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Acoustic DOA estimation using space alternating sparse Bayesian learning
EURASIP Journal on Audio, Speech, and Music Processing volume 2021, Article number: 14 (2021)
Abstract
Estimating the directionofarrival (DOA) of multiple acoustic sources is one of the key technologies for humanoid robots and drones. However, it is a most challenging problem due to a number of factors, including the platform size which puts a constraint on the array aperture. To overcome this problem, a highresolution DOA estimation algorithm based on sparse Bayesian learning is proposed in this paper. A group sparse prior based hierarchical Bayesian model is introduced to encourage spatial sparsity of acoustic sources. To obtain approximate posteriors of the hidden variables, a variational Bayesian approach is proposed. Moreover, to reduce the computational complexity, the space alternating approach is applied to push the variational Bayesian inference to the scalar level. Furthermore, an acoustic DOA estimator is proposed to jointly utilize the estimated source signals from all frequency bins. Compared to stateoftheart approaches, the highresolution performance of the proposed approach is demonstrated in experiments with both synthetic and real data. The experiments show that the proposed approach achieves lower root mean square error (RMSE), false alert (FA), and missdetection (MD) than other methods. Therefore, the proposed approach can be applied to some applications such as humanoid robots and drones to improve the resolution performance for acoustic DOA estimation especially when the size of the array aperture is constrained by the platform, preventing the use of traditional methods to resolve multiple sources.
1 Introduction
Acoustic directionofarrival (DOA) estimation is a key technology in audio signal processing where it enables source localization for humanoid robots [1, 2], drones [3, 4], teleconferencing [5, 6], and hearing aids [7]. The goal of acoustic DOA estimation is to obtain the bearing angle of acoustic waves generated by sound sources using a microphone array. According to the Rayleigh criterion [8], the resolution of traditional DOA estimation approaches (e.g., the classical beamforming (CBF)^{Footnote 1} based approach and the steeredresponse power phase transform (SRPPHAT) method [9]) is limited by the array aperture. Therefore, for some applications like humanoid robots and drones with a small platform size, the traditional approaches suffer in scenarios with multiple sources simultaneously present. Although methods such as the minimum variance distortionless response (MVDR) [8, 10], multiple signal classification (MUSIC) [11], and estimation of signal parameters via the rotational invariance technique (ESPRIT) [12] can offer a highresolution performance, they are sensitive to calibration errors and errors in the assumed or estimated signal statistics [13, 14]. The robustness of the MVDR and MUSIC methods have been studied in the presence of array errors [15–17]. However, these studies rely on asymptotic properties, i.e., high signaltonoise ratio (SNR) scenarios and large numbers of snapshots. Thus, these studies do not apply when only a small number of snapshots is available.
Sparse signal recoverybased DOA estimation methods have enjoyed much success in recent decades by exploiting the sparsity of sources in the spatial domain [18, 19]. These approaches are attractive because (1) they offer robustness against noise and limitations in data quality [18], (2) they have a good performance with a small number of snapshots [20], (3) they offer a higher resolution performance than MVDR and MUSIC methods [21, 22], and (4) they have the capability to resolve coherent sources [23]. In [18], the source localization problem was first formulated as an overcomplete basis representation problem. To estimate the source amplitudes, an l_{1}norm based singular value decomposition (SVD) method was proposed. In [24], a complex least absolute shrinkage and selection operator (cLASSO) method was proposed for DOA estimation. In [25], a reweighted regularized sparse recovery method was proposed for DOA estimation with unknown mutual coupling. All these methods are based on convex optimization theory, that is, the signals are recovered by solving a regularized optimization problem. They have a good performance with a properly chosen regularization factor, but the regularization factor needs to be determined empirically [26].
Because of its selfregularization nature and its ability to quantify uncertainty, the sparse Bayesian learning (SBL)based methods have attracted a lot of attention in sparse signal recovery and compressed sensing. The SBL principle was originally proposed in [27] for obtaining sparse solutions to regression and classification tasks. The SBL algorithm was applied to the compressed sensing in [28], and an SBLbased Bayesian compressed sensing method using Laplace priors was proposed in [29]. More recently, a scalable meanfield SBL was proposed in [30]. In [31], an SBLbased DOA estimation method with predefined grids was proposed. In that paper, the DOA estimation is formulated as a sparse signal recovery and compressed sensing problem. To obtain refined estimates of the DOA, an offgrid DOA estimation method was proposed in [32]. In [21], a multisnapshot SBL (MSBL) method was proposed for the multisnapshot DOA estimation problem. The method was further applied to sound source localization and speech enhancement in [22]. To reduce the computational complexity of the wideband approach, a computationally efficient DOA estimation method was proposed in [33] based on a sparse Bayesian framework. Additionally, some of our previous works are related to this paper. In [34], we proposed an SBL method with compressed data for sound source localization. The results show that the SBL method offers an excellent estimation accuracy for sound source localization even with low data quality. In [35], we proposed an SBLbased acoustic reflector localization method, which models the acoustic reflector localization problem as a sparse signal recovery problem. It shows that the SBLbased method offers a more robust performance for basis mismatch compared to the stateoftheart methods. However, a common drawback of these approaches is that the traditional SBLbased approaches are computationally complex due to the matrix inversion operation required for updating the covariance matrix of the source signals.
