INTRODUCTION
The bearingonly target motion analysis (TMA) is a classical singlesensor
passive location algorithm (Nardone and Graham, 1997;
Moon and Nordone, 2000) but its application is limited
because the observer must do maneuver movement for moving emitter location (Jauffret
et al., 2010; Hua et al., 2010).
Several researchers have investigated the methods of estimating the motion
trajectory by using measurements of time of arrival (TOA) and direction of arrival
(DOA) which is also a kind of TMA. The characteristics of the constantspeed
motion and the periodicity of signal are commonly used in TMA. In literature
of Yang and Zheng (1996), the nonlinear observation
equation is linearized according to Taylor expansion method and Weighted Least
Square (WLS) and Weighted Extended Kalman Filter (WEKF) are used in positioning
the moving target without the sensor movement. ZhongKang
et al. (2008) realized the target location by estimating the navigation
angle, range and velocity. The observability condition of target position based
on the measurements of TOA and DOA can be analyzed through the linearization
technique of nonlinear observation equation (Li et al.,
2004) and the observability analysis principle of nonlinear system (Xie
et al., 2007).
The TDMA system is a kind of time synchronization one, that is to say, the
time of TDMA terminal is synchronous with the system (Liu
et al., 2007). The sending time difference between different slots
of TDMA terminals is an integer multiple of the slot signal cycle. A TDMA target
transmits slot signals with slot interval as a benchmark but whether it can
transmit signals is restricted by the slot allocation rules (Lee
and Chang, 2004; Qin et al., 2009). A target
cannot transmit signals in each slot. So, the slot signal has quasiperiodic
feature.
The closedform TDMA target location algorithm with the variables of position coordinates, velocity and range is proposed based on the measurements of TOA and DOA in this study by parametric target motion analysis. The observability condition of target position is also analyzed according to the rank concept of matrix. The proposed algorithm simplifies the target localization process through the parametric target motion analysis.
SINGLESENSOR LOCATION MODEL OF TDMA TARGET
Quasiperiodic slot signal of TDMA system: Each terminal shares wireless channel with frameslot mode in the TDMA system, as shown in Fig. 1.
The time interval of frame is the minimum distribution cycle (T_{f})
of TDMA system, suppose that T_{f} is a constant and the sending time
of No. n frame is set to be t_{fu}, there by:

Fig. 1: 
Frames and slots in TDMA system 
One frame is divided into N_{s} slots and the slot is the minimum allocation unit for every terminal. Each slot comprises synchronization header, data and protective time. Suppose that the slot duration is not changed.
s (n, m) is used for denoting the No. m slot within No. n frame, the sending time is t_{tnm}, the slot duration time is T_{s} = T_{f}/N_{s}, there by:
It can be known from Eq. 1 and 2 that if the sending time of any one frame is known, the sending time of any slot in subsequent frame can be calculated.
Suppose that the sensor receives the signals of No. m slot in No. n frame, the relationship among the receiving time t_{rnm}, sending time t_{tnm} and range r between sensor and TDMA terminal is as follows:
The sending time of any slot can be confirmed if the TDMA system synchronization time is known. For this synchronized TDMA system, the range r can be estimated according to the receiving time of slot signal. If the range r is known, the slot sending time can be confirmed through the slot receiving time, then the system synchronization time can be calculated through the Fig. 1 and 2, namely, the sending time of any slot can be determined.
Suppose the sensor receives two slot signals, their sending times, receiving
times and ranges are, respectively represented as
and ,
and ,
and .
Before the synchronization relationship of the TDMA system is confirmed, n_{1},
m_{1}, n_{2} and m_{2} are unknown, the
and
and the difference between the two times can not be confirmed. Since the sending
time difference of any two slots is the integer multiple (represented with Δm)
of the slot period, there by:
The following relationship of receiving times can be obtained by Eq. 3:
The electromagnetic wave propagation time is taken into account during designing
the TDMA system and protection time is preserved for each slot. In the above
equation,
represents the range difference of two slot signals to the sensor that is generally
far smaller than the diffusion distance of the electromagnetic wave within the
signal cycle T_{s}. If the following condition is met:
Δm can be calculated by:
where, round () is a mathematic round function. From (5), the following is provided:
Therefore, the range difference between the different slots to sensor can be calculated according to the times of arrival (TOA) of slot signals.
Singlesensor location model: The twodimensional target is taken as
an example to illustrate the position principle. Assume that a TDMA target makes
constantspeed movement in twodimensional space with the velocity of (v_{x},v_{y})
and the sensor is on coordinate origin as shown in Fig. 2.
If the sensor receives N+1 signals provided with corresponding receiving time
t_{r(ki)} and azimuth β_{ki} (i = 0, 1,..., N) which
are, respectively sent by the moving target on the positions T_{k},
T_{k1}, ... and T_{kN} . The singlesensor location algorithm
aims at realizing the estimation of twodimensional coordinate (x, y) according
to the receiving time t_{r(ki)} and azimuth β_{ki} (i
= 0, 1,...,N) of the N+1 signals.
An actual target cannot make straight constantspeed movement always but its trajectory can be considered as a piecewise straight constantspeed movement one. For positioning a target of specific TDMA system, suppose that the cycle T_{s} of slot signal is known.
The target moving time between any two positions is the sending time difference of corresponding signals, namely:
According to Eq. 4,
Where:
According to Eq. 8, the range difference between the target positions at T_{ki} and T_{kj} to the sensor is:
Where the range equation is given by:
TARGET LOCATION ALGORITHM
As shown in Fig. 2, the following relationship exists for the target position T_{ki}:
According to Eq. 12, use r_{k} to express r_{ki}, then:
Equation 14 is deformed as:
In the same way, the following formula is derived for xaxis:
According to Eq. 16 and 17, 2 (N+1) equations
can be established for N+1 positions. The matrix form of equations is:
where, Δt_{00} = 0, Δr_{00} = 0,
The solvable necessary and sufficient condition of Eq. 18 is A to be a column full rank matrix, i.e.:
Suppose that A_{i} expresses the column vector of column i of matrix A. When the target makes the radial motion, the azimuth of target maintains invariable expressed as β_{0}. According to Eq. 18, the following relationship exists:
Equation 20 explains that the 5th column vector is linearly
correlation with the 1st and 2nd columns, i.e., Eq. 19 is
untenable. So, Eq. 18 has no solution when the target makes
the radial motion.
Equation 18 is get under the condition of straight constantspeed
movement. If the target makes a circle movement, the range difference Δr_{0i}
along trajectory for any subscript i is equal to zero and the target position
coordinates cannot be obtained because B in (18) is a zerovector.
Above observability condition of target position is the same as literatures
of Li et al. (2004) and Xie
et al. (2007).
The position coordinates velocity components and range can be estimated simultaneously through Eq. 18.
THEORETIC ACCURACY ANALYSIS OF TARGET LOCATION
The differential of Eq. 16 is:
where, dy denotes the differential of variable y.
According to Eq. 10:
According to Eq. 12:
Eq. 21 can be expressed as:
In the same way, the differential of Eq. 17 is:
The 2 (N+1) equations can be established for N+1 positions. The matrix form of equations is:
Where:
The solution of Eq. 26 is given by:
Where:
Equation 27 illuminates that the target location estimation
error is related with the measurement errors of location system, synchronization
error of TDMA system and target states in twodimensional space. The measurement
errors of location system include ones of azimuth (dβ) and TOA measurements
(dt_{r}) of N+1 signals. The synchronization error of TDMA system is
a jitter one of the slot signal period (dT_{s}).
The target states related to location error include range, velocity, azimuth and navigation angle. Matrix C contains range r_{ki} and azimuth β_{ki}. Vector E contains velocity components v_{x} and v_{y}. The navigation angle θ of target movement is related to the components v_{x} and v_{y} which relation is as follows:
Suppose that the errors of azimuth, TOA and synchronization are mutually independent white noises with zero mean and respective variances of σ_{β}^{2}, σ_{r}^{2} and σ_{s}^{2}. The covariance matrix of dX is:
Where:
I _{(N+1) x(N+1)} is a unit matrix.
The GDOP (Geometric Dilution of Precision) of location precision analysis is:
SIMULATION ANALYSIS OF TARGET LOCATION
In the simulation, the singlesensor location accuracy of TDMA target is analyzed along the target trajectory and with different amount of measurements.
Assume that a target makes constantspeed movement from left to right with the constant flight velocity of 200 m sec^{1} and sends a signal every 5 sec. The trajectory is parallel to the Xaxis in twodimensional space.
Suppose that the RMS (Root Mean Square) error of range difference is 30 m which corresponds to the TOA measurement error, that of target system synchronization 1 microsecond and that of azimuth 0.5 degree.

Fig. 3: 
Relationship of actual and estimated trajectory 

Fig. 4: 
Location precision analysis of different amount of data 
The TOA and DOA measurements of 3 positions (N = 2), 6 positions (N = 5) and 12 positions (N = 11) are used in analyzing the location accuracy.
Figure 3 show the Actual and estimated trajectories. The more the amount of data is used, the closer the estimated trajectory approaches to the actual one.
Figure 4 show the simulation results of location accuracy. GDOPi is used in indicating the location accuracy using the measurements of i positions. According to Fig. 4, the following conclusions could be obtained:
• 
The more the amount of data is used, the better the location
accuracy will be. The location accuracy using the measurements of 12 positions
is obviously better than that of 6 and 3 positions 
• 
The closer the target approaches to sensor, the better the location accuracy
will be 
• 
The positioning accuracy is asymmetric at the same distance on the left
and right of sensor during the entire trajectory. This is because the current
target position is estimated using previous measurements. The positioning
accuracy on the right is better than that on the left at the same distance 
CONCLUSION
According to the quasiperiodic characteristic of slot signal of TDMA target,
this paper makes the parametric target motion analysis to realize singlesensor
target location based on the measurements of TOA and DOA. The main conclusions
are as follows:
• 
The TDMA target can be positioned using the TOA and DOA measurements
of more than two target positions. The more the amount of data is used,
the better the location accuracy will be 
• 
The target location estimation error is related with the measurement errors
of location system, synchronization error of TDMA system and target states
in twodimensional space 