Semidefinite Relaxation Method for Target Localization by MIMO Radar Using Bistatic Ranges

Abstract

We address the target localization problem by using bistatic range (BR) measurements in widely separated multiple-input multiple-output (MIMO) radar network. The BR information defines a set of elliptic equations from which the target location can be estimated. By applying the semidefinite relaxation (SDR), we transform the nonconvex BR-based localization problem into a convex semidefinite programming (SDP) problem, whose solution is guaranteed to be globally optimal without initial estimate. Moreover, we extend this method to robustly solve the localization problem in the presence of antenna position errors. Simulation results demonstrate that the proposed SDR method provides superior estimation performance over the existing method.

1. Introduction

Target localization problem by using multiple-input-multiple-output (MIMO) radar with widely separated antennas (WSA) structure has received significant attentions [1–4]. Unlike the direct multilateration, which requires high computational cost [3], the indirect multilateration involving estimating position of target by jointly utilizing multiple bistatic range (BR) measurements from a subset of distributed receivers is more practical. In this letter, we focus on the target localization problem in MIMO radar based on BR measurements.

In principle, the maximum-likelihood estimation (MLE), which is asymptotically optimum, is a straightforward solution to this problem. However, the realization of MLE involves intensive computations. Moreover, owing to the high nonlinearity and nonconvexity of this maximum-likelihood (ML) problem [3], it cannot guarantee convergence to the global solution. To circumvent the nonlinearity drawback, a method based on the concept of best linear unbiased estimator (BLUE) has recently been proposed in [4], which linearizes the BR elliptic equations using Taylor series expansion and hence obtains a closed-form solution. Nevertheless, the BLUE method requires a good initial estimate and still cannot ensure global convergence. A least squares (LS) estimation method, which lends itself to a closed-form without requiring any initial condition, has been developed in [5]. However, the LS solution is based on the noise-free measurement model which is unreachable in practical situations.

Recently, convex optimization, particularly the semidefinite relaxation (SDR) methodology, has been well applied in localization applications with noisy time of arrival (TOA) or time difference of arrival (TDOA) measurements [6–9]. Aiming at achieving a tradeoff between high accuracy and global convergence of location estimate, we intend to relax the nonconvex BR-based localization problem to a semidefinite programming (SDP) problem [6]. The contribution of this letter is twofold: first, we start from reformulating the ML problem as an equivalent constrained optimization problem and then relax it to obtain a convex SDP problem by applying SDR technique; second, we devise a robust SDR method to tackle the BR-based localization problem when antenna positions are subject to errors, which has not been presented in the literature for the BR-based localization methods. Finally, we evaluate the performance of the proposed SDR method compared with the Cramêr-Rao lower bound (CRLB) by numerical simulations.

Notations. $I_{M}$ and $1_{M}$ denote the $M \times M$ identity matrix and $M \times 1$ all-one column vector, respectively. $Tr (A)$ means the trace of A. $diag (\cdot)$ , $blkdiag (\cdot)$ , and $vec (\cdot)$ construct diagonal matrix, block diagonal matrix, and vector from input arguments, respectively. $A ⪰ B$ means that $A - B$ is positive semidefinite. Superscript $(\cdot)^{T}$ stands for the transpose operator. $∥ \cdot ∥$ denotes the Euclidean norm of a vector, and ⊙ denotes the Hadamard product.

2. Problem Statement

Consider a BR-based two-dimensional (2D) localization problem employing M transmitters and N receivers at known locations $p_{k}^{o} = [x_{p, k}^{o}, y_{p, k}^{o}]^{T} (k = 1, \dots, M)$ and $q_{l}^{o} = [x_{q, l}^{o}, y_{q, l}^{o}]^{T} (l = 1, \dots, N)$ , respectively. Let $u^{o} = [x^{o}, y^{o}]^{T}$ be the true location of the target and let $u = [x, y]^{T}$ be the variable. The BR measurement at the $(k, l)$ th channel is modeled as follows:

\begin{matrix} r_{k l} = ∥ u^{o} - p_{k}^{o} ∥ + ∥ u^{o} - q_{l}^{o} ∥ + ϵ_{k l}, \end{matrix}

(1)

where

ϵ_{k l}

is the measurement noise. Collect the measurement noise

ϵ_{k l} (k = 1, \dots, M, l = 1, \dots, N)

into vector

ϵ

. Assume that

ϵ

follows a Gaussian distribution with zero mean and covariance matrix

Q_{ϵ}

, that is,

ϵ ~ N (0, Q_{ϵ})

