稀疏贝叶斯学习提升离网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"这一研究领域。在当前的信号处理技术中，传统的空间离散网格方法在进行方向到达(DOA)估计时存在显著局限性，这限制了信号的精确表示和算法的实用性。预定义的离散网格假设往往无法捕捉到实际信号可能存在的连续或非均匀特性，导致估计精度下降。为了克服这一问题，作者提出了一种新的方法，即在DOA估计框架中引入了一个偏置参数，从而形成了一种扰动的稀疏信号表示模型。这种方法旨在提供更灵活的空间分辨率，使得算法能够适应离网信号（即不在预设网格中的信号）的方向估计。在这个新模型的基础上，作者开发了一种名为Perturbed Sparse Bayesian Learning (PSBL)的算法，该算法结合了稀疏贝叶斯学习的理论优势。PSBL利用贝叶斯统计的原理，通过后验概率分布来寻找最有可能的方向到达向量，并考虑到噪声、模型不确定性以及信号的稀疏性。它能够自适应地估计信号的方向，同时处理非正则化带来的复杂性，提高了估计的稳定性和准确性。理论分析部分，文章深入探讨了PSBL算法的收敛性、稳健性和计算效率，以及如何通过调整偏置参数优化估计性能。此外，还可能涉及了与传统DOA估计方法（如 MUSIC、 ESPRIT等）的对比分析，展示了PSBL在解决离网信号到达方向估计问题上的优势。总结来说，这篇研究论文为离网信号的DOA估计提供了一种创新且具有潜力的解决方案，通过稀疏贝叶斯学习的策略，提高了估计精度和适用范围，为无线通信、雷达信号处理等领域中的信号分析带来了新的思路和技术突破。未来，该方法可能被广泛应用于需要高精度和灵活性的信号处理场景中。

1530-437X (c) 2015 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/JSEN.2015.2508059, IEEE Sensors

Journal

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

), ··· , x(t

)] is the array output matrix,

S = [

s(t

), ··· ,

s(t

)] is the expanded signal matrix and V =

[v(t

), ··· , v(t

)] 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 = 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

) 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 caused by a

dense grid set.

Let a(θ

) = (1−ρ

)a(ϑ

)+ρ

a(ϑ

), where ρ

is deﬁned

as the bias parameter, ϑ

and ϑ

are the directions adjacent

to the true DOA θ

from the left and right, respectively. Then

the modiﬁed signal model can be rewritten as,

X =

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

, ··· , ρ

N−1

]

(5)

where A(i : j) means the subset of A consisting of the ith co-

lumn through the jth column of A. By deﬁning ∆ = diag(ρ),

= [I

N−1

, 0

N−1)×1

]

, I

= [0

N−1)×1

, I

N−1

]

, (5) can be

rewritten compactly as,

A = AI

N−1

− ∆) + AI

∆

= AI

+ A(I

− I

)∆

= A

+ A

∆,

(6)

where A

= AI

, A

= A(I

− I

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 differ-

ence results in a distinction about the estimation performance.

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 ℓ

norm-based algorithm [28] and the perturbed greedy algorithm

[35]. However, it has been proven that the ℓ

-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 Γ, namely,

s(t

) ∼ CN(0, Γ) (7)

where Γ = diag(γ) with γ = [γ

, ··· , γ

]

is the covariance

matrix of the lth column of

S. Note that γ

(n = 1, ··· , N) is

a nonnegative hyperparameter controlling the row sparsity of

S, i.e., when γ

= 0, the associated row of

S becomes zero.

With this assumption, we can obtain the probability density

function (PDF) of

S with respect to Γ as follows,

S; Γ) = |πΓ|

−

exp



−tr



−



. (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

) ∼ CN(0, σ

), where σ

is the noise

power. For the SMV model (4), the Gaussian likelihood is

p(X|

S; σ

, ρ) ∼ CN(

S, σ

I). (9)

Using the Bayes rule we obtain the posterior PDF of

S as,

S|X; Γ, σ

, ρ) ∼ CN(µ

¯s

, Σ

¯s

) (10)

with mean

¯s

= Γ

−1

X (11)

and the covariance matrix

¯s

= (Γ

−1

+ σ

−2

−1

= Γ − Γ

−1

AΓ

(12)

where Σ

= σ

I +

AΓ

To ﬁnd the hyperparameters Θ =



γ, σ

, ρ



, we employ

the EM method to minimize −log p(X,

S; Θ) while treating

S as hidden variables. It is equivalent to maximizing

Q(Θ) = E

S|X;Θ

old



log p(X,

S; Θ)



= E

S|X;Θ

old



log p(X|

S; σ

, ρ)



+ E

S|X;Θ

old



log p(

S; Γ)



(13)

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