Abstract

Direction of arrival (DOA) estimation has recently been developed based on sparse signal reconstruction (SSR). Sparse Bayesian learning (SBL) is a typical method of SSR. In SBL, the two-layer hierarchical model in Gaussian scale mixtures (GSMs) has been used to model sparsity-inducing priors. However, this model is mainly applied to real-valued signal models. In order to apply SBL to complex-valued signal models, a general class of sparsity-inducing priors is proposed for complex-valued signal models by complex Gaussian scale mixtures (CGSMs), and the special cases correspond to complex versions of several classical priors are provided, which is helpful to analyze the connections with different modeling methods. In addition, the expression of the SBL form of the real- and complex-valued model is unified by parameter values, which makes it possible to generalize and improve the properties of the SBL methods. Finally, the SBL complex-valued form is applied to the offgrid DOA estimation complex-valued model, and the performance between different sparsity-inducing priors is compared. Theoretical analysis and simulation results show that the proposed algorithm can effectively process complex-valued signal models and has lower algorithm complexity.

1. Introduction

Direction of arrival (DOA) estimation of spatial signals is an important content of array signal processing research. In recent years, compressive sensing (CS) [1] and sparse signal reconstruction (SSR) [2] have been introduced into the field of DOA estimation owing to their ultra-high resolution, good robustness to noise, and low dependence on the number of snapshots. One representative method is the L1-SVD algorithm proposed in [3], which uses L1-norm to construct an SSR model, reduces the complexity of the algorithm by singular value decomposition (SVD) under multisnapshots, and has a high resolution for the correlated sources as well. Sparse Bayesian learning (SBL) [4–7] or Bayesian compressed sensing (BCS) [8, 9] is another representative method for the SSR in CS. In SBL, the signal recovery problem is formulated from a Bayesian perspective, while the sparsity information is exploited by assuming a sparse prior for the signal of interest. One merit of SBL is its flexibility in modeling sparse signals, which can improve the sparsity of its solution [10]. Therefore, research work on DOA estimation based on SBL has been gaining momentum in recent years [11, 12]. While these methods have shown advantages over the conventional ones, however, there exists still difficulties in practical applications.

For these SSR methods mentioned above, which are called ongrid methods, true DOAs are assumed to lie exactly on a set of fixed sampling grids, and the existing sparse representation techniques can be directly applied. When the assumption that the true DOAs are located on a sampling grid fails, the performance of on-grid methods deteriorates due to the problem of mismatch. Although the offgrid (the distance from the true DOA to the nearest grid) error can be reduced by a dense sampling grid, the computational complexity will increase significantly. Furthermore, a dense grid may lead to the correlation between two steering vectors becoming high, which means the performance cannot be improved by the theory of CS.

Accordingly, an offgrid DOA estimation model was first studied in [13], where the true DOAs are no longer constrained in the sampling grid, and they proposed a sparsity cognizant total least-squares (S-TLS) method for the perturbed compressive sensing under sparsity constraints. It has been shown in [13] that the S-TLS can yield a MAP (maximum a posteriori) optimal estimate if the matrix perturbation caused by the basis mismatch is Gaussian. But the Gaussian condition cannot be satisfied in all DOA estimations. In [14], a new algorithm named offgrid sparse Bayesian inference (OGSBI) is proposed for the offgrid case. Firstly, the true DOA steering vector is approximated by a first-order Taylor expansion to handle the off-grid problem, and it is assumed to have a uniform distribution for off-grid parameters. Based on the OGSBI method, a linear interpolation between two adjacent grids is adopted in [15] to approximate the true DOA steering vector, and they proposed a perturbed sparse Bayesian learning (PSBL) algorithm to solve the offgrid DOA estimation problem.

Both ongrid and offgrid methods are grid-based methods. Another method to deal with the offgrid problem is the gridless method [16–18], which directly operates in the continuous domain so that can completely avoid the grid mismatch problem. However, this kind of method needs strong theoretical guarantees. In addition, it can only be applied to uniform or sparse linear arrays [19].

The SBL algorithm mainly deals with real-valued signals. In practical applications, such as DOA estimation, the experimental signals are often in complex-valued form. The results obtained in the real-valued model cannot be directly applied to the complex-valued model. In order to apply SBL to the complex-valued model, in [20–22], the real and imaginary parts of the complex-valued signal were separated to construct a new real-valued model, and then the SBL real-valued model was applied for processing. But in [20, 21], the real and imaginary parts of the complex-valued signal were treated as two independent variables, and the different hyperparameters were specified by ignoring the correlation between the real and imaginary parts of the complex-valued signal, which has obvious theoretical defects, and the reconstruction performance of the algorithm is low. It was improved in [22], where the performance of the algorithm was improved by assigning the same hyperparameters to the real and imaginary parts of the complex-valued signal by taking advantage of the fact that they have the same sparse structures. However, by decomposing the complex-valued signal into real and imaginary parts, the dimensionality of the signal is doubled, and the sensing matrix is tripled, so the complexity of the algorithm will increase significantly.