The BR from the kth transmitter to the lth receiver through u is $d_{k l} (u) = d_{p, k} + d_{q, l}$ , where

\begin{matrix} d_{p, k} = ∥ u - p_{k}^{o} ∥, d_{q, l} = ∥ u - q_{l}^{o} ∥ . \end{matrix}

(2)

By defining the following vectors,

\begin{matrix} r = {[r_{1}^{T}, r_{2}^{T}, \dots, r_{M}^{T}]}^{T}, \\ r_{k} = {[r_{k 1}, r_{k 2}, \dots, r_{k N}]}^{T}, (k = 1, \dots, M), \\ d (u) = {[d_{11} (u), d_{12} (u), \dots, d_{M N} (u)]}^{T}, \end{matrix}

(3)

we can rewrite (1) in vector form:

\begin{matrix} r = d (u^{o}) + ϵ . \end{matrix}

(4)

According to (4), the MLE can be obtained as [10]

\begin{matrix} \hat{u} = \arg \min_{u} {(r - d (u))}^{T} Q_{ϵ}^{- 1} (r - d (u)) . \end{matrix}

(5)

3. SDR Method for Target Localization without Antenna Position Errors

The optimization problem of (5) is highly nonlinear and nonconvex, implying that the global minimum can hardly be obtained. In this section, we present an SDR method to approximately solve the MLE problem (5). To facilitate the development of the SDR approximation for MLE problem, we first focus on reformulating the original MLE problem. Then we show how the reformulated MLE problem can be approximated in a convex manner using SDR technique.

Firstly, (1) can be rewritten as

\begin{matrix} r_{k l} - ∥ u^{o} - p_{k}^{o} ∥ = ∥ u^{o} - q_{l}^{o} ∥ + ϵ_{k l} . \end{matrix}

(6)

Squaring both sides of (6) yields

\begin{matrix} r_{k l}^{2} + {∥ u^{o} - p_{k}^{o} ∥}^{2} - 2 r_{k l} d_{p, k} = {∥ u^{o} - q_{l}^{o} ∥}^{2} + 2 ϵ_{k l} d_{q, l} + ϵ_{k l}^{2} . \end{matrix}

(7)

Substituting $d_{q, l}$ by $r_{k l} - d_{p, k} - ϵ_{k l}$ to reduce the number of variables, we obtain

\begin{array}{l} r_{k l}^{2} + {p_{k}^{o}}^{T} p_{k}^{o} - {q_{l}^{o}}^{T} q_{l}^{o} \\ = 2 {(p_{k}^{o} - q_{l}^{o})}^{T} u^{o} + 2 r_{k l} ϵ_{k l} \\ + 2 r_{k l} d_{p, k} - 2 ϵ_{k l} d_{p, k} - ϵ_{k l}^{2}, \end{array}

(8)

which can be written in matrix form as [8]

\begin{matrix} z = H θ, \end{matrix}

(9)

where

\begin{array}{c} z = [r_{11}^{2} + {p_{1}^{o}}^{T} p_{1}^{o} - {q_{1}^{o}}^{T} q_{1}^{o}, r_{12}^{2} + {p_{1}^{o}}^{T} p_{1}^{o} \\ {- {q_{2}^{o}}^{T} q_{2}^{o}, \dots, r_{M N}^{2} + {p_{M}^{o}}^{T} p_{M}^{o} - {q_{N}^{o}}^{T} q_{N}^{o}]}^{T}, \\ H = {[2 A, 2 P, - 2 I_{M N}, - I_{M N}]}_{M N \times (3 M N + M + 2)}, \\ A = {[p_{1}^{o} - q_{1}^{o}, p_{1}^{o} - q_{2}^{o}, \dots, p_{M}^{o} - q_{N}^{o}]}_{M N \times 2}^{T}, \\ P = {[diag (r), blkdiag (r_{1}, r_{2}, \dots, r_{M})]}_{M N \times (M N + M)}, \\ θ = {[{u^{o}}^{T}, v^{T}, t_{1}^{T}, t_{2}^{T}]}^{T}, \\ t_{1} = (F ⊙ V_{1 : M N, M N + 1 : M (N + 1)}) 1_{M}, \\ F = blkdiag (\underset{M}{\underset{︸}{1_{N}, 1_{N}, \dots, 1_{N}}}), \\ t_{2} = vec (diag (V_{1 : M N, 1 : M N})), \\ V = v v^{T}, \\ v = {[ϵ_{11}, ϵ_{12}, \dots, ϵ_{M N}, d_{p, 1}, d_{p, 2}, \dots, d_{p, M}]}^{T} . \end{array}

(10)

