Research ArticleDOA Estimation for a Mixture of Uncorrelated and CoherentSources Based on Hierarchical Sparse Bayesian Inference with aGauss-Exp-Chi2 Prior
Pinjiao Zhao ,1 Weijian Si,2 Guobing Hu,1 and Liwei Wang3
1Department of Electronic and Information Engineering, Jinling Institute of Technology, Nanjing 211169, China2Department of Information and Communication Engineering, Harbin Engineering University, Harbin 150001, China3No. 8511 Research of Institute of CASIC, Nanjing 210007, China
Correspondence should be addressed to Pinjiao Zhao; [email protected]
Received 29 November 2017; Accepted 31 May 2018; Published 10 July 2018
Direction of arrival (DOA) estimation algorithms based on sparse Bayesian inference (SBI) can effectively estimate coherentsources without recurring to extra decorrelation techniques, and their estimation performance is highly dependent on theselection of sparse prior. Specifically, the specified sparse prior is expected to concentrate its mass on the zero and distributewith heavy tails; otherwise, these algorithms may suffer from performance degradation. In this paper, we introduce a newsparse-encouraging prior, referred to as “Gauss-Exp-Chi2” prior, and develop an efficient DOA estimation algorithm for amixture of uncorrelated and coherent sources under a hierarchical SBI framework. The Gauss-Exp-Chi2 prior distributionexhibits a sharp peak at the origin and heavy tails, and this property makes it an appropriate prior to encourage sparsesolutions. A three-layer hierarchical sparse Bayesian model is established. Then, by exploiting variational Bayesianapproximation, the model parameters are estimated by alternately updating until Kullback-Leibler (KL) divergence between thetrue posterior and the variational approximation becomes zero. By constructing the source power spectra with the estimatedmodel parameters, the number and locations of the highest peaks are extracted to obtain source number and DOA estimates. Inaddition, some implementation details for algorithm optimization are discussed and the Cramér-Rao bound (CRB) of DOAestimation is derived. Simulation results demonstrate the effectiveness and favorable performance of the proposed algorithm ascompared with the state-of-the-art sparse Bayesian algorithms.
1. Introduction
Direction of arrival (DOA) estimation has been a crucialissue in various application areas involving radar, wirelesscommunication, and navigation [1–3]. Multiple signal classi-fication (MUSIC) [4] and estimation of signal parameter viarotational invariance technique (ESPRIT) [5], known as thetwo most classical subspace-based DOA estimation algo-rithms, have been proposed to resolve uncorrelated sources.However, in multipath propagation environments, sourcesfrom an identical target may undergo reflection from varioussurfaces, and thus, the received sources may be a mixture ofuncorrelated and coherent sources. In such environments,
the aforementioned subspace-based algorithms would sufferfrom serious performance deterioration or even fail [6].
In order to solve the aforementioned problem, severalpreprocessing techniques are developed for decorrelation.In the related studies, these preprocessing techniques aremainly classified into two categories: spatial smoothing(SS) [7] and matrix reconstruction (MR) [8]. Theoretically,the SS technique is implemented by dividing the wholearray into multiple subarrays to combat the rank deficiency,while the MR technique is performed by rearranging therank-deficient matrix into Hankel matrix, Toeplitz matrix,or other matrices to restore the rank. By combing the SS orMR techniques with subspace-based algorithms, several
HindawiInternational Journal of Antennas and PropagationVolume 2018, Article ID 3505918, 12 pageshttps://doi.org/10.1155/2018/3505918
DOA estimation algorithms have been developed to han-dle the scenarios where uncorrelated and coherent sourcescoexist [9–13]. More specifically, the oblique projectionspatial smoothing (OPSS) [9] and moduli property spatialsmoothing (MPSS) [10] are the typical SS-based algo-rithms, and the oblique projection Toeplitz matrix recon-struction (OPT-MR) [11], symmetric non-Toeplitz matrixreconstruction (SNT-MR) [12], and real-valued Hankelmatrix reconstruction (RVH-MR) [13] are the popularMR-based algorithms.