Computationally efficient SBL algorithms have also been proposed in various applications. For example, in [36], a basis adding/deleting scheme based on the marginal distribution was proposed. In [37], an inverse free SBL method was proposed by relaxing the evidence lower bound. In [38], a space alternating variational estimation (SAVE) algorithm was proposed to push the variational Bayesian inference (VBI) based SBL to a scalar level. The experimental results show that the SAVE approach has a faster convergence and a lower minimum mean square error (MMSE) performance than other fast SBL algorithms.
Based on this, we propose a space alternating SBLbased acoustic DOA estimation method for highresolution estimation in this paper. A hierarchical Bayesian framework with group sparse priors is built to model multiple measurement vector (multisnapshot) signals. As direct calculation of the posterior distribution is not possible, variational Bayesian inference is applied to infer all hidden variables in the proposed model. Furthermore, we extend the SAVE method [38] to the multiple measurement vector (MMV) case to reduce the computational complexity of the algorithm. The proposed algorithm can be applied to each frequency bin independently. To jointly utilize the recovered signals from all frequency bins, a complex Gaussian mixture model (CGMM) based expectation–maximization (EM) algorithm is proposed. We refer to the proposed method as the SAVEMSBLEM method.
The rest of this paper is organized as follows: In Section 2, we pose the narrowband acoustic DOA estimation problem as a sparse signal recovery problem with an overcomplete dictionary. Moreover, under the assumption that the DOAs of all sources do not change in a frame, a hierarchical Bayesian framework is built by exploiting the group sparsity of the MMV source signals. In Section 3, the SAVEMSBL algorithm is proposed to infer all the hidden variables in the hierarchical Bayesian model for one frequency bin. Then, the CGMMbased EM algorithm is formulated to deal with the wideband acoustic DOA estimation. In Section 4, we evaluate the performance of the proposed algorithm using both synthetic data and real data. Finally, we provide our conclusions in Section 5.
Note that vectors and matrices are represented using bold lowercase and uppercase letters, respectively. The superscripts (·)^{T} and (·)^{H} denote the transpose and conjugate transpose operator, respectively. Moreover, L×L identity matrix is denoted as I_{L}. The l_{p} norm and Frobenius norm are represented using ∥·∥_{p} and ∥·∥_{F}, respectively.
2 Signal model
2.1 Problem formulation
The problem considered in this paper can be stated as follows. We consider the scenario that P sound sources exist in the farfield of an arbitrary microphone array with M microphones which are used to record the signals. The center point of the microphone array is denoted as O. All the microphones are assumed to be omnidirectional and synchronized. As it is shown in [18, 22, 33], the DOA estimation problem can be formulated as a sparse signal recovery problem using an overcomplete dictionary with basis vectors containing the DOA information. Let θ=[θ_{1},θ_{2},⋯,θ_{K}]^{T} denote a set of candidate DOAs, where K denotes the total number of candidate DOAs. The signal model for the fth (1≤f≤F) frequency bin of one frame can be expressed as
where
F is the total number of frequency bins, X_{f}\(\in \mathbb {C}^{M\times L}\) is a collection of signal snapshots in the frequencydomain with x_{f,l,m} being the signal at the fth frequency bin, lth snapshot, and mth microphone. We refer to the matrix X_{f} as one frame and x_{f,l}\(\in \mathbb {C}^{M}\) as one snapshot, l∈[1,2,⋯,L] is the index of the snapshots^{Footnote 2}. The matrix A_{f}\(\in \mathbb {C}^{M\times N}\) is the dictionary for the fth frequency bin with the basis vector a_{f,k}\(\in \mathbb {C}^{M}\) representing the array response for the direction θ_{k},ω_{f} is the fth angular frequency, and τ_{km} is the relative time delay of source k between microphone m and the array center point O. Moreover, S_{f}\(\in \mathbb {C}^{K\times L}\) is a collection of the source signals with s_{f,k} being the kth row. The noise matrix N_{f}\(\in \mathbb {C}^{M\times L}\) is defined similarly to S_{f}. Assuming that several sound sources are active in one frame, let θ_{s} (θ_{s}⊂θ) denote the true DOA set and k_{s}(k_{s}⊂[1,2,⋯,K]) denote the true index set. Based on the above definition and the signal model in (1), S_{f} is an allzero matrix except for the elements of the rows within the ground truth index set k_{s}. An example is given in Fig. 1, which uses a uniform linear array (ULA). In this example, the target space is sampled uniformly with an interval of 10^{∘}. Two sources are located at −30^{∘} and 40^{∘}, respectively. Thus, when the two sources are active simultaneously, only the elements in the two rows of S_{f} corresponding to the bearing angles −30^{∘} and 40^{∘} are nonzero.
Based on (1), to obtain the DOA estimator, we can first recover the source signal, S_{f}, given the MMV, X_{f}, and the predefined dictionary, A_{f}, using MMV sparse signal recovery methods, and then find the row index set of the nonzero elements, which indicates the acoustic DOAs. We assume that the sound sources are static or move slowly such that the direction of the sound sources do not change within the snapshots in a frame. We further assume that the number of active sound sources P is very small compared to the number of candidate DOAs K, i.e., P≪K. As a result, the sound source signal, S_{f}, is a signal matrix with group sparsity and the algorithms for sparse signal recovery can be applied [18, 19]. In this paper, we propose a space alternating MSBL method to improve the estimation performance by exploiting the group sparsity of S_{f}.
2.2 Probabilistic models
