivity and to optimize the resources management in a wide range of sectors[1], [2]: industrial, medical, home-automation or military. For example, tracking the stocks in a warehouse, locating medical equipment and personnel in a hospital, monitoring the assembly line in a factory or developing intelligent audio guides that select narration according to the position of the visitor in a limitations of both GPS and WLAN, and are more suitable for indoor location-based applications. The conventional methods for wireless positioning are based on the estimation of TOA, TDOA (time difference of arrival) or DOA, once multiple timing or angles of arrival are measured at a convenient number of base stations. In [3], a high resolution and low complexity subspace method has been applied to the delay estimation for DS-UWB system. In [5], a low complexity frequency domain TOA estimation
1875-3892 © 2012 Published by Elsevier B.V. Selection and/or peer review under responsibility of ICMPBE International Committee. doi:10.1016/j.phpro.2012.05.153
906
FuCheng Cao and MingJing Li / Physics Procedia 33 (2012) 905 – 911
method has been proposed for the UWB signals. A joint estimation method of TOA and DOA for a single UWB source is performed on low complexity frequency domain approach in [4], on which this paper deals with the positioning estimation parameters based. An additional potentiality of the tracking problem that has not been explored enough is the mobility of the moving targets, especially for the complexity surroundings. The use of a kalman tracker helps in reducing the variability of the positioning by optimally averaging in time. In [6], the use of extended a moving terminal in nonlinear system. In [8], a robust algorithm based on the H a linear system. However, under the circumstances of the system model and observation error are often no-Gaussian and/or biased, and the statistical properties of the errors are often unknown or not fully known, both algorithms are not able to deal with. In this paper, the problem of joint estimation of TOA and DOA is introduced for a single UWB source. The UWB pulse position modulation signal is considered, and a frequency domain MUSIC algorithm is introduced. This algorithm can estimate the TOA and DOA of the moving target accurately. Considering the uncertainties of the target system and the measurement noise statistics, a robust algorithm based on the extended H -time location and velocity. The effectiveness of this positioning and tracking strategy was also evaluated by the computer simulation. The outline of this paper is as follows: Section II introduces the UWB signals model. Section III introduces the TOA and DOA joint estimation scheme. Section IV, a robust tracking algorithm is presented. Finally, computer simulation results and conclusions are discussed in Section V and VI, respectively. 2.System model We considered an IR-UWB system where transmission of an information symbol is typically N implemented by the repetition of f pulses of very short duration, The transmitted signal is expressed as, Nf 1
p(t T jk ) ,
s(t )
where
Tjk
(kN f
j)Tf
(1)
j 0
k
c jTc bkqT . Pulse Position Modulation (PPM) is assumed with { bk }
being the information symbols taking values {0, 1} with equal probability. p(t ) refers to the single pulse waveform, typically a Gaussian monocycle or its derivatives, and Tp is the repetition pulse period. Tp
is the frame period. N f is the number of frames per symbol,
Tf
is the chip period,
Tc
T
is the
PPM modulation time shift and { c j `is the time hopping sequence. A typical model for the multipath fading channel is given by, h(t )
L 1
hl (t Tl )
(2)
l 0
, Tl denote the fading coefficient and the time delay by the l th path respectively. Let UWB source s (t ) impinge upon an M element antenna array. The signal received by the antenna at time t is
where
hl
xm (t )
s (t ) * h(t ) L 1 l 0
hl s(t
vm (t ) l ,m
(Tl , l )) vm (t )
m th
(3) (4)
907
FuCheng Cao and MingJing Li / Physics Procedia 33 (2012) 905 – 911
where vm (t ) denotes the m th antenna additive Gaussian noise, s (t ) represents the UWB transmitted signal, and l ,m (Tl , l ) refers to the transmitted signal propagation delay at the m th antenna and the l -th arriving path. For a ULA the propagation delay associated to the m th antenna and the l th arriving path for the transmitted signal is given by l ,m
(Tl , l ) Tl
d m sin( l ), c
0
m
M
(5)
1
With d being the distance between antenna elements in the array, c the speed of light and direction of arrival of the transmitted signal via the l th arriving path.
