FFT自适应二维DOA估计：任意阵列结构的高效跟踪
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本文主要探讨了一种基于快速傅立叶变换（Fast Fourier Transform, FFT）的自适应二维方向到达角（Direction-of-Arrival, DOA）估计方法，特别适用于各种阵列结构的应用。该研究针对传统的多重信号分类（Multiple Signal Classification, MUSIC）方法存在的搜索性质和计算复杂度问题进行了改进。
MUSIC方法在阵列处理中是一种常见的DOA估计算法，但其依赖于搜索策略，当需要同时估计多个目标的方位角（Azimuth）和仰角（Elevation）时，计算效率较低。为了克服这一限制，作者提出了一种新颖的二维方向跟踪框架，结合了子空间跟踪技术（Subspace Tracking）和Manifold Separation Technique (MST）。
首先，子空间跟踪技术被用于实时更新阵列数据的新样本到达时的特征向量基，这使得算法能够动态适应环境变化，提高了估计的准确性。这种方法通过连续地维护一个有效的信号子空间，有效地减小了搜索空间，降低了计算负担。
其次，利用移位矩阵，文中找到了一个快速的一次性操作来更新系数矩阵，实现了并行处理，进一步提升了算法的执行效率。相比于逐元素操作，这种并行处理方式极大地提高了处理大规模数据的能力，尤其在处理高维数据时，性能优势更为明显。
最后，2-D FFT算法被巧妙地应用到整个过程中，它能够一次计算出二维的空间谱，显著减少了计算复杂度，同时保持了频率域分析的优势，即能高效地捕捉到多径信号的特性。通过这种结合，作者设计的算法能够在保证精度的同时，实现对任意阵列结构下多个移动目标的实时、高效跟踪。
总结来说，这项研究提出了一个创新的自适应2-D DOA估计方法，通过改进的子空间跟踪和FFT技术，克服了传统MUSIC方法在计算效率和适应性方面的局限，对于提高多目标定位的实时性和实用性具有重要的理论和实践价值。这对于无线通信、雷达系统以及智能传感器网络等领域具有广泛的应用前景。
FFT-based Adaptive 2-D DOA Estimation for
Arbitrary Array Structures
Jie Zhuang, Hao Xiong, Wei Wang, and Xianglin Cai
School of Communication and Information Engineering,
University of Electronic Science and Technology of China (UESTC), Chengdu, 611731, China
Abstract—Direction-of-arrival (DOA) estimation is a ubiq-
uitous task in array processing. The conventional MUltiple
SIgnal Classification (MUSIC) method is search-based and often
computationally expensive, particularly in the application of joint
azimuth and elevation estimation. In this paper, we propose
an adaptive 2-dimensional direction finding framework to track
multiple moving targets for arbitrary array structures by using
the manifold separation technique (MST). First, we employ the
subspace tracking technique to update the eigenbasis recursively
on the arrival of a new data snapshot. In addition, by using
the shift matrix a fast one-step operation is found to update
the coefficient matrix in parallel. Finally, 2-D fast Fourier
transform (FFT) algorithm is performed to compute the 2-D
spatial spectrum at once. In comparison with the traditional
MUSIC or root-MUSIC methods, the proposed method can
reduce the computational complexity considerably.
I. INTRODUCTION
A ubiquitous task concerned in array processing is direction-
of-arrival (DOA) estimation which has been widely used in
wireless communication, radar, sonar, acoustics, astronomy,
medical imaging, and other areas. Let us consider an array of
arbitrary geometry operating in the presence of M uncorre-
lated narrowband signals via unknown directions. The signal
vector received at the time instant t can be expressed as
x(t) = A(t)m(t) + n(t) (1)
where n(t) represents the additive white Gaussian noise with
covariance σ
2
n
I
N
(σ
2
n
is the noise power). The notation I
N
denotes an N -by-N identity matrix (where N is the sensor
number). The matrix A(t) collects the steering vectors of the
M signals, i.e.,
A(t) = [a(θ
1
(t), φ
1
(t)), . . . , a(θ
M
(t), φ
M
(t))] (2)
where θ ∈ [0, 2π) and φ ∈ [0, π] denote the azimuth and
elevation angle respectively. The vector m(t) has elements
the M complex narrowband signal envelopes. The DOA
estimation problem that we will deal with in this paper can
now be briefly stated as follows: adaptively estimate the time-
varying azimuth and elevation angles using arbitrary array
configurations.
In practical applications, the time-varying covariance matrix
of the received vector x(t) can be obtained as follows
b
R(t) =
t
X
k=1
β
t−k
x(k)x
H
(k) = β
b
R(t − 1) + x(t)x
H
(t) (3)
where β ∈ (0, 1] is commonly known as the forgetting factor.
Eq. (3) tells us that all the received sample vectors available
in the time interval 1 ≤ k ≤ t are involved in the estimation
and the data in the distant past should be downweighted. Then
performing eigenvalue decomposition on
b
R(t) produces
b
R(t) = E
s
(t)Λ
s
(t)E
H
s
(t) + E
n
(t)Λ
n
(t)E
H
n
(t) (4)
where E
s
(t) ∈ C
N×M
is the eigenvectors associated with the
largest M eigenvalues and E
n
(t) ∈ C
N×(N −M)
represents the
eigenvectors corresponding to the remaining small eigenval-
ues. Commonly E
s
(t) and E
n
(t) are referred to as the signal-
subspace eigenvectors and noise-subspace eigenvectors. The
diagonal matrices Λ
s
(t) and Λ
n
(t) have diagonal elements
associated with the signal and noise eigenvalues respectively.
By exploiting the orthogonality between the signal and
noise subspace, the well-known MUltiple SIgnal Classification
(MUSIC) method searches the continuous array manifold
vector over the area of θ and φ to find the M minima of
the following null-spectrum cost function
(
b
θ,
b
φ) = arg min
θ,φ
{a(θ, φ)
H
E
n
(t)E
H
n
(t)a(θ, φ)}. (5)
However, such spectral search process may be unaffordable
for some real-time implementations, which can be explained
by the following fact. To obtain each search point, a matrix
product of a(θ, φ)
H
E
n
(t)E
H
n
(t)a(θ, φ) has to be computed.
This drawback becomes particularly apparent in joint estima-
tion of azimuth and elevation since we have to search over
two dimensions. Moreover, eigen-decomposition operation is
also computationally expensive and thus in many real-time
applications we cannot conduct it when each new snapshot
arrives.
In this paper, we propose a computation attractive 2-D DOA
estimator for arbitrary arrays. First, we convert the 2D-MUSIC
cost function into standard two-dimensional discrete Fourier
transform (DFT) form by using the manifold separation tech-
nique (MST). In doing so, the 2-D spatial spectrum can be
efficiently computed by using the 2D fast Fourier transform
(FFT) algorithms. For the sake of computation efficiency,
we employ existing signal subspace tracking methods to
update the noise subspace when each snapshot arrives, thereby
avoiding the computationally expensive eigen-decomposition
process. Moreover, we find the coefficient matrix can be
updated in parallel. The proposed method not only alleviates
the computation burden of root-MUSIC or MUSIC; it is also
easy for hardware implementation.
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