The SBL method is a widely used sparse signal reconstruction method. It is a probabilistic parameter estimation approach based on a hierarchical Bayesian framework. It learns the sparse signal from the overcomplete observation model, resulting in a robust maximum likelihood estimation method [27, 39]. Like other Bayesian algorithms, SBL estimates model parameters by maximizing the posterior with a sparse prior. However, instead of adding a specialized model prior, SBL encourages sparsity by using a hierarchical framework that controls the scaling of Gaussian priors through updating individual parameters of each model [27, 40].
2.2.1 Sparse signal model
Following the SBL method proposed in [27], a hierarchical Bayesian framework is used to model the signal matrix, S_{f}. For the sake of brevity, we omit the dependency of random variables on the subscript, f, where appropriate. First, we assume that the candidate sources are independent to each other. Then, a multivariate complex Gaussian distribution is used to describe the kth candidate source signal s_{k} with zero mean and a covariance matrix \(\lambda _{k}^{1}\mathbf {I}_{L}\), i.e.,
where λ=[λ_{1},λ_{2},⋯,λ_{K}]^{T} is the hyperparameter vector, λ_{k} is the hyperparameter related to the amplitude of the kth candidate source signal s_{k}, e.g., the amplitude of s_{k} is 0 when λ_{k}→∞. Moreover, I_{L} is the L×L identity matrix, \(\mathcal {CN}(\cdot)\) denotes the complex Gaussian distribution and λ_{k} is the precision of s_{k}. Note that, for each candidate DOA (e.g., the kth DOA), an individual precision λ_{k} is used, but the precision λ_{k} is set to the same for the signal in different snapshots, thereby encouraging group sparsity [41].
The motivation is that the DOAs of the sound sources, as well as the set of active sources, are assumed to not change within a frame. For different candidate DOAs, different precisions are used to encourage the sparsity (see [18, 19] for further details).
In the second layer of the hierarchy, we assume that the precision variables are independent and follow gamma distributions, i.e.,
where \(\mathcal {G}(a,b)\) denotes the gamma distribution with the shape parameter a and the rate parameter b. There are two reasons for this particular choice of prior distribution: (1) the gamma distribution is a conjugate prior for the variable λ_{k} in the complex Gaussian distribution, leading to a tractable posterior, and (2) the marginal distribution \(\int p(\boldsymbol {S}\boldsymbol {\lambda }) p(\boldsymbol {\lambda }\boldsymbol {\gamma }) d\boldsymbol {\lambda }\) is a Student’s t distribution encouraging sparsity [27].
To facilitate the inference of γ, we further assume that the variables in γ=[γ_{1},⋯,γ_{k},⋯,γ_{K}]^{T} follow i.i.d. gamma distributions, i.e.,
where a and b are model parameters that will be treated as hyperparameters.
2.2.2 Likelihood function and noise model
Under the assumption of circular symmetric complex Gaussian noises, the likelihood function can be written as
where ρ denotes the noise precision.
For tractability, we assume that ρ follows a gamma distribution as follows
where c and d are modeling parameters.
The hierarchical Bayesian model is built using (2), (3), (4), (5) and (6), and the graphical model is shown in Fig. 2.
3 Bayesian inference using space alternating variational estimation
3.1 Variational Bayesian inference
Let Θ={S,λ,γ,ρ} denote the set of hidden variables. Based on the graphical model shown in Fig. 2, the joint pdf can be written as
A closedform expression of the full posterior p(ΘX) requires computation of the marginal pdf (X), which is intractable. In this paper, VBI is therefore applied to obtain an approximation of true posterior using a factorized distribution [42, 43]
where q(Θ) is an approximation of the full posterior p(ΘX). For notational simplicity, the dependency of the approximated posterior on the observed signal X is omitted. Note that, instead of pursuing the full posterior q(S) of the source signals, a factorial form of the posterior \(\prod _{k=1}^{K}q({\boldsymbol {s}}_{k})\) is used to reduce the computational complexity. This is an extension to the SAVE proposed in the single measurement vector (SMV) scenario [38]. When L=1, the proposed approximation model (8) reduces to the model in SAVE. We also assume that the approximate posteriors have the same functional forms as the priors for all the hidden variables. For example, both the prior p(s_{k}λ_{k}) and posterior q(s_{k}) are complex Gaussian. The VBI approach minimizes the Kullback–Leibler (KL) divergence between p(ΘX) and q(Θ) by maximizing the following variational objective:
where E_{q}[·] denotes the expectation operator over the distribution q, i.e., \(\mathrm {E}_{q(x)}[p(x)]=\int q(x)p(x)\mathrm {d}x\).
Since the prior and likelihood of all nodes of the model shown in Fig. 2 fall within the conjugate exponential family, the VBI can be written as [42, 43]
where C is a constant and Θ_{i} denotes one of the variables in the factorized distribution (8), such as s_{k}. The notation \(\boldsymbol {\Theta }_{\bar {i}}\) denotes the hidden variable set Θ excluding Θ_{i}.
3.2 The logarithm of the joint distribution
As shown in (9), the logarithmic form of the joint distribution is required for VBI. Substituting (2), (3), (4), (5), and (6) into (7), we have
where ∥·∥_{F} denotes the Frobenius norm. Next, we present the approximate posterior by substituting (10) into (9).
3.2.1 Update of s _{k}
The approximate posterior of s_{k} can be written as ^{Footnote 3}
where
and <·> is the shorthand of the expectation operator E_{q}[·]. Moreover, tr[·] denotes the trace operator, a_{k} denotes the kth column of \(\boldsymbol {A}, \boldsymbol {A}_{{\bar {k}}}\) is the matrix A with the kth column a_{k} being removed, and \(\boldsymbol {S}_{\bar {k}}\) is the matrix S with the kth row \({\boldsymbol {s}}_{k}^{\mathrm {T}}\) being removed. From (11), it can be shown that \(q({\boldsymbol {s}}_{k})=\mathcal {CN}\left ({\boldsymbol {s}}_{k}\boldsymbol {\mu }_{k},\sigma ^{2}_{k}\mathbf {I}\right)\), where
where the property \(\boldsymbol {a}_{k}^{\mathrm {H}}\boldsymbol {a}_{k}=M\) is used. Note that the mean {μ_{k}} is updated based on the space alternating approach [38, 44], where the newest estimates are always used.
3.2.2 Update of λ,γ and ρ
The approximate posteriors for λ,γ and ρ can be derived in a similar way as s_{k}, and we only give the results here.