Unlike traditional subspace-based DOA estimation algo-rithms, the emerging sparse source reconstruction (SSR)algorithms [14–19], including matching pursuit (MP) algo-rithm [14], lp-norm optimization algorithms [15, 16], andsparse Bayesian inference (SBI) algorithms [17–19], providea new perspective for DOA estimation. Since SSR-basedalgorithms realize DOA estimation via sparse source recon-struction, instead of calculating the covariance matrix, theycan resolve the coherent sources directly without extra pre-processing techniques. Among these SSR-based algorithms,both the MP and the lp-norm optimization algorithms arerestricted to solve an optimization problem and are reliedon the point estimate, so that these algorithms ignore theuncertainty of the underlying source in the process of sourcereconstruction. By contrast, SBI-based algorithms speciallyconsider the uncertainty of the underlying source andestimate the source via choosing an appropriate prior,which yield favorable reconstruction performance [20].Moreover, they can provide good estimation performancein the case of low SNR or small number of snapshots [21].In general, SBI-based algorithms first specify a sparsity-encouraging prior to the unknown source model and thenthe model parameters are estimated via Bayesian inference.Since the exact Bayesian inference is intractable, two main-stream approximation inference algorithms were presentedto estimate the model parameters, in which one is evidenceprocedure [22] and the other is variational Bayesian approx-imation [23]. For the evidence procedure, some unknownhyperparameters with respect to the hierarchical prior modelare estimated iteratively by maximizing the evidence. Forthe variational Bayesian approximation, the posterior dis-tribution is approximated as the product of several tracta-ble distributions, and the model parameters keep updatingto minimize the Kullback-Leibler (KL) divergence betweenthe true posterior and the variational approximation, whichhas attractive computational efficiency along with high esti-mation performance [24]. Note that these approximationinference algorithms operate under the premise that anappropriate sparse prior has been imposed on the sourcemodel for the purpose of encouraging sparse solutions. Manysparsity-encouraging prior models have been investigated inthe SBI framework [25–27]. In [25], Bayesian compressedsensing (BCS) was proposed with a two-layer hierarchicalGaussian-inverse-gamma prior (or Student’s-t prior), wherethe first layer is a Gaussian probability density function(pdf) and the second layer is a gamma pdf. Babacan et al.[26] proposed an equivalent Laplace prior that is generatedby a Gaussian prior and an Exponential prior. In [27], a nor-mal product (NP) prior was developed with two algorithms:
NP-0 (using one-layer source model) and NP-1 (using two-layer source model). However, these priors are concentratednear the origin with relatively light tails, which may causeovershrinkage of the incident sources [28].
In this paper, we develop a new sparse-encouraging prior(called Gauss-Exp-Chi2 prior) whose pdf distributionexhibits a sharp peak at the origin and heavy tails. With theproposed prior, the DOA estimation for a mixture of uncor-related and coherent sources is performed under the hierar-chical SBI framework using a uniform linear array (ULA).A three-layer hierarchical Bayesian model is establishedbased on the Gauss-Exp-Chi2 prior. Subsequently, accordingto the variational Bayesian approximation, the model param-eters (including the mean and variance of sparse sources andhyperparameters) keep alternately updating until the KLdivergence between the true posterior and the variationalapproximation tends to be zero. By exploiting the estimatedmodel parameters, the source power spectra is constructed,from which the number and locations of the highest peaksare extracted to obtain source number and DOA estimates.Simulation results show that the proposed algorithm hassuperior estimation performance. Now we briefly summarizethe contributions of this work as follows:
(i) To encourage sparse solutions, we develop a newsparse-encouraging prior, called Gauss-Exp-Chi2
prior, whose pdf distribution has a sharp peak atthe origin and heavy tails.
(ii) By constructing the source powers of all the poten-tial directions in the angular space, both sourcenumber and DOA estimates are obtained.
(iii) Variational approximations are adopted for the esti-mation of the hierarchical sparse Bayesian modelparameters.
(iv) Several implementation details for algorithm opti-mization including Woodbury matrix identity fordimension-reduction, pruning a basis functionand the third kind Bessel function approximationare discussed, and the CRB of DOA estimationis derived.
The remainder of this paper is organized as follows. TheDOA estimation model for mixed sources is formulated inSection 2. Section 3 presents the Gauss-Exp-Chi2 prior andDOA estimation algorithm for a mixture of uncorrelatedand coherent sources within the hierarchical SBI framework.The algorithm optimization and CRB of DOA estimation arediscussed in Section 4. Section 5 presents the simulationresults of the proposed algorithm. Conclusions are drawnin Section 6.
Notations. Vectors and matrices are denoted by lowercaseand uppercase boldface letters, respectively. ⋅ T , ⋅ H ,⋅ −1, and ⋅ represent transpose, conjugate transpose,
inverse, and the statistical expectation, respectively. I is anidentity matrix. ⋅ 2 denotes the Euclidean norm, and diag
2 International Journal of Antennas and Propagation
⋅ denotes a diagonal matrix. Additionally, bavdv repre-
sents the integral of v from a to b.