l
the
3.Toa and doa joint estimation Considering a N-DFT sample frequency vector, the Fourier transform of the received signal vector at the q th antenna element can be expressed by [4], X m ( wk )
hl S ( wk )e
jwk
l , m (Tl , l
)
Vm ( wk )
(6)
where X m , S and Vm denote the Fourier transform of the received signal, transmitted signal and noise respectively. Rearranging equation (6) into a matrix notation, the received signal is given by, L 1
Xm
hl Se
l,m
(7)
l 0
where the elements of the delay signature vector include the dependency with the angle of arrival in e
l,m
[1
e
jw0
l,m
e
jw0
( N 1)
l ,m
]T
l, m
,
(8)
The estimation of the correlation matrix R from the frequency domain samples, R
1 Ns
NS
Z (S )Z H (S )
(9)
S 1
Where N S is the number of snapshots, Z contains the DFT components of the received signal at the s-th snapshot, rearranged in a way that the columns of Z are associated to a single frame. That is, M 1 hm p(t jT f Tm,q ) v j (t )} , v j (t ) denotes Z (S ) [ X1 (S ) X 2 (S ) X j (S ) X N (S )] , where X j ( S ) DFT { E P m 0
the noise contribution.
RE E where
diag (
0 , 1,
N S ),
0
1
NS
, E is the eigenvector, let
En
(10) be the noise subspace
whose columns are the eigenvectors corresponding to the smallest eigenvalue. The problem of interest is to estimate the TOA and DOA of UWB transmitted signal, i.e , , which can be achieved by the MUSIC pseudo-spectrum P( , )
a H ( , )a( , ) H
a ( , ) En EnH ( , )
(11) where (·) denotes Hermitian-transpose operation. The TOA and DOA of target user is indicated by the peak of (11). H
908
FuCheng Cao and MingJing Li / Physics Procedia 33 (2012) 905 – 911
4.Tracking algorithm Kalman minimum mean square error solutions for the state estimation problem, considering that both the process and the measurement noises of the target system are assumed as Gaussian with known statistical properties. However, in practical situations, the uncertainties and nonlinearities of the target system and the measurements normally do not satisfy the Gaussian assumption, and the noise statistics is usually not available. A extended H is presented, which is an combination of the linear H lter and the [7]. Considering B reference nodes. The measurement vector is formed z ( k ) [ zTOA z DOA ]T (12) ^
where
c[T01,
zTOA
^
TDB ]
,
the transition equation can b
t by As(k ) w(k ) (13)
s(k 1)
Where
s(k ) [ x(k ) y(k )vx (k )v y (k )]T
is the state vector. A 1 0 0 1
A
w(k )
0 0 wx
wy
0 0
0 0 1 0 0 0
0 1
(14)
is the disturbance transition vector with covariance matrix Q, where w x and w y
are the speed noise vectors.Measurements of TOA and DOA are nonlinear in the position state variables: (15) z(k ) f (s(k )) v(k ) Where z (k ) [ zTOA z DOA ]T
where matrix
0B
2
v(k ) [vTOA vDOA ]T . C (k )
0 CTOA 0 C DOA
G (k )
GTOA (k ) 0 B GDOA (k ) 0 B
(16) 2
(17)
2
is B 2 matrix with elements of zero, v(k ) is the measurement noise, with covariance From the nonlinear measurement equation to the linear measurement equation
C (k ) .
f s ^
where
s (k k 1)
18
^
s s ( k k 1)
is the state vector prediction, equal to the conditioned mean ^
s(k k 1)
E{s(k ) z(k 1)} (19)
The estimated output vector Y (k )
The output matrix
L(k )
L(k ) s(k ) (20)
is selected by the user according to the different applications. In our problem, t constitute the system state, so here
909
FuCheng Cao and MingJing Li / Physics Procedia 33 (2012) 905 – 911 L(k ) Y (k )
^
Y (k )
is speci using the measurement z (k ) .
of
has to minimize the following cost function [8], Y J
s up w, v [ 0, ]
2
1
Qk
2
(21)
2
wW
vV
k
k
where Qk Wk Vk are the weighting matrices for the estimation error, the process noise and the measurement noise. x
2 Qk
xkT Qk xk .