Update q(λ_{k}): \(q(\lambda _{k})=\mathcal {G}({\alpha _{\lambda _{k}}},\beta _{\lambda _{k}})\), where
Update q(γ_{k}): \(q(\lambda _{k})=\mathcal {G}\left ({\alpha _{\gamma _{k}}},\beta _{\gamma _{k}}\right)\), where
Update q(ρ): \(q(\rho)=\mathcal {G}({\alpha _{\rho }},\beta _{\rho })\), where
where \(\boldsymbol {\Sigma }=\text {diag}[\sigma ^{2}_{1},\cdots,\sigma ^{2}_{2},\cdots,\sigma ^{2}_{K}]\) and diag[·] denotes a diagonal matrix.
We refer to the proposed algorithm as SAVEMSBL. By using the space alternating approach, the computationally complex matrix inversion operation of the traditional MSBL [19] can be avoided. Moreover, instead of using the above formulas directly, we can further reduce the computational complexity by introducing a temporary matrix \(\widehat {\boldsymbol {X}}\), which can be seen as an approximation of X. By removing or adding the terms \(\boldsymbol {a}_{k}{\boldsymbol {\mu }}_{k}^{T}\), the two terms \(\boldsymbol {A}_{\bar {k}}\left <\boldsymbol {S}_{\bar {k}}\right >\) and A<S> in (13) and (16) can be updated using \(\boldsymbol {a}_{k}{\boldsymbol {\mu }}_{k}^{T}\), resulting in a computationally efficient implementation. The pseudocode for the proposed method is shown Algorithm 1. Note that the proposed SAVEMSBL algorithm can be applied to each frequency bin independently.
3.3 CGMMbased acoustic DOA estimator
Up to this point, the posteriors of the source signals (i.e., {q(s_{f,k})}) from all the frequency bins are obtained independently. The source signals s_{f,k} can be estimated using the MMSE estimator, i.e.,
where \(\widehat {\boldsymbol {s}}_{f,k}\) denotes the estimate of the source signal. In this section, we propose an acoustic DOA estimator, jointly utilizing the estimated source signals from all the frequency bins, based on the CGMM model. By fitting the observations and estimates of the source signals to the CGMM model, the weighting parameters can be obtained using the EM algorithm. The weighting parameter of each mixture component in the CGMM can be seen as the probability that there is an active acoustic source at the corresponding candidate location. With a known number of sources, the DOA estimates for all the sources can be obtained using peakpicking on the weighting parameters.
Inspired by the Gaussian mixture model [45, 46] and the probabilistic steeredresponse power (SRP) model [47, 48], we assume that x_{f,l} follows a CGMM distribution with estimated source signals s_{f,k}, i.e.,
where η is an empirically chosen small value, and w_{k}≥0 is the weighting parameter for the kth complex Gaussian component with the constraint \(\sum _{k=1}^{K} w_{k}=1\). Then, the distribution of the observation set for all frequency bins can be expressed as
where \(\boldsymbol {Y}=\{\boldsymbol {X}_{f}\}_{f=1}^{F}\) is the observation set for all frequency bins. Once (18) is maximized, each weight w_{k} represents the probability of an acoustic source being active in the direction θ_{k}. However, it is intractable to maximize the function in (18) due to its high dimensionality. Therefore, an EM procedure is applied to deal with this maximization problem. Following [42], we introduce a set of hidden variables \(\boldsymbol {z}=\{\boldsymbol {r}_{f}\}_{f=1}^{F}\). The r_{f} contains binary random variables with only one particular element r_{f,k} being 1 while the others are all zeros. The variable r_{f,k} can be seen as an indicator associated with the acoustic source from the direction θ_{k} at the fth frequency bin. Assuming p(r_{f,k}=1)=w_{k}, we can write the joint distribution as follows:
The conditional distribution of the observation set Y given z is
Then, the joint distribution can be derived from (19) and (20) using Bayes’ rule, i.e.,
3.3.1 Estep
In the Estep, we use the current parameter \(\hat {\boldsymbol {w}}^{\text {old}}\) to update the posterior mean of the hidden variable denoted as \(\mathrm {E}[r_{f,k}\boldsymbol {Y};\hat {\boldsymbol {w}}^{\text {old}}]\). From (21), the Estep can be written as
where
where μ_{f,k,l} is obtained using Algorithm 1.
Therefore, the expected value \(\mathrm {E}[r_{f,k}\boldsymbol {Y};\hat {\boldsymbol {w}}^{\text {old}}]\) is given by [42, 49]
3.3.2 Mstep
In the Mstep, the required parameter w is updated through a constrained maximization of (22), i.e.,
Therefore, the Mstep can be stated as
Given an initial value for the parameter w, the EM algorithm iterates between the Estep in (23) and the Mstep in (25) until convergence. The EM algorithm is summarized in Algorithm 2.
4 Results and discussion
In this section, we first investigate the computational complexity of the proposed SAVEMSBLEM method. Then, we test the performance of our proposed SAVEMSBLEM algorithm using both synthetic data and real data from the LOCATA dataset^{Footnote 4}. The performance of the different methods are tested in three different scenarios. In the first scenario, we test the recovery accuracy and the resolution performance using narrowband sources and a ULA. In the second part, we consider a complicated scenario with closely spaced sources in a virtual room. Last, the proposed method is tested using real data.
4.1 Computational complexity analysis
We first analyze the computational complexity of the proposed SAVEMSBL algorithm by counting the number of mathematical multiplication/division operations in each iteration. As can be seen from Algorithm 1, in each “for” loop, the complexity of the proposed algorithm mainly depends on the update of the temporary matrix \(\bar {\boldsymbol {X}}\) and μ_{k}, which is \(\mathcal {O}(ML)\). The computational complexity of updating <ρ> is \(\mathcal {O}(ML)\). Therefore, the computational complexity of the proposed algorithm for each iteration is \(\mathcal {O}(KML)\). If we consider the variational Bayesian inference without the space alternating approach, the computational complexity is \(\mathcal {O}(M^{3}L^{3})\). Thus, the space alternating approach leads to a significant reduction on the computational complexity. Moreover, the computational complexity of MSBL proposed in [19] is \(\mathcal {O}(KM^{2})\). Therefore, the proposed method is faster than the MSBL method when L<M. Since the SVD approach can be utilized for data reduction [18], the condition L<M is met in most cases. For the EM algorithm, the computational complexity is \(\mathcal {O}(KML)\) for one frequency bin. Thus, the computational complexity of the proposed SAVEMSBLEM method is \(\mathcal {O}(KML)\) for each frequency bin.