Note that $V_{M N + k, M N + k} (k = 1, \dots, M)$ is related to $u^{o}$ as

\begin{matrix} V_{M N + k, M N + k} = d_{p, k}^{2} = U - 2 {u^{o}}^{T} p_{k}^{o} + {p_{k}^{o}}^{T} p_{k}^{o}, \end{matrix}

(11)

where

U = {u^{o}}^{T} u^{o}

is introduced to make the equation linear. Inspired by the idea in [8], we let

\tilde{θ}, \tilde{V}, \tilde{U}

be the variables for

θ, V, U

, respectively. Then, the optimization problem (5) is reformulated as

\begin{array}{l} \min_{\tilde{θ}, \tilde{V}, \tilde{U}} Tr (Q_{ϵ}^{- 1} {\tilde{V}}_{1 : M N, 1 : M N}) \\ subject to z = H \tilde{θ}, \\ {\tilde{θ}}_{2 + M N + k} \geq 0, (k = 1, \dots, M), \\ \tilde{V} = \tilde{v} {\tilde{v}}^{T}, \\ {\tilde{V}}_{M N + k, M N + k} = \tilde{U} - 2 u^{T} p_{k}^{o} + {p_{k}^{o}}^{T} p_{k}^{o}, \\ (k = 1, \dots, M), \\ \tilde{U} = u^{T} u . \end{array}

(12)

Relaxing the nonconvex constraints in (12) gives the SDR formulation for BR-based localization:

\begin{array}{l} \min_{\tilde{θ}, \tilde{V}, \tilde{U}} Tr (Q_{ϵ}^{- 1} {\tilde{V}}_{1 : M N, 1 : M N}) \\ subject to z = H \tilde{θ}, \\ {\tilde{θ}}_{2 + M N + k} \geq 0, (k = 1, \dots, M), \\ \tilde{V} ⪰ \tilde{v} {\tilde{v}}^{T}, \\ {\tilde{V}}_{M N + k, M N + k} = \tilde{U} - 2 u^{T} p_{k}^{o} + {p_{k}^{o}}^{T} p_{k}^{o}, \\ (k = 1, \dots, M), \\ \tilde{U} \geq u^{T} u . \end{array}

(13)

This optimization problem in (13) can be solved using standard convex optimization methods such as SeDuMi [11].

4. SDR Method for Target Localization with Antenna Position Errors

In practice, the radar antenna positions are typically not exactly known, and the localization performance can be significantly improved by taking antenna position errors into account. Here, we extend the SDR method to address the localization problem because of the receiver antenna position uncertainties (APU). The case concerning localization in the presence of transmitter APU will follow the similar steps which is omitted here. Let $q_{l} (l = 1, \dots, N)$ denote the estimated position of the lth receiver antenna. We assume that $q_{l} = q_{l}^{o} + Δ q_{l}$ , where $Δ q_{l}$ is the position error. We further assume that $Δ q_{l}$ follows a zero-mean Gaussian distribution with covariance matrix $ϕ_{l}$ and is mutually independent of the BR measurement noise $ϵ$ .

Substituting $q_{l}^{o} = q_{l} - Δ q_{l}$ into (7) yields

\begin{array}{l} r_{k l}^{2} + {p_{k}^{o}}^{T} p_{k}^{o} - {q_{l}}^{T} q_{l} \\ = 2 {(p_{k}^{o} - q_{l})}^{T} u^{o} + 2 r_{k l} ϵ_{k l} \\ + 2 r_{k l} d_{p, k} - 2 ϵ_{k l} d_{p, k} - ϵ_{k l}^{2} \\ + Δ q_{l}^{T} Δ q_{l} + 2 Δ q_{l}^{T} u^{o} - 2 {q_{l}}^{T} Δ q_{l} . \end{array}

(14)

We let $Y = E^{T} E$ where $E = [Δ q_{1}, Δ q_{2}, \dots, Δ q_{N}, u^{o}]$ , which means $Y_{m, n} = Δ q_{m}^{T} Δ q_{n}, 1 \leq m, n \leq N$ and $Y_{N + 1, N + 1} = {u^{o}}^{T} u^{o} = E_{1 : 2, N + 1}^{T} E_{1 : 2, N + 1}$ . Let $Ξ_{l} = Δ q_{l} Δ q_{l}^{T}$ . Note that $Ξ_{l}$ and $Y_{l, l}$ are related by

\begin{matrix} Tr (Ξ_{l}) = Δ q_{l}^{T} Δ q_{l} = Y_{l, l}, (l = 1, \dots, N) . \end{matrix}

(15)