2. Problem Formulation
Consider a total of K far-field narrowband sources imping-ing on the uniform linear array (ULA) consisting of Momnidirectional sensors with the interspacing between adja-cent sensors d being a half of the carrier wavelength λ, thatis, d = λ/2. Then, the array output vector at the tth snapshotcan be expressed as
x t = 〠K
k=1a θk sk t + n t , t = 1, 2,… , T , 1
where T is the number of snapshots, a θk =1, e−jπ sin θk ,… , e−jπ M−1 sin θk
Tis the steering vector corre-
sponding to the direction θk of the kth impinging sourcesk t , and n t denotes the noise vector. Without loss of gen-erality, the impinging source sk t K
k=1 is a mixture ofuncorrelated and coherent sources. Note that there existspower fading among the coherent source, and fading coef-ficients are used to describe the degree of power fading[5]. Specifically, consider K far-field narrowband sourcessk t K
k=1 impinging on the ULA, in which the first Kusources s1 t , s2 t ,… , sKu
t are uncorrelated, and the lastKcsources sKu+1 t , sKu+2 t ,… , sK t are coherent. Then,
the uncorrelated source set is denoted as sk t Kuk=1 and
the coherent source set is denoted as sk t Kk=Ku+1 =
1, ρ1, ρ2,… , ρKc−1TsKu+1 t with ρ1, ρ2,… , ρKc−1are the
fading coefficients.Divide the entire angular space into J sampling grids
θ = θ 1, θ 1,… , θ JT, where J represents the grid number
and generally satisfies J ≫M > K . Assume that θ jk ∈
θ 1, θ 1,… , θ JTis the nearest grid to true direction θj ∈
θ1, θ2,… , θK T ; thus, we have hj t = sjk t δ j − jk . Thus,x t can be rewritten in a sparse form
x t = A h t + n t , t = 1, 2,… , T , 2
where A = α θ 1 , α θ 2 ,… , α θ J and h t =h1 t , h2 t ,… , hJ t
T . Due to the fact that h t hasK non-zero elements in J elements, it is a sparse vector. Inthe case of multiple snapshots, the sparse sources at all thesnapshots share the same support, and the array outputmatrix of T snapshots can be represented by
X = A H +N, 3
where X = x 1 , x 2 ,… , x T , H = h 1 , h 2 ,… , h T ,and N = n 1 n 2 … n T . The goal of this paper is toprovide the DOA estimation under the coexistence ofuncorrelated and coherent sources from a sparse Bayesianperspective.
3. DOA Estimation
In this section, a DOA estimation algorithm for a mixtureof uncorrelated and coherent sources is proposed withinthe hierarchical SBI framework. A Gauss-Exp-Chi2 prioris developed to encourage sparse solutions, and then theparameters of three-layer hierarchical Bayesian model areestimated via variational Bayesian approximations. By con-structing source power spectra, the source number andDOA estimation are obtained.
3.1. Bayesian Model. In the Bayesian model, the pdf of a jointdistribution p X,H, α, η with respect to all the unknownand observed quantities is required to be known. In thispaper, the joint distribution can be expressed as
p X,H, α, η = p X H, σ2 p H α p α τ, η p η v , 4
where α and η are referred to as the hyperparameters; τ ≥ 0and v ≥ 0 are, respectively, referred to as the rate parameterand shape parameter. It is assumed that the componentsn t , t = 1, 2,… , T are the independently zero-mean station-ary Gaussian noise with known variance σ2. Thus, the pdf ofthe noise matrix N is given by
p N σ2 = ∏T
t=1N n t 0, σ2I 5
Combining (3) and (5), the Gaussian likelihood model isobtained as follows:
p X H, σ2 = ∏T
t=1N x t A h t , σ2I 6
From a Bayesian perspective, the pdf distribution of anassigned prior is appealing to exhibit a sharp peak at the ori-gin and heavy tails, which favors strong shrinkage of noisesources and avoids overshrinkage of the interest sources.This property is generally considered as a desirable propertyfor enforcing sparsity and variable selection [24]. Some typ-ical sparse priors, such as Gaussian-inverse-gamma priorand Laplace prior, are widely used in the relevant research[25, 26], which, however, are concentrated near the originwith relatively light tails [28]. To alleviate this problem, wehere develop a three-layer hierarchical prior, referred to asGauss-Exp-Chi2 prior, for H. In the first layer of prior, weadopt a zero-mean Gaussian prior
p H α = ∏T
t=1N h t 0,Λ−1
= ∏T
t=1∏J
j=1N hj t 0, αj ,
7
with being Λ−1 = diag α = diag α1, α2,… , αJ . In the sec-ond layer of prior, an exponential hyperprior is imposedon α
p α τ, η = ∏J
j=1Exp αj τη , 8
3International Journal of Antennas and Propagation
where Exp ⋅ denotes the exponential distribution. In thethird layer of prior, a chi-square (Chi2) hyperprior is con-sidered over η, that is,
p η v = χ2 η v = Γ v2
−12−v/2ηv/2−1 exp −
η
2 , 9
where Γ ⋅ denotes the gamma function.The first two layers (7) and (8) of the proposed prior
result in a generative Laplace (Gaussian-Exponential) distri-bution [26], and the last layer (i.e., a Chi2 distribution) isembedded to obtain the proposed three-layer Gauss-Exp-Chi2 prior. Thus, the proposed prior has more free parame-ters to control the degree of sparseness as compared withthe Laplace prior [29].