We further measure the computational complexity using the “cputime” function provided by MATLAB. The computer is equipped with an i78700 processor. The clock rate is 3.19 GHz. The operation system is Windows 10. The software is MATLAB 2019a. We test the computational complexity for one frequency bin. The number of iterations is fixed to 100, the number of candidate DOAs is set to 41, the number of microphones is set to 15, the number of snapshots is set to 10, and the number of MonteCarlo experiments is set to 1000. For a single frequency bin, the time consumption of the proposed SAVEMSBLEM method and the MSBL proposed in [19] are 0.08 and 0.25 s, respectively, i.e., the proposed method is faster than the MSBL method by a factor of ∼3. Note that, in practice, the time consumption for the acoustic DOA estimation algorithm is proportional to the number of frequency bins.
4.2 Experimental results
The methods used for comparison in this section are summarized as following: CBF refers to classical beamforming based method which is widely used in practice; SRPPHAT is another widely used method for sound source localization especially in reverberant environments [9]; and MVDR is a method offering highresolution performance [10]. Note that the implementation of the MVDR method is based on the observed signal statistics. Moreover, MSBL refers to the multiple snapshots SBL method for narrowband signals proposed in [19]. MSBLEM is an acoustic DOA estimator which combines the MSBL algorithm and proposed EM algorithm. Furthermore, SAVEMSBL is the proposed method for narrowband signals and SAVEMSBLEM is the proposed method for acoustic DOA estimation. For the MSBL method, the threshold for stopping the iteration err_{max} is set to 1e−10. For the proposed SAVEMSBLEM method, the modeling parameters a, b, c, and d are all set to 1e−3, the parameter η is set to 0.1, the threshold for the SAVEMSBL algorithm err_{max} is set to 1e−10, and the threshold for the EM algorithm err_{0} is set to 1e−3.
4.2.1 Recovery performance analysis using a ULA
In this section, we test the recovery performance of the proposed SAVEMSBL algorithm using four acoustic sources comprising pure sinusoidal signals. Two assumptions are made in this simulation: (1) all the acoustic sources are located in the farfield of the microphone array and (2) the power of all the acoustic sources are equal. The frequencies of all the sources are set to 1 kHz. For each source, the initial phase is generated randomly. Assume that a ULA with 15 omnidirectional microphones is used to receive the signals. The distance between adjacent microphones is set to 0.05 m in this simulation. The microphone array data are generated by assigning different time delays according to the true bearing angles of the sources. White Gaussian noise is added to the clean array data and the SNR is set to 10 dB. The sampling frequency is set to 16 kHz. The timedomain data are converted to the frequencydomain using the shorttime Fourier transform (STFT). The temporal length of the snapshot is set to 1024. The length of the increment for the snapshots is set to 256, i.e., the overlap is 75%. The length of the FFT is set to 2048. The number of snapshots is set to 10. As the frequencies of all sources are 1 kHz, only the frequency bin whose center frequency is 1kHz is used for the estimation. We define the fanshaped horizontal plane in the range from −60^{∘} to 60^{∘} as the target space (see Fig. 1). The target space is uniformly separated with a grid interval of 3^{∘}, i.e., the number of grid points is 41 and the array response matrix (dictionary) is precomputed according to these grid points. Moreover, the bearing angles of four pure sinusoidal sources are −33^{∘},−27^{∘},−12^{∘}, and −3^{∘}, respectively. Figure 3 shows the estimation results of the CBF, MVDR, SRPPHAT, and SAVEMSBL methods.
It can be seen that the CBF and SRPPHAT methods fail to separate the two sources located at −33^{∘} and −27^{∘}, but the MVDR and proposed SAVEMSBL methods still work in this case.
We now proceed to test the performance of the proposed method with respect to the number of snapshots. The number of MonteCarlo runs is 1000. The recovery accuracy is measured by the rootmeansquareerror (RMSE), defined as
where \(\hat {\boldsymbol {S}}\) is the recovered signal, S is the true signal, ∥·∥_{F} denotes the Frobenius norm, L is the number of snapshots, and N_{MC} is the number of MonteCarlo experiments. We compare the proposed method with the CBF method in [6] and one of the widely used MSBL algorithms proposed in [19]. The results of the RMSEs of the recovered signals are illustrated in Fig. 4. It can be seen that the recovery performance of all the methods improve dramatically as the number of snapshots increases in the range from 1 to 3. Moreover, the simulation result shows that the proposed SAVEMSBL method achieves better recovery accuracy compared with the CBF and MSBL methods.
4.2.2 Simulation with virtual room
In this part, we test the resolution performance of the proposed method with respect to different intervals of bearing angles between two sources. The synthetic array data are generated using the “signalgenerator”^{Footnote 5} with a virtual room. Note that the “signalgenerator” is designed for the moving source scenario. The room setup is summarized in Table 1.
In this virtual room, a uniform circular array (UCA) with 32 omnidirectional microphones is used to record the signals. The center position of the UCA is (5,3.5,3) m. The radius of the UCA is set to 0.25 m. Two acoustic sources are used. Both of them play uninterrupted harmonic signals. The fundamental frequencies of the two sources are 300 Hz and 350 Hz, respectively. The spectrograms of the two sound sources are shown in Fig. 5.
