稀疏贝叶斯学习提升离线信号到达方向估计精度
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本文主要探讨了"基于稀疏贝叶斯学习的离网信号到达方向估计"这一主题，发表于2016年4月1日的IEEE SENSORS JOURNAL第16卷第7期。传统上，空间离散网格在信号到达方向估计（DOA）方法中存在固有的局限性，这限制了基于稀疏信号表示（SSR）的DOA估计算法的精度和实用性。作者Xiaohuan Wu、Wei-Ping Zhu（IEEE高级会员）和Jun Yan提出了一个解决策略。
首先，他们提出了一种改良的SSR模型，通过引入偏差参数来缓解固定网格带来的约束。这个模型允许对非网格点的信号方向进行更精确的估计，从而提高了DOA估计框架的灵活性。在此基础上，他们开发了一种名为PSBL（Perturbed Sparse Bayesian Learning，扰动稀疏贝叶斯学习）的算法，这是一种专门针对DOA问题设计的优化方法。
PSBL算法的优势在于理论分析支持其有效性和鲁棒性。文章接着介绍了两种改进方法：一是基于输出协方差矩阵的扰动协方差矩阵（PCM）算法，用于提升PSBL的收敛速度；二是改进的PCM（IPCM），旨在进一步优化性能。实验结果显示，PSBL在有限采样、低信噪比、相关信号以及邻近信号的情况下表现出最高的估计精度，证明了其在实际应用中的优越性。
这篇研究论文提供了一种创新的DOA估计方法，通过稀疏贝叶斯学习和扰动策略，有效地解决了离网信号方向估计的问题，为无线通信、信号处理等领域提供了有效的解决方案，尤其是在复杂环境下的信号检测和定位。该工作不仅提升了DOA估计的精度，还扩展了稀疏表示技术的应用范围，对于提高信号处理系统的性能具有重要意义。
2006 IEEE SENSORS JOURNAL, VOL. 16, NO. 7, APRIL 1, 2016
where A(ϑ) =[a(ϑ
1
), ···, a(ϑ
N
)] can be simply denoted
as A for brevity, ¯s(t) =[¯s
1
(t), ···, ¯s
N
(t)]
T
isasparsevector
where the non-zero entries indicate the true source signals.
Hence the signal directions can be determined easily once
¯s(t) is obtained using numerous existing algorithms for the
basic SMV model. When L snapshots are collected, the SMV
model (2) can be easily extended to an MMV model as,
X = A
¯
S + V , (3)
where X =[x(t
1
), ···, x(t
L
)] is the array output matrix,
¯
S =[¯s(t
1
), ···, ¯s(t
L
)] is the expanded signal matrix and
V =[v(t
1
), ···, v(t
L
)] is the measurement noises matrix of
the array.
It has been proven that, the DOA estimation performance
of the MMV model is always better than that of the SMV
model [33]. Also it is shown in [34] that, under certain mild
assumptions the recovery rate increases exponentially with the
number of measurement vectors.
Remark 1: A key assumption of the MMV model is that
each column of
¯
S shares the identical sparse structure, i.e.,
the non-zero entries of ¯s(t
l
)(l = 1, ···, L) should appear in
the same rows of
¯
S [33]. This is valid only if the directions
of incident signals change a little or even are invariant during
the acquisition of
¯
S. Unfortunately, the signal directions are
often time-varying in practice, hence a small L in the MMV
model is required from practical application point of view.
We assume the maximum of L is 150 in this paper.
Now we take the off-grid case into consideration, where
a bias exists between the true DOA and its nearest grid.
No matter how dense we divide the angle space, the bias
always exists. In general, the more dense the grid set is,
the more computational cost will be. Furthermore, a very
dense grid set may lead to a high correlation between a(ϑ
n
)
making many of the compressive sensing (CS) reconstruction
algorithms fail. We incorporate a bias parameter into the MMV
model (3) to avoid or alleviate the performance degradation
causedbyadensegridset.