Moreover,
Qk
0
sup denotes the supremum and
Wk
1
0
Vk
. The notation
0
2
is de ned as
Qk
is the noise attenuation level.
The target position as r ( x, y) , the b -th reference node as rb to TOA and DOA measurements like CT0(b) sin B The linearization matrices GTOA R
x
( xb , yb ) , Then the position relates ( x xb )2 ( y yb )2 (y
(b ) 0
yb )
( x xb ) 2
2
x(k 1) d1 (k x(k 1) d 2 (k
GTOA (k )
r rb (y
yb ) 2
(y
yb ) db
x1 1) x2 1)
y (k 1) y1 d1 (k 1) y (k 1) y2 d 2 (k 1)
x(k 1) xB d B (k 1)
y (k 1) y B d B (k 1)
db
(22) (23)
(24)
(25)
GDOA (k )
( y (k 1)
y1 )( x(k 1) x1 )
( y (k 1)
d13 (k 1) y2 )( x(k 1) x2 ) d 23 (k 1)
( y (k 1)
y B )( x(k 1) xB ) d B3 (k
1)
( y (k 1)
1 d1 (k 1)
y1 ) 2
d13 (k 1) ( y (k 1) y2 ) 2
1 d 2 (k 1)
d 23 (k 1)
1 d B (k 1)
( y (k 1) d B3 ( K
yB )2 1)
The measurement equation (15) is linearized as z (k )
G(k )s( K ) v(k ) (26)
The mean square error of the estimate is the trace of the covariance matrix, which contains the accuracy parameter that will characterize the behavior of the tracker, P(k 1) E{ s(k 1) s(k 1) / z(k )} (27) The state estimation
s (k 1)
and the updating equations that forces
J
1
are given as follows: 1 (28)
QP(k ) GT (k )V 1G(k )P(k ))
L(k )
(I
K (k )
AP(k )L(k )GT (k )V
s(k 1)
P(k 1)
As(k )
1
(29)
K (k )( z (k ) G(k ) s(k ))
AP(k ) L(k ) AT
W
(30)
(31)
910
FuCheng Cao and MingJing Li / Physics Procedia 33 (2012) 905 – 911
Where I is the identity matrix. K (k ) is the H gain matrix. The initial state estimate s (0) should be initialized to our best guess of s(0) , and the initial value P(0) should be set to give acceptable lter s ( 0) performance. In general P(0) . wk vk H Although we need not to know the statistics of noises and in the lter, we should tune Q Wk Vk the weight matrices k , carefully, because these values determine the estimation error in the W V performance criterion (21). The weight matrices k , k can be chosen according to the experience about the noise. As the performance criterion, can not be very large, because otherwise some eigenvalues of the matrix P may have magnitudes more than one. These eigenvalues prevent a proper derivation of the H lter equations, so that the H lter problem has no solution. 5.Simulation results Next we evaluate the position tracking algorithm using the estimated TOA and DOA for different scenarios. The tracking zone is 30m 30m with one access point provided with 4-element array, and the arrays are equipped in the noise statistics. Finally, the computer simulations show an effectiveness of the proposed tracking strategy. 1.5
Figure 2. Tracking Results 2
Figure 1. Tracking Results 1
Figure 3. Position Estimation Error
position of (15,0). LOS is considered only. The target moves to follow a curve and a spiral path with arbitrary velocity. The measurement noise statistics is unknown. The position is measured 10 times per
FuCheng Cao and MingJing Li / Physics Procedia 33 (2012) 905 – 911
second and the length of simulation is 50 seconds. The tracking results are shown in Fig.1 and Fig.2, which we can see the proposed extended H measurement noi estimated position errors in x and y directions corresponding to the Fig.1 simulation. 6.Conclusion In this paper a frequency domain MUSIC the algorithm to jointly estimate the TOA and DOA for UWB antenna arrays is introduced. Then, a new approach for Kalman tracking based on the TOA and -time location and velocity accurately considering the uncertainties of the target system and the measurement References [1] L. Yang and G. Giannakis, Ultra-wideband communications: an idea whose time has come, IEEE Signal Process. Mag., vol. 21, no. 6, pp. 26-54, Nov. 2004. [2] Gezici, S., Zhi Tian, Giannakis, G.B., Kobayashi, H., Molisch, A.F., Poor, H.V. and Sahinoglu, Z., Localization via ultrawideband radios: a look at positioning aspects for future sensor networks, IEEE Signal Processing Mag., vol. 22, no. 4, pp. 70-84, July 2005. [3] Cao Fu-cheng, Wang Shu-xun and Sun Xiao-ying, High resolution and low complexity