"基于子空间的多目标跟踪与二维方向估计研究"

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This paper presents a novel approach for tracking the two-dimensional direction-of-arrivals (DOAs) of multiple moving targets with crossover points on their trajectories, based on subspace techniques. The proposed method, outlined in the IEEE Transactions on Aerospace and Electronic Systems, aims to improve the accuracy and efficiency of tracking multiple targets in complex environments. The key innovation of this approach lies in the use of subspaces to estimate and track the DOAs of multiple targets simultaneously. By decomposing the received signals into subspaces associated with individual targets, the proposed method is able to effectively separate and track the targets even in challenging scenarios where the trajectories intersect or overlap. The paper outlines the theoretical foundation of the subspace-based approach, highlighting the advantages of using subspaces for DOA estimation and tracking. In particular, the method leverages the inherent structure of the signal subspace to improve the accuracy of DOA estimation and mitigate the effects of interference and noise. Experimental results demonstrate the effectiveness of the proposed approach in tracking multiple targets with crossover points on their trajectories. The method achieves superior performance compared to conventional tracking algorithms, especially in scenarios with limited sensor measurements or high levels of interference. Overall, the subspace-based two-dimensional direction estimation and tracking technique presented in the paper offer a promising solution for tracking multiple targets in complex environments. By leveraging the power of subspaces, the method enables robust and accurate tracking of targets with crossover points, advancing the state-of-the-art in target tracking technology.

IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS, VOL. XX, NO. YY, AUGUST 2014 3

180

Fig. 2. The permissible regions for (a) the elevation angle θ

(n) and the

azimuth angle ϕ

(n) and (b) the elevation angle θ

(n) and the projected

azimuth angle

In this paper, the following basic assumptions are made on

the data model.

A1) The mathematical model of array response matrices

(i.e., A(θ) and A(ϕ)) is known, and the sensor spacing

d satisﬁes 0 < d ≤ λ/2 to avoid angle ambiguity.

A2) For facilitating the theoretical performance analysis,

the incident signals {s

(n)} are temporally complex

white Gaussian random processes with zero-mean and

the variance are given by E{s

(n)s

∗

(t)} = r

n,t

and E{s

(n)s

(t)} = 0 ∀n, t for 1 ≤ k ≤ p.

A3) The additive noises {w

(n)} and {w

(n)} are tem-

porally and spatially complex white Gaussian ran-

dom processes with zero-mean and the covariance

matrices are given by E{w

(n)w

(t)} = E{w

(n)

·w

(t)} = σ

n,t

and E{w

(n)w

(t)} =

E{w

(n)w

(t)} = O

M×M

∀n, t, and they

are statistically independent with each other, i.e.,

E{w

(n)w

(t)} = O

M×M

A4) The additive noises {w

(n)} and {w

(n)} at two

ULAs are statistically independent with the incident

signals {s

(n)}.

A5) The number of incident signals p is known or estimated

by number detection techniques in advance (cf. [42],

[43], and it satisﬁes the inequality that p < M.

From the relationship cos(ϕ

(n)) = cos(

(n)) sin(θ

(n))

[10], [37], we easily ﬁnd the permissible region for θ

(n) and

(n) and that for θ

(n) and

(n) as shown in Fig. 2(a)

and 2(b), respectively, while the geometry restrictions require

that the parameters θ

(n) and ϕ

(n) lie in the square region

deﬁned by

−θ

(n) + 90

◦

≤ ϕ

(n) ≤ θ

(n) + 90

◦

for 0

◦

≤ θ

(n) ≤ 90

◦

(n) − 90

◦

≤ ϕ

(n) ≤ −θ

(n) + 270

◦

for 90

◦

≤ θ

(n) ≤ 180

◦

(3)

The classical 1-D subspace-based direction estimation meth-

ods with eigendecomposition (e.g., MUSIC [26], estimation

of signal parameters via rational invariance techniques (ES-

PRIT) [44]) and the computationally simple 1-D subspace-

based direction estimators without eigendecomposition (e.g.,

propagator method (PM) [45], subspace-based method without

eigendecomposition (SUMWE) [39]) can be applied to each

ULA to obtain the reliable estimates of azimuth and elevation

angles separately. However, in general, there are p! possible

combinations between the estimates {

(n)} and {

(n)}.

Consequently the crux of 2-D direction estimation is the

pair-matching of the azimuth and elevation angles estimated

independently, which can require a tremendous computational

burden when the number of incident signals is large. Except

for the CODE [40], most of the existing techniques for pair-

matching or automatic pairing involve the computationally

extensive eigendecomposition process [5]–[8], [10], [29], [30],

[32]–[34], [36], [37]. Even though the pair-matching is accom-

plished correctly, the separate estimation of the azimuth and

elevation angles may cause the estimated angles

(n) and

(n) lie outside the permissible region shown in Fig. 2(a),

or equivalently we may have |cos(

(n))/sin(

(n))| > 1 for

sin

= 0, and obviously the estimate of the conventional

azimuth angle

(n) is unavailable with the relation

(n) =

arccos( cos(ϕ

(n))/sin(θ

(n)) ). Thus the estimation failure

occurs in this situation [10]. Unfortunately, the estimation

failure has not been resolved yet for the L-shaped array.

Therefore in order to overcome the aforementioned pairing

and estimation failures, we focus our attention on the joint

azimuth-elevation DOA estimation without eigendecomposi-

tion process and pair-matching procedure.

Remark 1: For decoupling the 2-D DOA estimation prob-

lem into 1-D estimation problems and fully exploiting the

well-known property of the ULA in a more straightforward

way, we parameterize the 2-D direction of the incident signals

by (θ

(n), ϕ

(n)) instead of (θ

(n),

(n)). In fact, by re-

deﬁning the parameter θ

(n) as the angle of the signal s

(n)

with respective to the y axis, the proposed CODEC method is

also applicable to the L-shaped array placed in the x–y plane.

However in order to compare the performance of the CODEC

method with the existing methods, we still concentrate on the

problem of 2-D DOA estimation with L-shaped array placed

in the x–z plane in this paper.

III. BATCH METHOD FOR 2-D DOA E STIMATION

A. Estimation of Elevation Angles

Firstly by assuming the elevation and azimuth angles be

time-invariant, i.e., θ

(n) = θ

and ϕ

(n) = ϕ

, we can pro-

pose a new computationally efﬁcient batch 2-D DOA estima-

tion method without eigendecomposition and pair-matching,

which is suitable for DOA tracking. Under Assumption A5, we

can divide the ULA along the z axis into two nonoverlapping

forward subarrays with p or M − p sensors, and the received

signal vector z(n) in (1) can be rewritten as

z(n) = [

(n),

(n)]

= [A

(θ), A

(θ)]

s(n) + [w

¯z

(n), w

¯z

(n)]

(4)

where

(n)

△

= [z

(n), z

(n), · · · , z

p−1

(n)]

(n)

△

(n), z

p+1

(n), · · · , z

M−1

(n)], w

¯z

(n)

△

= [w

(n), w

(n),

· · · , w

p−1

(n)]

, and w

¯z

(n)

△

= [w

(n), w

p+1

(n), · · · ,

M−1

(n)]

, while A(θ) in (1) is divided into two sub-

matrices A

(θ) and A

(θ) with the columns given by

(θ

)

△

= [1, e

jα

, · · · , e

j(p−1)α

]

and a

(θ

)

△

= [ e

jpα

j(p+1)α

, · · · , e

j(M−1)α

]

. Then under the assumptions of

data model, from (1), (2) and (4), we easily obtain the cross-

correlation matrix R

between the received signals of two

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