Based on the above analysis, the directed graph of thesparse Bayesian model is shown in Figure 1, where arrowsare used to point to the generative model. Note that the firstfive blocks from the left (corresponding to the variables v,η, τ, α1,… , αJ , and h1 t ,… , hJ t ) depict the three-layerhierarchical prior mentioned above.
Since p αj ∣ τv ∝ ταj + 1/2 − v/2+1 , j = 1, 2,… , J , it isobvious that most αj for j = 1, 2,… , J are favored to zero,which leads to the fact that most hj t are favored to zero.Thus, the proposed three-layer hierarchical Gauss-Exp-Chi2
prior is a sparsity-encouraging prior, which can help toimprove the performance of source reconstruction [30]. Bycombining the three layers of the proposed prior, the gener-ated prior can be computed via
p H ; τ, v = ∏T
t=1∏J
j=1p hj t ; τ, v , 10
where
p hj t ; τ, v =+∞
0
+∞
0p hj t αj p αj τ, η p η v dαjdη
11
Note that p αj ∣ τv = +∞0 p αj ∣ τη p η ∣ v dη; thus, p hj
t ; τ, v can be computed by marginalizing p hj t ∣ αj
p αj ∣ τv over αj (see Appendix for detail).
p H ; τ, v = ∏T
t=1∏J
j=1−
τ
πΓ v/2 + 1 Γ v/2 + 1/2
Γ v/2
Uchfv2 + 1
2 ,12 , h
2j t τ ,
12
where Uchf ⋅ , ⋅ , ⋅ is the confluent hypergeometric functiondefined by
Uchf a, a − b + 1, c = 1Γ a
+∞
0ta−1 1 + t −b exp −ct dt
13
The reason for choosing the Gauss-Exp-Chi2 prior istwofold: its pdf distribution has a sharp spike at the originand heavy tails; it forms in a conjugate manner since theexponential and Chi2 distributions belonging to the expo-nential distribution family are chosen as the 2nd layer andthe 3rd layer of this prior, which significantly simplifiesthe form of posterior distributions [24]. The tail to spikebehavior for Gauss-Exp-Chi2 prior, Laplace prior, Stu-dent’s-t prior, and Gaussian prior is illustrated in Figure 2,wherein the parameter selection of these priors is accordingto the standard derivation [25, 26]. It is observed that theproposed Gauss-Exp-Chi2 prior is a sparsity-encouragingprior which has sharper peak and heavier tails than theexisting priors.
3.2. Variational Bayesian Approximations. It is well knownthat Bayesian inference operates on the basis of the posteriordistribution p H, α, η ∣X = p H, α, η,X /p X . However,this posterior distribution is intractable for the reason thatp X = p H, α, η,X dHdαdη cannot be calculated ana-lytically. In order to approximate this posterior distribution,we resort to mean-field variational Bayesian approximation[23], whose task is to seek several analytically approximatedistributions that minimize the KL divergence between theposterior p H, α, η ∣X and its approximation distributionqH, α, η . According to [23], q H, α, η can be factorized asthe product of three variational distributions q H , q α ,and q η ; thus, we have
p H, α, η X ≈ q H, α, η = q H q α q η 14
Denote a set ζ = H, α, η , the distribution of each vari-able ζk is expressed as ln q ζk ∝ <ln q ζk,X >q ζ/ζk , whereζ/ζk represents the subset excluding ζk. More specifically,the posterior approximation of H, α, and η in this paperare, respectively, given by
ln q H ∝ <ln p X,H, α, η >q α , 15
ln q α ∝ <ln p X,H, α, η >q H q η , 16
ln q η ∝ <ln p X,H, α, η >q α 17
τv 𝜂 𝜎2
xJ(t)hJ(t)
h1(t) x1(t)𝛼1
𝛼j
Figure 1: Directed graph of the sparse Bayesian model.
4 International Journal of Antennas and Propagation
(1) Update of q H . Keeping the terms of q H onlydepend on H, we have
ln q H ∝ <ln p X H p H α >q α
∝ 〠T
t=1σ−2hH t Α
HΑ x t −
12σ
−2hH t ΛHh t ,
18
with Λ = diag 1/α1, 1/α2,… , 1/αJ . By gathering togetherthe similar terms, we have
ln q H ∝ 〠T
t=1−12 h
H t σ−2ΑHΑ + <Λ > h t
+ σ−2 〠T
t=1hH t Α
Hx t ⇒q H
∝ 〠T
t=1exp −
12 h
H t Σ−1h t + hH t Σ−1μ t ,
19
which implies that q H is the product of multiple Gaussiandistributions N h t ∣ μ t Σ with the mean μ t andcovariance Σ being
μ t = σ−2Σ−1ΑHx t , 20
Σ = σ−2ΑHΑ + <Λ >
−121
(2) Update of q α . Similarly, according to (16), q α isderived as
−5 −4 −3 −2 −1 0 1 2 3 4 50
0.1
0.2
0.3
0.4
0.5
0.6
0.7
x
p(x)
Gauss−Exp−Chi2Laplace
Student′s−tGaussian
(a)
5 6 7 8 9 10 11 12 13 14 15x
Gauss−Exp−Chi2Laplace
Student′s−tGaussian
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
p(x)
(b)
Figure 2: Four kinds of pdf curves with the standard derivation: (a) compares four distributions at origin and (b) compares four distributionsin tail.