subspace based delay estimation for DS-UWB system, Journal of Jilin University(Engineering and Technology Edition), China, vol. 38, no. 2, pp. 471-475, March 2008. [4] Lagunas, E., Najar, M. and Navarro, M., UWB joint TOA and DOA estimation, ICUWB 2009, IEEE International Conference on, Vol., no., 9-11, pp. 839-843, 2009. [5] Cao Fu-cheng, Wang Shu-xun and Jiang Hong, Low complexity frequency domain TOA estimation for UWB sources, Journal of Jilin University(Engineering and Technology Edition), China, vol. 39, no. 3, pp. 792-796, May 2009. [6] Najar, M., Huerta, J.M., Vidal, J. and Castro, J.A., Mobile location with bias tracking in non-line-of-sight,IEEE Acoustics, Speech, and Signal Processing, 2004. Proceedings. Vol. 3, iii956-9, May 2004. [7] Simon, Dan,
nals, http://www.embedded.com/columns/ showArticle.jhtml?
articleID=190400581, Embedded Systems Design, EE Home-Page, July 2006. [8] Xiang Li and Andreas Zell,
Filtering for a Mobile Robot Tracking a Free Rolling Ball, Lecture Notes in Computer
Science, Springer-Verlag, Germany, Num. 4434, pp. 296-303, 2007.
911
Physics Procedia 33 (2012) 905 – 911
2012 International Conference on Medical Physics and Biomedical Engineering
An Algorithm for UWB Signals Tracking Based on Extended H Filter FuCheng Cao*, MingJing Li College of Electronic Information and Engineering,Changchun University ,Changchun 130022, China Email: [email protected]
Abstract In this paper, we present a positioning and tracking method for UWB signals. By the low complexity array processing, the time of arrival (TOA) and direction of arrival (DOA) is estimated. A robust algorithm based on the extended H -time location and velocity. This method is effective to the no-Gaussian or biased system model and the unknown or not fully known observation errors statistical properties.
©2012 2011Published Published by Elsevier Ltd. Selection and/or peer-review under responsibility of [name Committee. organizer] © by Elsevier B.V. Selection and/or peer review under responsibility of ICMPBE International Keywords:UWB; TOA; DOA; extended H
lter; tracking
1.Introduction Ultra-wideband (UWB) signals promise interesting perspectives for location estimation and object cation in short range environments. The use of location and tracking information is an excellent tool to improve productivity and to optimize the resources management in a wide range of sectors[1], [2]: industrial, medical, home-automation or military. For example, tracking the stocks in a warehouse, locating medical equipment and personnel in a hospital, monitoring the assembly line in a factory or developing intelligent audio guides that select narration according to the position of the visitor in a limitations of both GPS and WLAN, and are more suitable for indoor location-based applications. The conventional methods for wireless positioning are based on the estimation of TOA, TDOA (time difference of arrival) or DOA, once multiple timing or angles of arrival are measured at a convenient number of base stations. In [3], a high resolution and low complexity subspace method has been applied to the delay estimation for DS-UWB system. In [5], a low complexity frequency domain TOA estimation
1875-3892 © 2012 Published by Elsevier B.V. Selection and/or peer review under responsibility of ICMPBE International Committee. doi:10.1016/j.phpro.2012.05.153
906
FuCheng Cao and MingJing Li / Physics Procedia 33 (2012) 905 – 911