We assume the sound sources are moving on a horizontal plane where the microphone array is located in. The horizontal plane is separated into 73 grid points from 0^{∘} to 360^{∘} with an angle interval 5^{∘}, where 0^{∘} is in the positive direction of the xaxis and 90^{∘} is in the positive direction of the yaxis. For simulation 1, the trajectories of the two sources are illustrated in Fig. 6. The first source moves along the negative direction of yaxis while the second source moves along the negative direction of xaxis. The original positions of the first and second sound sources are (3.5,5,3) m and (6,5.5,3) m. The end positions are (3.5,3,3) m and (4,5.5,3) m, respectively. The true DOA trajectories of the two sources with respect to the microphone array are shown in Fig. 7(a).
According to the simulation setup, the timedomain array signals can be generated using the “signalgenerator.” Then, the received array signals are first segmented into a batch of snapshots with 87.5% overlap. By applying the fast Fourier transform (FFT) on each snapshot, the timedomain array signals are converted to the frequencydomain array data. Then, the frequencydomain array data is segmented into several frames with L consecutive snapshots grouped as one frame. In the first and second simulations, L is set to 15. The effect of L is discussed in the last part of this subsection. Note that the SVD approach is used for data reduction in this paper. After applying acoustic DOA estimation methods for each frame, we find the peaks for each frame and label these peaks according to the ground truth DOAs of the two sources. The error range is set to 15^{∘}, i.e., if the minimum error between the estimated angle and all ground truth angles is larger than 15^{∘}, we label the peak as a false estimate. In this paper, we use the black and red circles to denote estimates of the first source and the second source, respectively. Moreover, we use magenta triangles to denote false estimates.
To quantitatively show the difference of the resolution performance between the proposed SAVEMSBLEM method and other methods, the RMSE, the false alarm (FA) rate, and the missdetected (MD) rate are used to measure the recovery performance. The RMSE is defined as
where N_{c} is the total number of correct estimates, \(\tilde {\theta }_{i}\) is the ith correct estimate, and θ_{i} is the ith true bearing angle. Following [50], the FA rate is defined as the percent of sources that are falsely estimated out of the total number of sources and the MD rate is defined as the percent of sources that are missdetected out of the total number of sources, i.e.,
where N_{F} is the number of sources with false estimation, N_{T} is the total number of sources for all frames, and N_{M1} and N_{M2} are the missdetected number of the first source and the second source, respectively. Note that two continuous harmonic sound signals are used in this simulation. Thus, two active sources exist in each frame.
We consider two reverberation conditions for all the methods: the freefield (no reverberation) and lowreverberation conditions (RT60 = 0.25 s). For the CBF, MVDR, and SRPPHAT methods, the estimation results are shown using the spatial spectrums of all frames. For the proposed SAVEMSBLEM method, the estimation results are shown using the weight, w, of all frames. For comparison, all the data are normalized frame by frame and displayed using color maps.
In simulation 1, the estimation results of the CBF method in freefield and lowreverberation environments are shown in Figs. 7b and c, respectively. The estimation results of the different methods in both the free field and low reverberation conditions are shown in Fig. 7b–i. The RMSE, FA, and MD are shown in Table 2. Note that “FF” refers to the freefield condition and “RB” refers to the reverberation environment. It can be seen that all the methods perform well under the freefield condition. In the presence of reverberation, the good accuracy performance of the CBF, SRPPHAT, and proposed SAVEMSBLEM method are retained but the MVDR method degrades considerably.
To further verify the performance of the proposed SAVEMSBLEM method in terms of resolution, another scenario is considered. In this case, all of the setup remains the same except the trajectories of the two sources. We refer to this simulation as simulation 2. The original position of the first source is (2.5,5.5,3) m while the second is (7.5,5.5,3) m. The end positions are (4,7,3) m and (6,7,3) m, respectively. Figure 8 shows the trajectories of the two sources in the virtual room.
The true bearing angles of the two sources with respect to the microphone array are illustrated in Fig. 9a. The estimation results of the CBF, SRPPHAT, MVDR, and SAVEMSBLEM methods in the freefield environment are shown in Figs. 9b, d, f, and h, respectively, while the results for the low reverberation condition are shown in Figs. 9c, e, g, and i, respectively. The RMSE, FA, and MD are summarized in Table 3.
From Figs. 9b, c, d, e, f, and g, it can be seen that the performance of the CBF, SRPPHAT and MVDR methods degrade dramatically as two sound sources move closer. However, the proposed SAVEMSBLEM method retains an accurate estimation performance for the acoustic DOA estimation. In this case, the proposed SAVEMSBLEM method offers higher resolution performance than other methods.
We then test the performance of the proposed method and MVDR method using static sources and the results are shown in Fig. 10.
The microphone array signals are generated using the “rirgenerator”^{Footnote 6}. The distance between the sound sources and the microphone array center is set to 3 m. We tested the FA rate with different bearing intervals between the two sound sources in the low reverberation condition (RT60 = 0.25 s). Figure 10(a) depicts the FA rates of the MVDR method and the proposed algorithm.
It can be seen that the proposed SAVEMSBLEM algorithm has a lower FA rate in the interval range from 15^{∘} to 40^{∘}. Figure 10(b) shows he MD rates of two algorithms. Compared with the MVDR method, the proposed method has a lower MD rate in the range from 15^{∘} to 40^{∘}. From Figs. 7, 9 and 10, we can thus conclude that the proposed SAVEMSBLEM method provides a better resolution performance than the CBF, SRPPHAT, and MVDR methods in both freefield and lowreverberation conditions.
