International Journal of Antennas and Propagation

Volume 2019, Article ID 4943872, 7 pages

https://doi.org/10.1155/2019/4943872

## Joint Position and Velocity Estimation of a Moving Target in Multistatic Radar by Bistatic Range, TDOA, and Doppler Shifts

Electronic Information School, Wuhan University, Wuhan 430072, China

Correspondence should be addressed to Huotao Gao; moc.361@368thgoag

Received 5 June 2019; Revised 25 September 2019; Accepted 6 November 2019; Published 23 November 2019

Academic Editor: Felipe Cátedra

Copyright © 2019 Lijuan Yang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

In this paper, an estimation algorithm for the position and velocity of a moving target in a multistatic radar system is investigated. Estimation accuracy is improved by using bistatic range (BR), time-difference-of-arrival (TDOA), and Doppler shifts. Multistatic radar system includes several independent receivers and transmitters of time synchronization. Different transmitters radiate signals of different frequencies, and receivers detect the Doppler shifts of the received signals. These estimation parameters, BR, TDOA, and Doppler shifts, are readily available. The proposed algorithm combines different estimated parameters and optimizes estimation accuracy by two-step weighted least squares minimisations (WLS). This estimation algorithm is analysed and verified by simulations, which can reach the Cramer–Rao lower bound (CRLB) performance under mild Gaussian noise when the measurement error is small. Numerical simulations also demonstrate the superior performance of this method.

#### 1. Introduction

The classical problem of position and velocity estimation in the multistatic radar system receives great attentions in recent years due to its importance to detection, parameter estimation, tracking, surveillance, and navigation systems [1–3]. Generally, there are two modes for the multistatic radar system [4, 5], one is that a multistatic radar system consists of one transmitter and N receivers (T-R^{N}) [6] and the other is that the multistatic radar system which is composed of M identical transmitters and N identical receivers, radiating in a common coverage area (T^{M}-R^{N}) [3]. The multistatic radar system structure of T^{M}-R^{N} has the advantages in antistealth and antidestruction. In this structure, transmitters radiate signals of different frequencies, and those signals are received by receivers to detect the Doppler shifts. More transmitters and receivers would obtain more estimation parameters. Therefore, the multistatic radar system offers more flexibility and better system robustness as well as improves estimation accuracy of position and velocity.

Several localization algorithms about the multistatic system, including radar and sonar, have been developed such as time-of-arrival (TOA) [7], time-difference-of-arrival (TDOA) [8, 9], Doppler shift [10, 11], bistatic range (BR) [2], angle-of-arrival (AOA) [8, 12], and received signal strength (RSS) [13]. In the multistatic system, AOA needs a complex estimation algorithm. BR is equivalent to ellipse localization, which is relied on the propagation time of the transmitted signals travelling from transmitters to the target and reflected back to receivers. The transmitters and receivers are located at different sites [14]. Every single BR can form an ellipse where the target lies on, and the corresponding transmitter and receiver serve as its foci. The intersection of multiple ellipses can determine the approximate position of the target. TDOA, which is the time difference of the signal reflected from the target to different receivers, induces a hyperbola locus for the target to be located, with the associated different receivers as its foci. The intersection of the hyperbola from multiple transmitter-receiver pairs gives the target position. Ellipse and hyperbola can be obtained based on time synchronization of transmitters and receivers. Doppler shifts can be detected through receivers. A moving target creates Doppler shifts which depend on the location and speed of a target. Thus, it can be exploited to improve the localization accuracy in the multistatic radar system [12], and then the accuracy of velocity estimation can be improved.

Single parameter can be used to estimate the position or velocity in some cases, but the accuracy needs to be improved. Several localization algorithms have been researched using two types of measurements, such as BR and Doppler shifts [15], BR and bearing measurements [16], and ellipse and hyperbola [17]. Also, there is an estimation algorithm using time delay, Doppler shifts, and AOA measurements [18]. Using two or three kinds of estimated parameters improves the position or velocity accuracy than only one estimated parameter.

In this paper, we present a joint position and velocity estimation algorithm, which uses Doppler shifts, BR, and TDOA, to estimate the position and velocity of a moving target. This algorithm is optimized based on two-step weighted least squares minimisations (WLS) [9, 18]. In the first step, a set of pseudolinear BR, TDOA, and Doppler shift equations are established by introducing nuisance parameters, rewriting pseudolinear equations to a linear equation that can utilize the calculation formula of WLS, and then the initial estimation of position and velocity is obtained. In the second step, the error term of the first-stage solution is estimated by using the relationship between initial estimated terms and nuisance variables. Similarly, error terms are calculated by WLS, and then the final estimation of position and velocity is obtained through subtracting the error terms from initial values. Simulation results show that it has a better localization and velocity accuracy not only than the method of ellipse and hyperbola but also than BR and Doppler shifts.

We shall use the common convention that bold upper and lower case letters denote matrix and column vector. , , and represent an identity matrix, a zero vector with the length *m*, and zero matrix, respectively. The notations and denote transpose and inverse of matrix A. The norm is denoted by . The symbol stands for Kronecker products, and denotes the true value of the variable .

#### 2. Problem Formulation for the Multistatic Radar System

We consider the localization of a moving target in a three-dimensional (3D) scenario using the multistatic radar system network with M transmitters and N receivers, whose positions are denoted by , and , . The target position , and the target velocity . The geometry structure of the multistatic radar system is shown in Figure 1.