Expressing (14) in matrix form, we have

\begin{matrix} z = H θ + 2 A E_{1 : 2, N + 1}, \end{matrix}

(16)

where

\begin{array}{c} z = [r_{11}^{2} + {p_{1}^{o}}^{T} p_{1}^{o} - {q_{1}}^{T} q_{1}, r_{12}^{2} + {p_{1}^{o}}^{T} p_{1}^{o} \\ {- {q_{2}}^{T} q_{2}, \dots, r_{M N}^{2} + {p_{M}^{o}}^{T} p_{M}^{o} - {q_{N}}^{T} q_{N}]}^{T}, \\ H = [2 P, - 2 I_{M N}, - I_{M N}, D, 2 D, \\ {- 2 D blkdiag (q_{1}^{T}, q_{2}^{T}, \dots, q_{N}^{T})]}_{M N \times (3 M N + M + 4 N)}, \\ A = {[p_{1}^{o} - q_{1}, p_{1}^{o} - q_{2}, \dots, p_{M}^{o} - q_{N}]}_{M N \times 2}^{T}, \\ P = {[diag (r), blkdiag (r_{1}, r_{2}, \dots, r_{M})]}_{M N \times (M N + M)}, \\ D = {[I_{N}, I_{N}, \dots, I_{N}]}_{M N \times N}^{T}, \\ θ = {[v^{T}, t_{1}^{T}, t_{2}^{T}, t_{3}^{T}, Y_{1 : N, N + 1}^{T}, ve c^{T} (E_{1 : 2,1 : N})]}^{T}, \\ t_{1} = (F ⊙ V_{1 : M N, M N + 1 : M (N + 1)}) 1_{M}, \\ F = blkdiag (\underset{M}{\underset{︸}{1_{N}, 1_{N}, \dots, 1_{N}}}), \\ t_{2} = vec (diag (V_{1 : M N, 1 : M N})), \\ t_{3} = vec (diag (Y_{1 : N, 1 : N})), \\ V = v v^{T}, \\ v = {[ϵ_{11}, ϵ_{12}, \dots, ϵ_{M N}, d_{p, 1}, d_{p, 2}, \dots, d_{p, M}]}^{T} . \end{array}

(17)

Using the same relaxation technique as in the previous section, we define $\tilde{θ}, \tilde{V}, \tilde{Y}, \tilde{E}, {{\tilde{Ξ}}_{l}}$ as the variables for $θ, V, Y, E, {Ξ_{l}}$ , respectively, and we have the new SDP problem as follows:

\begin{array}{l} \min_{\tilde{θ}, \tilde{V}, \tilde{Y}, \tilde{E}, {{\tilde{Ξ}}_{l}}} Tr (Q_{ϵ}^{- 1} {\tilde{V}}_{1 : M N, 1 : M N}) + \sum_{l = 1}^{N} ‍ Tr (ϕ_{l}^{- 1} {\tilde{Ξ}}_{l}) \\ subject to z = H \tilde{θ} + 2 A {\tilde{E}}_{1 : 2, N + 1}, \\ {\tilde{θ}}_{M N + k} \geq 0, (k = 1, \dots, M), \\ {\tilde{θ}}_{3 M N + M + N + 1 : 3 M N + M + 2 N} = {\tilde{Y}}_{1 : N, N + 1}, \\ {\tilde{θ}}_{3 M N + M + 2 N + 1 : 3 M N + M + 4 N} = vec ({\tilde{E}}_{1 : 2,1 : N}), \\ \tilde{V} ⪰ \tilde{v} {\tilde{v}}^{T}, \\ {\tilde{V}}_{M N + k, M N + k} = {\tilde{Y}}_{N + 1, N + 1} \\ - 2 {\tilde{E}}_{1 : 2, N + 1}^{T} p_{k}^{o} + {p_{k}^{o}}^{T} p_{k}^{o}, \\ (k = 1, \dots, M), \\ [\begin{bmatrix} \tilde{Y} & {\tilde{E}}^{T} \\ \tilde{E} & I_{2} \end{bmatrix}] ⪰ 0_{(N + 3) \times (N + 3)}, \\ {\tilde{Y}}_{N + 1, N + 1} \geq {\tilde{E}}_{1 : 2, N + 1}^{T} {\tilde{E}}_{1 : 2, N + 1}, \\ [\begin{bmatrix} {\tilde{Ξ}}_{l} & Δ {\tilde{q}}_{l} \\ Δ {\tilde{q}}_{l}^{T} & 1 \end{bmatrix}] ⪰ 0_{3 \times 3}, (l = 1, \dots, N), \\ Tr ({\tilde{Ξ}}_{l}) = {\tilde{Y}}_{l, l}, (l = 1, \dots, N) . \end{array}

(18)

The optimization problem in (18) can now be solved efficiently using standard SDP tools.