To test the effect of the frame (window) length L on the localization performance, we conduct a simulation for different number of snapshots L. The simulation setup is the same as that of simulation 2, that is, the trajectories of the two sources and the true bearing angles of the two sources with respect to the microphone array are shown in Figs. 8 and 9a, respectively. The simulation is conducted in the reverberation environment (RT60 = 0.25 s). The results are illustrated in Fig. 11. The RMSE, FA, and MD are shown in Table 4.
It can be seen that the proposed method works for all snapshot numbers. However, the localization performance degrades if the number of snapshots is small, e.g., the FA and MD in Figs. 11a and b are higher than the FA and MD in Figs. 11c and d.
4.2.3 Real data experiments
The LOCATA dataset provides a series of microphone array data recorded in the Computing Laboratory of the Department of Computer Science of Humboldt University Berlin [51]. The room size is 7.1×9.8×3 m, with the reverberation time RT60 = 0.55 s. In this paper, we use the “benchmark2” microphone array data in task #6 to test the highresolution performance of the proposed method. The number of microphones of the ‘’benchmark2” array is 12. Two speakers are moving and continuously speaking with short pauses. The spectrograms of the two sources recorded with one microphone are illustrated in Fig. 12.
In this experiment, we just consider the azimuth angle estimation with the elevation angle fixed at 90^{∘}. The target plane is uniformly separated into 73 grid points from −180^{∘} to 180^{∘} with a uniform interval of 5^{∘}. The true positions and sound source signals of two sources are provided by the LOCATA dataset. We applied a voice activity detector [52] to these source signals to obtain groundtruth voice activity information of the two sound sources. Figure 13a shows the true trajectories of the two sources. We also applied the voice activity detector to the microphone array signals to obtain the voice activity information of each frame. Similar to the simulation part, we find two peaks for each voice active frame and label these peaks according to the true source position. Note that a threshold δ is set to judge the existence of peaks, i.e, if the amplitude of peaks is less than δ, this estimated peak is considered as an invalid estimate. The black circles and red circles denote the true DOAs of the first and second sources, respectively. The magenta triangles denote the false estimates.
The estimation results of the CBF, MVDR, SRPPHAT, and MSBLEM methods are shown in Figs. 13b, c, d, and e, respectively. Moreover, the estimation results of the proposed SAVEMSBLEM method is shown in Fig. 13f. From Figs. 13b–d, it can be seen that the two sources can hardly be separated in the time range from 6 to 10 s using the CBF, SRPPHAT, and MVDR methods. However, the proposed SAVEMSBLEM method can separate two sources successfully, indicating a higher resolution than the CBF, SRPPHAT, and MVDR methods (see Fig. 13f). Comparing Fig. 13e and f, it can be seen that the proposed SAVEMSBLEM method achieves better recovery performance than MSBLEM method in the time range from 8 to 10 s. To evaluate the performance of all the methods, the MD rate versus FA rate is computed by varying the peak selection threshold (see Fig. 14). For all the curves in Fig. 14, the closer to the leftbottom the better. It can be seen that the proposed SAVEMSBLEM method achieves better performance than stateoftheart methods.
We further report the estimation result for a fixed peak selection threshold δ=−40 dB (see Table 5). It can be seen that the proposed SAVEMSBLEM method outperforms other methods especially for the FA rate and RMSE. The reason is that the proposed method successfully resolves the two sources while the others are failing in the range from 6 to 10 s. The results indicate that the proposed SAVEMSBLEM method provides a higher resolution performance than stateoftheart methods also in real conditions where all assumptions of the proposed method might not hold.
5 Conclusion
In this paper, we propose a space alternating MSBL method for acoustic DOA estimation that offers a highresolution performance. First, we build a group sparse prior based hierarchical Bayesian framework for the MMV signal model by exploiting the group sparsity of candidate source amplitude matrix. Then, the computational efficient SAVEMSBL algorithm is proposed to infer all hidden variables in the Bayesian model. Moreover, an EM algorithm is proposed to deal with the acoustic DOA estimation problem. In the experimental parts, the performance of the proposed method is investigated using both synthetic and real data. The results show that the proposed method has lower RMSE and FA rate than stateoftheart methods in both freefield and lowreverberation conditions. As a result, the proposed method can be applied to some applications (e.g., humanoid robots and drones) to improve the resolution performance for acoustic DOA estimation.
6 Appendix A: Derivation of (11)
According to Eq. 9 and Eq. 10, the signal s_{k} can be updated using the space alternating approach as follows:
where Θ/s_{k} denotes the set of variables with s_{k} removed, C denotes a constant. Note that AS can be rewritten as \(\boldsymbol {A}_{{\bar {k}}}\boldsymbol {S}_{\bar {k}}+\boldsymbol {a}_{k}{\boldsymbol {s}}_{k}^{\mathrm {T}}\).
Availability of data and materials
The software for microphone array data generation is from “International Audio Laboratories Erlangen” and is online available: https://www.audiolabserlangen.de/home. The LOCATA data originates from the“IEEEAASP Challenge on Acoustic Source Localization and Trackin” and can be found under the following link: https://www.locata.lms.tf.fau.de/.
Notes
In this paper, the CBF is referred to as delay and sum beamforming.
Here, a snapshot refers to the array data in one observation window.
See Appendix A: Derivation of (11) for more derivation details.
The LOCATA dataset is publicly available at https://www.locata.lms.tf.fau.de/
The “signalgenerator” for synthetic array data generation is online available: https://www.audiolabserlangen.de/fau/professor/habets/software.
The RIR generator is publicly available at: https://www.audiolabserlangen.de/fau/professor/habets/software/signalgenerator.
Abbreviations
 DOA:

Directionofarrival
 CBF:

Classical beamforming
 SRPPHAT:

Steeredresponse power phase transform
 MVDR:

Minimum variance distortionless response
 MUSIC:

Multiple signal classification
 ESPRIT:

Estimation of signal parameters via rotational invariance technique
 SNR:

Signaltonoise ratio
 SVD:

Singular value decomposition
 cLASSO:

Complex least absolute shrinkage and selection operator
 SBL:

Sparse Bayesian learning
 MSBL:

Multisnapshot sparse Bayesian learning
 SAVE:

Space alternating variational estimation
 VBI:

Variational Bayesian inference
 MMSE:

Minimum mean square error
 RMSE:

Rootmeansquareerror
 MMV:

Multiple measurement vector
 CGMM:

Complex Gaussian mixture model
 EM:

Expectation–maximization
 ULA:

Uniform linear array
 UCA:

Uniform circular array
 KL:

KullbackLeibler
 FFT:

Fourier transform
 FA:

False alarm
 MD:

Missdetected
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Acknowledgements
The authors would like to thank Zhilin Zhang for providing the source code of the MSBL approach.
Funding
This work was supported by the China Scholarship Council, grant ID.201806120176.
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Z. Bai and L. Shi conceptualized the study and run the experiments. M. G. Christensen, J. R. Jensen, and J. Sun edited the manuscript. All the authors read and approved the final manuscript.
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Bai, Z., Shi, L., Jensen, J.R. et al. Acoustic DOA estimation using space alternating sparse Bayesian learning. J AUDIO SPEECH MUSIC PROC. 2021, 14 (2021). https://doi.org/10.1186/s1363602100200z
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DOI: https://doi.org/10.1186/s1363602100200z