ln q α ∝ <ln p H α p α τ, η >q H q η
∝ <ln ∏T
t=1∏J
j=1
12παj
exp −h2j t
2αj
∏J
j=1τη exp −τηαj >q H q η
22
Denoting h2j =∑Tt=1h2j t and <h2j > = μ2j + Σjj, (22) can
be further simplified to
ln q α ∝ 〠J
j=1< ln α−1/2j exp −
h2j2αj
− τηαj >q H q η
⇒q α ∝ ∏J
j=1α−1/2j exp −
<h2j >2αj
− τ < η > αj
23
The posterior distribution of q α can be approximatelyequivalent to the product of a series of generalized inverseGaussian (GIG) distributions, that is, α ∼∏J
j=1GIG αj ∣ 1/2< h2j > 2τ < η > , and thus, we have
<αnj > =<h2j >2τ < η >
n/2 κn−1/2 2τ < η > <h2j >
κ−1/2 2τ < η > <h2j >, 24
where κp ⋅ is the third kind Bessel function with order p.The case of n = 1 in (24) gives the calculation of <αj > , whichcan be used in (25). The case of n = −1 gives <α−1j > used forthe evaluation of Σ in (21).
5International Journal of Antennas and Propagation
(3) Update of q η . For q η , we have
ln q η ∝ <ln p α τ, η p η v >q α
∝ <ln ∏J
j=1τη exp −τηαj
2−v/2Γ v/2 ηv/2−1 exp −
η
2
>q α ⇒q η ∝ ηJ+v/2−1 exp −τ〠J
j=1< αj > η −
η
225
Thus, it can be known that q η is a gamma distribution,that is, η ∼Gamma J + v/2, τ∑J
j=1 < αj > +1/2 with the fol-lowing mean
<η > = J + v/2τ〠J
j=1 < αj > +1/226
The estimation of model parameters is implemented byalternately updating the mean μ, variance Σ, and hyperpara-meters <αj > and <η > to minimize the KL divergencebetween the true posterior and the variational approxima-tion, and the main steps are summarized in Algorithm 1.
3.3. Source Power and DOA Estimation. The DOAs of theimpinging sources are estimated via the following two steps:(1) form spatial spectra using the estimated source powersof all the potential directions and (2) extract the numberand the corresponding locations of the peaks beyond a powerthreshold from the spectra. LetU= μ 1 , μ 2 ,… , μ T andΣ be the final estimates of U and Σ (the mean and variancewith respect to H) and consider H row by row, then we havehj ∼N Uj, Σjj
I . As outlined in [18], let Pojbe the source
power of direction θ j; then we have
Poj= 1T
Uj
2
2+ TΣ̂jj = 1
TUj
2
2+ Σ̂jj 27
Therefore, the source powers of all the potential direc-tions in the angular space are estimated by calculating
Po = Po1, Po2
,… , PoJ
T. Suppose that there are K peaks
exceeding the power threshold Pthres and the correspondinggrid indices are jk jk = 1, 2,… , K . Then, the estimated
source number and DOAs will be K and θ jkjk = 1, 2,… , K .
4. Discussions
4.1. Algorithm Optimization. For alleviating the computa-tional complexity and speeding up the update efficiency,some implementation details are adopted for algorithm opti-mization in this subsection.
4.1.1. Woodbury Matrix Identity for Dimension-Reduction.At each iteration, it is inevitable to calculate a J × J matrixinversion when updating Σ according to (21), which requiresO J3 computations. To reduce the computational complex-ity, we adopt the Woodbury matrix identity, and then (21)
can be rewritten as Σ = <Λ−1 > − <Λ−1 >AHG−1 A <Λ−1 >with an M ×M matrix G = σ−2I + A <Λ−1 >AH
. Thus, thecomputational complexity is reduced to O M3 .
4.1.2. Pruning a Basis Function. In order to speed up the
update efficiency, <αj > and the corresponding a θ j wouldbe pruned from the original sparse Bayesian model when<αj > is smaller than a certain threshold. This procedure isreferred to as “pruning a basis function”, and similarapproaches have been used in [29, 30].