method has been proposed for the UWB signals. A joint estimation method of TOA and DOA for a single UWB source is performed on low complexity frequency domain approach in [4], on which this paper deals with the positioning estimation parameters based. An additional potentiality of the tracking problem that has not been explored enough is the mobility of the moving targets, especially for the complexity surroundings. The use of a kalman tracker helps in reducing the variability of the positioning by optimally averaging in time. In [6], the use of extended a moving terminal in nonlinear system. In [8], a robust algorithm based on the H a linear system. However, under the circumstances of the system model and observation error are often no-Gaussian and/or biased, and the statistical properties of the errors are often unknown or not fully known, both algorithms are not able to deal with. In this paper, the problem of joint estimation of TOA and DOA is introduced for a single UWB source. The UWB pulse position modulation signal is considered, and a frequency domain MUSIC algorithm is introduced. This algorithm can estimate the TOA and DOA of the moving target accurately. Considering the uncertainties of the target system and the measurement noise statistics, a robust algorithm based on the extended H -time location and velocity. The effectiveness of this positioning and tracking strategy was also evaluated by the computer simulation. The outline of this paper is as follows: Section II introduces the UWB signals model. Section III introduces the TOA and DOA joint estimation scheme. Section IV, a robust tracking algorithm is presented. Finally, computer simulation results and conclusions are discussed in Section V and VI, respectively. 2.System model We considered an IR-UWB system where transmission of an information symbol is typically N implemented by the repetition of f pulses of very short duration, The transmitted signal is expressed as, Nf 1
p(t T jk ) ,
s(t )
where
Tjk
(kN f
j)Tf
(1)
j 0
k
c jTc bkqT . Pulse Position Modulation (PPM) is assumed with { bk }
being the information symbols taking values {0, 1} with equal probability. p(t ) refers to the single pulse waveform, typically a Gaussian monocycle or its derivatives, and Tp is the repetition pulse period. Tp
is the frame period. N f is the number of frames per symbol,
Tf
is the chip period,
Tc
T
is the
PPM modulation time shift and { c j `is the time hopping sequence. A typical model for the multipath fading channel is given by, h(t )
L 1
hl (t Tl )
(2)
l 0
, Tl denote the fading coefficient and the time delay by the l th path respectively. Let UWB source s (t ) impinge upon an M element antenna array. The signal received by the antenna at time t is
where
hl
xm (t )
s (t ) * h(t ) L 1 l 0
hl s(t
vm (t ) l ,m
(Tl , l )) vm (t )
m th
(3) (4)
907
FuCheng Cao and MingJing Li / Physics Procedia 33 (2012) 905 – 911
where vm (t ) denotes the m th antenna additive Gaussian noise, s (t ) represents the UWB transmitted signal, and l ,m (Tl , l ) refers to the transmitted signal propagation delay at the m th antenna and the l -th arriving path. For a ULA the propagation delay associated to the m th antenna and the l th arriving path for the transmitted signal is given by l ,m
(Tl , l ) Tl
d m sin( l ), c
0
m
M
(5)
1
With d being the distance between antenna elements in the array, c the speed of light and direction of arrival of the transmitted signal via the l th arriving path.
l
the
3.Toa and doa joint estimation Considering a N-DFT sample frequency vector, the Fourier transform of the received signal vector at the q th antenna element can be expressed by [4], X m ( wk )
hl S ( wk )e
jwk
l , m (Tl , l
)
Vm ( wk )
(6)
where X m , S and Vm denote the Fourier transform of the received signal, transmitted signal and noise respectively. Rearranging equation (6) into a matrix notation, the received signal is given by, L 1
Xm
hl Se
l,m
(7)
l 0
where the elements of the delay signature vector include the dependency with the angle of arrival in e
l,m
[1
e
jw0
l,m
e
jw0
( N 1)
l ,m
]T
l, m
,
(8)
The estimation of the correlation matrix R from the frequency domain samples, R
1 Ns
NS
Z (S )Z H (S )
(9)
S 1
Where N S is the number of snapshots, Z contains the DFT components of the received signal at the s-th snapshot, rearranged in a way that the columns of Z are associated to a single frame. That is, M 1 hm p(t jT f Tm,q ) v j (t )} , v j (t ) denotes Z (S ) [ X1 (S ) X 2 (S ) X j (S ) X N (S )] , where X j ( S ) DFT { E P m 0
the noise contribution.