In this paper, the extension of SBL from the real-valued model to the complex-valued model in conjunction with the DOA estimation complex-valued model is considered, and the complex multisnapshot SBL (CSBL) offgrid DOA estimation algorithm is proposed. The expression of the SBL form is unified under the two models by parameter values, which enables us to generalize and improve the properties of the SBL methods. Theoretical analysis and simulation results show that the proposed algorithm can effectively process complex-valued signals with lower complexity.

A comment on notation: we use boldface lowercase letters for vectors and boldface uppercase letters for matrices. , , , and denote the transpose, inverse, conjugation, and conjugation-transpose operations, respectively. and are the F-norm and 2-norm of a matrix, respectively. denotes the trace operation of a matrix. describes Hadamard product operator. represents the probability density function of random variables.

2. Offgrid DOA Estimation Model

Consider a uniform linear array (ULA) with isotropic sensors and interelement spacing as shown in Figure 1. Suppose that far-field narrowband signals of frequency illuminate on this ULA from directions . By defining the first sensor of the array as the reference, the received signals can be expressed aswhere , , , and is the number of snapshots. , , , , is the additive complex white Gaussian noise vector with zero mean and variance . When , equation (3) is the case of multiple measurement vector (MMV).

Figure 1 

The configuration of a ULA.

Let be a fixed sampling grid in the DOA range , where denotes the grid number and typically satisfies . Without loss of generality, let be a uniform grid with a grid interval . In practical situations, the target locations are randomly distributed in space, which leads to the offgrid problem.

Suppose and that is the nearest grid to , which satisfies . So, the steering vector can be approximated by the first-order Taylor [23],where is the first-order derivative of . Let , , and grid parameter . By considering the approximation error into the noise, the offgrid DOA estimation sparse model can be expressed aswhere is abbreviated as in the following paragraphs, , , , , , and .

In SBL, the sparse representation of is induced by designing a prior for . Instead of using the prior directly, the SBL typically uses a two-layer hierarchical prior model, which involves a conditional prior and a hyperprior with the hyperparameters , which controls the estimation accuracy and sparsity of .

3. CSBL Offgrid DOA Estimation Algorithm

3.1. Two-Layer Hierarchical Prior Model

The extension of SBL from the real-valued model is considered to the complex-valued model, and the SBL form is unified, which can be applied to both real-valued and complex-valued models with different values of parameters. Assumed that the signals are independent at different snapshots, i.e., are independent of each other, there exist the following conditional prior equations [24],

orwhere , and . is the variance of , and is the precision of . The parameter when is complex, and when is real. Next, the hyperprior for variance and hyperprior for precision are modeled, respectively.

The generalized inverse Gaussian (GIG) distribution is chosen [25, 26] as the hyperprior for variance with the following expression with three parameters , , and .where denotes the modified Bessel function of the second kind of order , and the value range of parameters , , and are described later. The GIG distribution includes three distributions as special cases: the Gamma distribution, the inverse Gamma distribution, and the IG distribution according to the different values of the three parameters shown in Figure 2.

Figure 2 

The relationship between different two-layer hierarchical priors.

For the hyperprior , from the inverse relationship between precision and variance , the probability density function transformation formula [26] is used, and the hyperprior is given by

In equations (6) and (7), the parameters satisfy the relationship as follows:

The marginal distribution of can be obtained by the two-layer hierarchical prior model, , where

or

The abovementioned setting of independent hyperparameters for each parameter to be estimated is the most significant feature of the SBL model, and it is also the fundamental reason for the sparsity of the model [27, 28]. When takes the value , the equation (9) corresponds to the generalized hyperbolic (GH) distribution of the real-valued model, where equation (10) is equivalent to (9). In this paper, corresponds to the GH distribution of the complex-valued model.

As mentioned above, the GIG distribution is selected as the hyperprior and , respectively, as it includes a fairly broad class of distributions commonly used as hyperpriors, and the resulting marginal distribution, the GH distribution, again covers a large number of distributions as special cases. Due to this generalization, the connections between different modeling strategies can be analyzed, and several special cases that correspond to standard priors commonly used in sparse modeling are summarized in Figure 2.