4.1.3. The Third Kind Bessel Function Approximation. It isknown that (24) involves the third kind Bessel function,and this integral process is relatively complicated. Accordingto [31], the third kind Bessel function of m can be approxi-mated as κp m ∼ 1/2Γ p m/2 −p, if p > 0, and κp m ∼ 1/2Γ −p m/2 p, if p < 0. Then, (24) can be simplified to
<αnj > =
<h2j >2τ < η >
1/2
, n = 1,
<h2j >2τ < η >
−1/2 2Γ 3/2Γ 1/2
12τ < η > <h2j >
, n = −1
28
Input: Α, XOutput: μ, Σ1. Initialize: <αj > , <α−1j > , <η > , τ, v, σ2, ε, imax2. while <αj> i − <α j> i−1 / <αj> i−1 > ε and i < imax do
3. update Σ i using (21)
4. update μ t i using (20)5. update <αj> i and <α−1j > i using (24)
6. update <η> i using (26)7. i = i + 18. end while
Algorithm 1: The main steps of the model parameter estimation.
6 International Journal of Antennas and Propagation
4.2. Cramér-Rao Bound (CRB). To evaluate the DOA esti-mation performance, the CRB of the DOA estimation fora mixture of uncorrelated and coherent sources is derivedin this subsection. Consider the sparse form of the array out-put vector x t mentioned in (2), the log-likelihood functionof x 1 , x 2 ,… , x T is firstly construct as
ln ℒ x 1 , x 2 ,… , x T
= const −MT ln σ − σ−1 〠T
t=1x∗ t − h∗ t A
∗
· x t − A h t
29
Let hr t and hi t be the real part and imaginary part ofh t , that is, hr t ≜ Re h t and hi t ≜ Im h t . Then,the partial derivatives of (29) with respect to hr t , hi t , σ,and θ are, respectively, given as
∂ ln ℒ∂hr t
= 2σ−1 Re A∗n t ,
∂ ln ℒ∂hi t
= 2σ−1 Im A∗n t ,
∂ ln ℒ∂σ
=MTσ−1 + σ−2 〠T
t=1n∗ t n t ,
∂ ln ℒ
∂ θ= 2σ−1 〠
T
t=1Re H∗
d t A∗Dn t ,
30
where H∗d t = diag h1 t , h2 t ,… , hJ t and AD =
da θ 1 /dθ 1, da θ 2 /dθ 2,… , da θ J /dθ J . Subsequently,define
ℵ = E∂ ln ℒ
∂ θ∂ ln ℒ
∂ θ
T
= 2σ−1 〠T
t=1Re H∗
d t A∗DADHd t ,
Ξmr = Re Ξm = E
∂ ln ℒ∂hr t
∂ ln ℒ
∂ θ
T
= 2σ−1 Re A∗ADHd t ,
Ξmi = Im Ξm = E
∂ ln ℒ∂hi t
∂ ln ℒ
∂ θ
T
= 2σ−1 Im A∗ADHd t ,
Qr = E∂ ln ℒ∂hr t
∂ ln ℒ∂hr k
T −1
= E∂ ln ℒ∂hi t
∂ ln ℒ∂hi k
T −1
= 2σ−1 Re A∗A δkt
−1,
Qi = −E∂ ln ℒ∂hr t
∂ ln ℒ∂hi k
T −1
= 2σ−1 Im A∗A δkt
−1
31
Then, the Fisher information matrix (FIM) [32] can beobtained as
FIM =ℵ − Ξ1rΞ1
i Ξ2rΞ2
i ⋯ ΞMr ΞM
i
=
Qr −Qi 0 ⋯ 0
Qi Qr −Qi ⋱ ⋮
0 Qi ⋱ ⋱ 0
⋮ ⋱ ⋱ Qr −Qi
0 ⋯ 0 Qi Qr
Ξ1r
Ξ1i
Ξ2r
Ξ2i
⋮
ΞMr
ΞMi
32
It is noted that the following equality holds
ΞTr ΞT
i
Qr −Qi
Qi Qr
Ξr
Ξi
= Re Ξ∗QΞ , 33
with Q = 2σ−1A∗A
−1and Ξ = 2σ−1A
∗ADHd t . Finally,
by combing (32) and (33), we have
CRB θ = FIM−1 = 0 5σ
〠T
t=1Re Hd t A∗
D I − A A∗A
−1A
∗ADHd t
−1
34
5. Simulation Results
In this section, several simulations are presented to illustratethe performance of the proposed algorithm as comparedwith the BCS [25] and NP-1 [27] algorithms. Consider a 9-sensor ULA with sensor interspacing being a half of the car-rier wavelength. In the proposed algorithm, hyperparameters
α and η are initialized as 1/T∑Tt=1A
Hx t and 0.1; thevalues of rate parameter τ, shape parameter v, and power
7International Journal of Antennas and Propagation
threshold Pthres are, respectively, set to be τ = 1 5, v = 1,and Pthres = −10 dB. Two hundred independent MonteCarlo trials are conducted for the following simulations,and the success rate and root mean squared error (RMSE)are two significant performance metrics for evaluating theDOA estimation performance. The success rate is definedas the rate between the number of successful estimatesand the total number of Monte Carlo trials, where a suc-cessful estimate is declared if the estimation error is withina certain small Euclidean distance of the true directions.The RMSE is defined by
RMSE = 1200K〠
200
i=1〠K
k=1θ k,i − θk
2, 35
where θ k,i is the estimates of θk in the ith Monte Carlo trial.In the first simulation, we evaluate the effectiveness of the
proposed algorithm. Assume that two uncorrelated sourcesfrom −32 2°, −11 5° and two coherent sources from 6 7°,23 4° with fading coefficients 1, 0 2891 − 0 7567j impingeon this ULA. The grid number, SNR, and number ofsnapshots are, respectively, set to be 361, 0 dB, and 200.Figure 3 plots the source power spectrum, which is a func-tion of DOA.