RE E where
diag (
0 , 1,
N S ),
0
1
NS
, E is the eigenvector, let
En
(10) be the noise subspace
whose columns are the eigenvectors corresponding to the smallest eigenvalue. The problem of interest is to estimate the TOA and DOA of UWB transmitted signal, i.e , , which can be achieved by the MUSIC pseudo-spectrum P( , )
a H ( , )a( , ) H
a ( , ) En EnH ( , )
(11) where (·) denotes Hermitian-transpose operation. The TOA and DOA of target user is indicated by the peak of (11). H
908
FuCheng Cao and MingJing Li / Physics Procedia 33 (2012) 905 – 911
4.Tracking algorithm Kalman minimum mean square error solutions for the state estimation problem, considering that both the process and the measurement noises of the target system are assumed as Gaussian with known statistical properties. However, in practical situations, the uncertainties and nonlinearities of the target system and the measurements normally do not satisfy the Gaussian assumption, and the noise statistics is usually not available. A extended H is presented, which is an combination of the linear H lter and the [7]. Considering B reference nodes. The measurement vector is formed z ( k ) [ zTOA z DOA ]T (12) ^
where
c[T01,
zTOA
^
TDB ]
,
the transition equation can b
t by As(k ) w(k ) (13)
s(k 1)
Where
s(k ) [ x(k ) y(k )vx (k )v y (k )]T
is the state vector. A 1 0 0 1
A
w(k )
0 0 wx
wy
0 0
0 0 1 0 0 0
0 1
(14)
is the disturbance transition vector with covariance matrix Q, where w x and w y
are the speed noise vectors.Measurements of TOA and DOA are nonlinear in the position state variables: (15) z(k ) f (s(k )) v(k ) Where z (k ) [ zTOA z DOA ]T
where matrix
0B
2
v(k ) [vTOA vDOA ]T . C (k )
0 CTOA 0 C DOA
G (k )
GTOA (k ) 0 B GDOA (k ) 0 B
(16) 2
(17)
2
is B 2 matrix with elements of zero, v(k ) is the measurement noise, with covariance From the nonlinear measurement equation to the linear measurement equation
C (k ) .
f s ^
where
s (k k 1)
18
^
s s ( k k 1)
is the state vector prediction, equal to the conditioned mean ^
s(k k 1)
E{s(k ) z(k 1)} (19)
The estimated output vector Y (k )
The output matrix
L(k )
L(k ) s(k ) (20)
is selected by the user according to the different applications. In our problem, t constitute the system state, so here
909
FuCheng Cao and MingJing Li / Physics Procedia 33 (2012) 905 – 911 L(k ) Y (k )
^
Y (k )
is speci using the measurement z (k ) .
of
has to minimize the following cost function [8], Y J
s up w, v [ 0, ]
2
1
Qk
2
(21)
2
wW
vV
k
k
where Qk Wk Vk are the weighting matrices for the estimation error, the process noise and the measurement noise. x
2 Qk
xkT Qk xk .
Moreover,
Qk
0
sup denotes the supremum and
Wk
1
0
Vk
. The notation
0
2
is de ned as
Qk
is the noise attenuation level.