3.2. Complex Sparse Bayesian Learning

As mentioned above, the model of the signal matrix has been defined by different two-layer hierarchical priors, and the SBL model characterization will be completed in the following by modeling the observation matrix in equation (3). Firstly, according to the different values in the model, the noise is modeled as Gaussian (real-valued model) or complex Gaussian (complex-valued model) with independent and identical distribution as follows:where is the noise precision with variance . The parameter when is complex, and when is real. Therefore, the following expression is obtained:

The noise precision is unknown, and a Gamma hyperprior is assumed for with , where obeys the Gamma distribution of parameters and . The offgrid parameter obeys a uniform distribution with . Taking the two-layer hierarchical prior , as an example, the relationship between the parameters of the SBL model is shown in Figure 3.

Figure 3 

SBL off-grid DOA estimation parameter model.

For equation (3) of the off-grid DOA estimation model, when the signal variance is used to model the two-layer hierarchical prior, the joint probability distribution of the complete SBL model is represented as

According to Bayesian theory, the posterior distribution of the parameters is estimated as

However, the exact posterior cannot be given in closed form due to the fact that is not calculated analytically [25]. Therefore, SBL needs an effective approximation . The sparsity-inducing property of the resulting estimator does not only depend on the two-layer hierarchical prior, but also the approximation method used.

In SBL, two widespread approximate approaches referred to as type I and type II estimation have been used. Type II is considered in this paper. In type II estimation [4–9], the impact of hyperparameter or is concerned on the model, and can be decomposed aswhere . Therefore, type II estimation needs to effectively approximate the posterior distribution , by using MAP to estimate hyperparameter as follows:and type II estimation is equivalent to maximizing the cost function of , thenwhere is the covariance matrix of the likelihood function . In the current widely studied real-valued model, it is usually assumed that a uniform hyperprior for , which leads to maximizing equivalently to maximizing the likelihood function . Therefore, type II estimation is also called as type II maximum likelihood (ML) estimation.

In [26], the results show that type II estimation has superior reconstruction performance than type I estimation. In subsequent paragraphs, type II estimation combined with the above two-layer hierarchical prior extension from the real-valued model is changed to the complex-valued model for the SBL derivation for offgrid DOA estimation.

In type II estimation, equation (15) is used to decompose the exact posterior , and the following expression can be acquired:where and are the covariance matrix and the mean of , respectively,

The signal variance and noise precision are estimated by using MAP as follows:where is the covariance matrix of , and . If and are known, type II estimated value of is obtained aswhere is a diagonal matrix composed of signal variance, and its diagonal elements control the row-sparsity of . As tends to a particularly small value (theoretically zero), the corresponding i-th row of becomes . DOA parameters (the closest grid to the true DOA , ) can be estimated from the position of nonzero rows in .

When signal precision is used to model two-layer hierarchical prior, similar expression results are acquired aswhere is a diagonal matrix composed of signal precision, as tends to a particularly large value (theoretically infinity), the corresponding i-th row of becomes .

In the above, the complete SBL model is analyzed with signal modeling variance or precision separately. Next, the iterative process of will be mainly discussed here, the iterative process of is similar and will not be repeated.

3.3. Two-Layer Hierarchical Prior , Type II Estimation

Maximizing the cost function of is equivalent to maximizing the following function:

Then the partial derivative of with respect to is

Let the partial derivative be zero, it will result in the hyperparameter with the following expression:

In this paper, the Gamma hyperprior for the noise precision is considered, the update of can be similarly obtained as follows:

3.4. Estimate of the Offgrid Parameter

As mentioned above, the updates of signal variance and noise precision by SBL mode are derived. In the following, the update of the important parameter is explained in the offgrid DOA estimation model. Considering that the only information is bounded, the update of cannot be obtained by MAP, so the maximum expectation algorithm is used to estimate the offgrid parameter .

The quantity is treated as a hidden variable and maximize , where denotes an expectation with respect to the posterior of as given in equation (18) using the current estimates of the hyperparameters, and is abbreviated as . For the offgrid parameter , the irrelevant items are ignored, and the value of is computed by

According to equations (12) and (14), maximizing is equivalent to minimizing , where is a constant independent of , is a positive semidefinite matrix, . The optimization problem of is given bywhich is a linear least-squares problem with boundary constraints. Let its partial derivative with respect to be zero as follows:

If is invertible, equation (29) has a unique solution, that is

Considering the boundedness, when , then . While one of the above two conditions is not satisfied, is updated element by element. When updating , other elements are fixed and the l-th equation with the partial derivative is zero, and the expression is given bywhere is the vector to remove the l-th element and is the l-th column in . The calculation formula of is described as