As can be seen from Figure 3, the power peaks in thevicinity of the true DOAs can be distinguished from otherpeaks by their relatively strong powers and the biasesbetween these peaks and the true DOAs are slight, whichdemonstrate the effectiveness of the proposed algorithm.
The second simulation compares the estimation per-formance of three algorithms, including the proposedalgorithm, BCS, and NP-1 algorithms. Consider one uncor-related source from 19 2° and two coherent sources from−25 5°, 33 3° with fading coefficients being 1, −0 5280 +
0 6010j . The grid number and SNR are, respectively, fixedat 361 and 0dB. The success rate versus number of snapshotsand SNR are depicted in Figures 4 and 5, whereas the RMSEversus number of snapshots and SNR are shown in Figures 6and 7. The results from Figures 4 and 5 illustrate that all thethree algorithms are able to estimate the DOA correctlyunder the condition that the SNR or number of snapshotsare relatively high, but for low SNR or small number of snap-shots, the proposed algorithm has a higher success rate thanthe BCS and NP-1 algorithms. Figures 6 and 7 show that theestimation accuracy of the three algorithms tends to improvewith the increase of SNR or the number of snapshots, and the
−80 −60 −40 −20 0 20 40 60 80−25
−20
−15
−10
−5
0
DOA (degrees)
Pow
er (d
B)
TruthEstimated
Figure 3: Spectra for the proposed algorithm with the fixed SNR0 dB and number of snapshots 200.
50 150 250 350 450 550 650 750 850 9500.65
0.7
0.75
0.8
0.85
0.9
0.95
1
Number of snapshots
Succ
ess r
ate
ProposedNP−1BCS
Figure 4: Success rates versus number of snapshots with the fixedSNR 0 dB.
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
Succ
ess r
ate
ProposedNP−1BCS
−10 −8 −6 −4 −2 0 2 4 6 8 10SNR (dB)
Figure 5: Success rates versus SNR with the fixed number ofsnapshots 200.
8 International Journal of Antennas and Propagation
proposed algorithm yields the best estimation accuracy ascompared to the other two algorithms. This is because theproposed Gauss-Exp-Chi2 prior is a three-layer sparsity-encouraging prior, which can help to improve the accuracyof source reconstruction.
The third simulation investigates the RMSE versus gridnumber. The simulation settings are the same as those ofthe second simulation, except that grid number in this simu-lation is ranged from 61 to 361. Figure 8 plots the RMSEcurves versus grid number with fixed SNR 10dB and numberof snapshots 500.
It is observed that the RMSE curves of the three algo-rithms rapidly decrease as grid number increases from 61to 181, and these curves tend to slowly decrease with the
increase of grid number when grid number is beyond 181.In addition, the proposed algorithm has minimum estima-tion errors among the three algorithms. Note that the threealgorithms recover sparse sources from a Bayesian perspec-tive; thus, they are not restricted to the restricted isometryproperty. This implies that the grid number can continue toincrease to further improve the estimation accuracy and thecomputational complexity would increase accordingly.Therefore, for the purpose of balancing estimation accuracyand computational complexity, the reasonable range of gridnumber is from 181 to 361.
In the last simulation, we test the angular resolutionof the three algorithms. Consider two coherent sources from
50 150 250 350 450 550 650 750 850 950
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Number of snapshots
RMSE
(deg
ree)
ProposedNP−1
BCSCRB
Figure 6: RMSE versus number of snapshots with the fixed SNR0 dB.
−10 −8 −6 −4 −2 0 2 4 6 8 10
0.3
0.4
0.5
0.6
0.7
0.8
SNR (dB)
RMSE
(deg
ree)
ProposedNP−1
BCSCRB
Figure 7: RMSE versus SNR with the fixed number of snapshots200.
60 90 120 150 180 210 240 270 300 330 360
0.4
0.6
0.8
1
1.2
1.4
1.6
Grid number
RMSE
(deg
ree)
ProposedNP−1
BCSCRB
Figure 8: RMSE versus grid number with the fixed SNR 10 dB andnumber of snapshots 500.
3 4 5 6 7 8 9 100.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Angular separation (degree)
RMSE
(deg
ree)
ProposedNP−1
BCSCRB
Figure 9: RMSE versus angular separation with the fixed SNR 10 dBand number of snapshots 500 for coherent sources.