The target position as r ( x, y) , the b -th reference node as rb to TOA and DOA measurements like CT0(b) sin B The linearization matrices GTOA R
x
( xb , yb ) , Then the position relates ( x xb )2 ( y yb )2 (y
(b ) 0
yb )
( x xb ) 2
2
x(k 1) d1 (k x(k 1) d 2 (k
GTOA (k )
r rb (y
yb ) 2
(y
yb ) db
x1 1) x2 1)
y (k 1) y1 d1 (k 1) y (k 1) y2 d 2 (k 1)
x(k 1) xB d B (k 1)
y (k 1) y B d B (k 1)
db
(22) (23)
(24)
(25)
GDOA (k )
( y (k 1)
y1 )( x(k 1) x1 )
( y (k 1)
d13 (k 1) y2 )( x(k 1) x2 ) d 23 (k 1)
( y (k 1)
y B )( x(k 1) xB ) d B3 (k
1)
( y (k 1)
1 d1 (k 1)
y1 ) 2
d13 (k 1) ( y (k 1) y2 ) 2
1 d 2 (k 1)
d 23 (k 1)
1 d B (k 1)
( y (k 1) d B3 ( K
yB )2 1)
The measurement equation (15) is linearized as z (k )
G(k )s( K ) v(k ) (26)
The mean square error of the estimate is the trace of the covariance matrix, which contains the accuracy parameter that will characterize the behavior of the tracker, P(k 1) E{ s(k 1) s(k 1) / z(k )} (27) The state estimation
s (k 1)
and the updating equations that forces
J
1
are given as follows: 1 (28)
QP(k ) GT (k )V 1G(k )P(k ))
L(k )
(I
K (k )
AP(k )L(k )GT (k )V
s(k 1)
P(k 1)
As(k )
1
(29)
K (k )( z (k ) G(k ) s(k ))
AP(k ) L(k ) AT
W
(30)
(31)
910
FuCheng Cao and MingJing Li / Physics Procedia 33 (2012) 905 – 911
Where I is the identity matrix. K (k ) is the H gain matrix. The initial state estimate s (0) should be initialized to our best guess of s(0) , and the initial value P(0) should be set to give acceptable lter s ( 0) performance. In general P(0) . wk vk H Although we need not to know the statistics of noises and in the lter, we should tune Q Wk Vk the weight matrices k , carefully, because these values determine the estimation error in the W V performance criterion (21). The weight matrices k , k can be chosen according to the experience about the noise. As the performance criterion, can not be very large, because otherwise some eigenvalues of the matrix P may have magnitudes more than one. These eigenvalues prevent a proper derivation of the H lter equations, so that the H lter problem has no solution. 5.Simulation results Next we evaluate the position tracking algorithm using the estimated TOA and DOA for different scenarios. The tracking zone is 30m 30m with one access point provided with 4-element array, and the arrays are equipped in the noise statistics. Finally, the computer simulations show an effectiveness of the proposed tracking strategy. 1.5
Figure 2. Tracking Results 2
Figure 1. Tracking Results 1
Figure 3. Position Estimation Error
position of (15,0). LOS is considered only. The target moves to follow a curve and a spiral path with arbitrary velocity. The measurement noise statistics is unknown. The position is measured 10 times per
FuCheng Cao and MingJing Li / Physics Procedia 33 (2012) 905 – 911
second and the length of simulation is 50 seconds. The tracking results are shown in Fig.1 and Fig.2, which we can see the proposed extended H measurement noi estimated position errors in x and y directions corresponding to the Fig.1 simulation. 6.Conclusion In this paper a frequency domain MUSIC the algorithm to jointly estimate the TOA and DOA for UWB antenna arrays is introduced. Then, a new approach for Kalman tracking based on the TOA and -time location and velocity accurately considering the uncertainties of the target system and the measurement References [1] L. Yang and G. Giannakis, Ultra-wideband communications: an idea whose time has come, IEEE Signal Process. Mag., vol. 21, no. 6, pp. 26-54, Nov. 2004. [2] Gezici, S., Zhi Tian, Giannakis, G.B., Kobayashi, H., Molisch, A.F., Poor, H.V. and Sahinoglu, Z., Localization via ultrawideband radios: a look at positioning aspects for future sensor networks, IEEE Signal Processing Mag., vol. 22, no. 4, pp. 70-84, July 2005. [3] Cao Fu-cheng, Wang Shu-xun and Sun Xiao-ying, High resolution and low complexity subspace based delay estimation for DS-UWB system, Journal of Jilin University(Engineering and Technology Edition), China, vol. 38, no. 2, pp. 471-475, March 2008. [4] Lagunas, E., Najar, M. and Navarro, M., UWB joint TOA and DOA estimation, ICUWB 2009, IEEE International Conference on, Vol., no., 9-11, pp. 839-843, 2009. [5] Cao Fu-cheng, Wang Shu-xun and Jiang Hong, Low complexity frequency domain TOA estimation for UWB sources, Journal of Jilin University(Engineering and Technology Edition), China, vol. 39, no. 3, pp. 792-796, May 2009. [6] Najar, M., Huerta, J.M., Vidal, J. and Castro, J.A., Mobile location with bias tracking in non-line-of-sight,IEEE Acoustics, Speech, and Signal Processing, 2004. Proceedings. Vol. 3, iii956-9, May 2004. [7] Simon, Dan,
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