5. Simulations

5.1. Simulation Setup

In this section, we provide two examples to demonstrate the performance of the proposed SDR target localization methods comparing with the BLUE method [4]. The proposed SDR methods are realized by utilizing the MATLAB toolbox YALMIP [12] where the solver SeDuMi [11] is employed. In our test, we place $M = 4$ transmitters in a 2D area at $p_{1}^{o} = [5,0]^{T} km$ , $p_{2}^{o} = [0,5]^{T} km$ , $p_{3}^{o} = {[- 5,0]}^{T} km$ , and $p_{4}^{o} = {[0, - 5]}^{T} km$ and $N = 4$ receivers at $q_{1}^{o} = {[3.535,3.535]}^{T} km$ , $q_{2}^{o} = {[- 3.535,3.535]}^{T} km$ , $q_{3}^{o} = [- 3.535, - 3.535]^{T} km$ , and $q_{4}^{o} = [3.535, - 3.535]^{T} km$ . The target is located at $u^{o} = [0,0]^{T} m$ . The measurement covariance matrix $Q_{ϵ}$ is $(c^{2} / 8 π^{2} B^{2}) diag (1 / SN R_{11}, 1 / SN R_{12}, \dots, 1 / SN R_{M N})$ [3] where $c = 3 \times 1 0^{8} m / s$ and bandwidth $B = 50 MHz$ and SNR of each channel are assigned to be identical for simplicity. The root mean square error (RMSE) is employed as the performance metric and all the results are based on averages of $1000$ Monte Carlo runs.

5.2. Localization Performance without APU

In this example, we consider no radar antenna position errors. We choose three different initial reference points for simulations of the BLUE method [4] and the results of efficiency comparison between the proposed approach and the BLUE method for different initial conditions are presented in Figures 1 and 2.

Figure 1

Comparison of mean estimate error using SDR and BLUE methods in the absence of antenna position errors.

Figure 2

Comparison of RMSEs using SDR and BLUE methods with different reference points in the absence of antenna position errors.

Since the BR measurements are highly nonlinear with respect to u in (4), according to the analysis of [13], there exists no efficient unbiased estimator for $u^{o}$ in the BR measurement model. Figure 1 shows the estimate error mean value versus signal-to-noise ratio (SNR) using the BLUE and the proposed SDR methods, respectively. We can see that the proposed SDR method obtains unbiased estimate when $SNR \geq 0 dB$ , while the BLUE method estimate is consistently biased which is attributed to the initial reference point selection. The RMSE results versus SNR are plotted in Figure 2 where we can see the superiority of the proposed SDR method over the BLUE method for large SNR conditions when the initial estimate is not sufficiently close to the global minimum, although the proposed SDR RMSE degrades the CRLB [3] by about $1.5 dBm$ because the third constraint in (13) is not tight.

5.3. Localization Performance with APU

Here, we consider localization in the presence of receiver APU. The RMSE is also used to evaluate the performance of the proposed algorithm. We assume $ϕ_{l} = σ_{β}^{2} I_{2}$ , $l = 1, \dots, N$ , where $σ_{β}^{2}$ is the antenna position error level and $SNR = 5 dB$ . The RMSEs and CRLB with APU [14] are shown in Figure 3.

Figure 3

Comparison of RMSE using SDR and CRLB in the presence of antenna position errors.

We see in Figure 3 that although the RMSE of the position estimates cannot attain the CRLB in the presence of APU, it is still close to RMSE without antenna position errors when $σ_{β}^{2} \leq - 25 dBm$ . Therefore, the proposed SDR method is consistently reliable when the level of antenna position errors is not large. The gap between RMSE and CRLB becomes larger when the level of antenna position errors increases.

6. Conclusion

We have proposed an SDR formulation to target localization problem in MIMO radar network based on the BR measurement model. Our method applies the SDR technique to the reformulated MLE problem to obtain an SDP. Accordingly, it guarantees global convergence without initial estimate. We also propose a robust SDR method to obtain location estimate when antenna positions are subject to errors. Simulation results demonstrate the performance advantage of the newly developed SDR methods for BR-based target localization over BLUE method, under various SNR conditions and in the presence of antenna position errors. The proposed SDR method can either serve to estimate target location or serve as an initialization to the original ML problem.

Footnotes

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The work was supported in part by Grant from National Natural Science Foundation of China under no. 61302142 and National Science Fund for Distinguished Young Scholars under no. 61025006.

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