9International Journal of Antennas and Propagation
10.1° and 10 1° + Δθ with the angular separation Δθ rangingfrom 3° to 10°. The fading coefficients are 1, 0 4469 −0 7696j , and the grid number is set to be 361. Figure 9depicts the RMSE versus angular separation with the fixedSNR 10dB and number of snapshots 500. The results fromFigure 9 show that both the BCS and NP-1 algorithms failto work when the angular separation is less than 4°, whereasthe proposed algorithm can still provide accurate DOA esti-mation as long as the angular separation is no less than 3°.Moreover, the proposed algorithm has smaller RMSE thanthe BCS and NP-1 algorithms at the same angular separation.Thus, we can say that the proposed algorithm has a higherangular resolution than the BCS and NP-1 algorithms.
6. Conclusion
In this paper, we develop a DOA estimation algorithm for amixture of uncorrelated and coherent sources using SBI. Inthe Bayesian framework, a Gauss-Exp-Chi2 prior is intro-duced to promote sparse solutions, and the correspondingthree-layer hierarchical Bayesian model is established. Then,the model parameters are estimated via variational Bayesianapproximations. Finally, we form the source power spectraby using the estimated model parameters, from which thenumber and the locations of the highest peaks are extractedto achieve source number and DOA estimates. Simulationresults show that the proposed algorithm can effectively esti-mate the DOAs of mixed sources and has better estimationperformance than the state-of-the-art BCS algorithm andNP-1 algorithm in terms of estimation accuracy, success rate,and angular resolution.
Appendix
A. The Derivation of the Probability DensityFunction for Gauss-Exp-Chi2 Prior
In this Appendix, we prove that (12) holds.
p H ; τ, v = ∏T
t=1∏J
j=1p hj t ; τ, v
= ∏T
t=1∏J
j=1
+∞
0p hj t αj p αj τ, v dαj
= ∏T
t=1∏J
j=1−
τ
πΓ v/2 + 1 Γ v/2 + 1/2
Γ v/2 Uchf
v2 + 1
2 ,12 , h
2j t τ
A1The expression on the right hand side of (12) can be
rewritten as
p hj t ; τ, v =+∞
0p hj t αj
+∞
0p αj τ, η p η v dη dαj
A2
Thus, we first calculate the integral over η
+∞
0p αj τ, η p η v dη
=+∞
0τη exp −τηαj
2−v/2Γ v/2 ηv/2−1 exp −
η
2 dη
= 2−v/2τΓ v/2
+∞
0ηv/2 exp −τηαj −
η
2 dη
= Γ v/2 + 1 2−v/2τΓ v/2 ταj +
12
− v/2+1
A3
By substituting (A3) into (A2), we have
p hj t ; τ, v =+∞
0
+∞
0p hj t αj p αj τ, η p η v dαjdη
= ϖ τ, v+∞
0
12παj
exp −h2j t
2αj
τα j +12
− v/2+1dαj,
A4
where ϖ τ, v = Γ v/2 + 1 2−v/2τ/Γ v/2 . Let βj = 1/αj (A4)can be expressed as
p hj t ; τ, v = ϖ τ, v2π
+∞
0β3/2j exp −
h2j t
2 βj
τβ−1j + 1
2− v/2+1
dβj
= ϖ τ, v2π
τ1/22 v/2−1/2+∞
0βj
v/2‐1/2
1 + βj
− v/2+1exp −h2j t τβj dβj,
A5
where βj = βj/2τ. According to the definition of the confluenthypergeometric function, (A5) can be further simplified to
p hj t ; τ, v = τ
π
Γ v/2 + 1 Γ v/2 + 1/2Γ v/2 Uchf
v2 + 1
2 ,12 , h
2j t τ
A6
Thus, we have
p H ; τ, v = ∏T
t=1∏J
j=1p hj t ; τ, v
= ∏T
t=1∏J
j=1
τ
π
Γ v/2 + 1 Γ v/2 + 1/2Γ v/2 Uchf
v2 + 1
2 ,12 , h
2j t τ
A7
10 International Journal of Antennas and Propagation
Conflicts of Interest
The authors declare that they have no competing interests.
Authors’ Contributions
Pinjiao Zhao proposed the main idea and conceived the pro-posed approach. Pinjiao Zhao and Weijian Si discussed anddesigned the proposed algorithm. Pinjiao Zhao performedthe experiments and wrote the paper. Guobing Hu and LiweiWang reviewed and revised the manuscript. All authors readand approved the manuscript.
Acknowledgments
This work was financially supported by the High LevelTalent Research Starting Project of Jinling Institute of Tech-nology (Grant no. jit-b-201724), the National Natural Sci-ence Foundation of China (Grant no. 61671168), and theNatural Science Foundation of the Jiangsu Province (Projectno. BK20161104) and the Six Talent Peaks Project of theJiangsu Province (Project no. DZXX-022).
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