Let a(θ
n
) = (1 −ρ
n
)a(ϑ
n
) +ρ
n
a(ϑ
n
),whereρ
n
is defined
as the bias parameter, ϑ
n
and ϑ
n
are the directions adjacent
to the true DOA θ
n
from the left and right, respectively. Then
the modified signal model can be rewritten as,
X =
¯
A
¯
S + V, (4)
where
¯
A is a new manifold matrix
¯
A = A(1 : N − 1)diag(1 − ρ) + A(2 : N)diag(ρ)
with ρ =[ρ
1
, ···,ρ
N−1
]
T
, (5)
where A(i : j) means the subset of A consisting of the
ith column through the jth column of A. By defining =
diag(ρ), I
f
=[I
N−1
, 0
N−1)×1
]
T
, I
b
=[0
N−1)×1
, I
N−1
]
T
,
(5) can be rewritten compactly as,
¯
A = AI
f
(I
N−1
− ) + AI
b
= AI
f
+ A(I
b
− I
f
)
= A
f
+ A
bf
, (6)
where A
f
= AI
f
, A
bf
= A(I
b
− I
f
).
Remark 2: The proposed off-grid model (6) is similar to
the off-grid model in [29] and [30]. In particular, the model
in [29] and [30] is a first-order approximation method while
our proposed model is a linear interpolation method. This
difference results in a distinction about the estimation per-
formance. A theoretical analysis of the proposed model will
be provided at the end of this section.
Remark 3: Some latest methods have been proposed for
the off-grid model presented in [29] such as the perturbed
1
-norm-based algorithm [28] and the perturbed greedy
algorithm [35]. However, it has been proven that the
1
-norm-
based algorithms often fail to obtain the sparsest solution and
the greedy ones are usually sensitive to the high correlation
between the columns of the manifold matrix [25]. Unlike these
two kinds of methods, the SBL does not rely on the restricted
isometry property (RIP) to guarantee reliable performance
and is convenient to incorporate proper priors to exploit the
signal’s structure. Hence, we employ the SBL algorithm to
solve our off-grid model (4) in this paper. In fact, the authors
of [29] have proposed a sparse Bayesian based algorithm with
a Gamma hyperprior assumption. However, this assumption is
likely to lead to the instability of the algorithm or even the
incorrect solution [36].
B. PSBL Algorithm
We assume that the columns of
¯
S are mutually independent,
and each column obeys a zero-mean Gaussian distribution with
variance , namely,
¯s(t
l
) ∼ CN(0, ) (7)
where = diag(γ ) with γ =[γ
1
, ···,γ
N
]
T
is the covariance
matrix of the lth column of
¯
S. Note that γ
n
(n = 1, ···, N)
is a nonnegative hyperparameter controlling the row sparsity
of
¯
S,i.e.,whenγ
n
= 0, the associated row of
¯
S becomes zero.
With this assumption, we can obtain the probability density
function (PDF) of
¯
S with respect to as follows,
p(
¯
S;) =|π|
−L
exp
−tr
¯
S
H
−1
¯
S
. (8)
Assume that the entries of the noise matrix V are mutually
independent and each row of V has a complex Gaussian
distribution, i.e., v
n
(t
l
) ∼ CN (0,σ
2
),whereσ
2
is the noise
power. For the SMV model (4), the Gaussian likelihood is
p(X|
¯
S;σ
2
, ρ) ∼ CN (
¯
A
¯
S,σ
2
I). (9)
Using the Bayes rule we obtain the posterior PDF of
¯
S as,
p(
¯
S|X;,σ
2
, ρ) ∼ CN (μ
¯s
,
¯s
) (10)
with mean
μ
¯s
=
¯
A
H
−1
x
X (11)
and the covariance matrix
¯s
= (
−1
+ σ
−2
¯
A
H
¯
A)
−1
= −
¯
A
H
−1
x
¯
A (12)
where
x
= σ
2
I +
¯
A
¯
A
H
.
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