Sparse Non-Uniform Linear Array-Based Propagator Method for Direction of Arrival Estimation

Abstract

A novel approach that does not require the number of sources as a priori is proposed to estimate the direction of arrival (DOA) based on a sparse non-uniform linear antenna array. To ensure the identifiability of the DOA, a specific configuration scheme of sparse array is designed. Based on this specific sparse array, firstly the fourth-order cumulant (FOC) is adopted to eliminate the impact imposed by Gaussian noise. Secondly, to circumvent eigenvalue decomposition or singular value decomposition, a propagator is constructed by using a Hermitian FOC matrix and a hyperparameter. Finally, a projection onto an irregular Toeplitz set is proposed to further improve estimation accuracy.

1. Introduction

Direction of Arrival (DOA) estimation technology has extensive applications in various domains, including antenna array signal processing, seismic exploration, wireless communication systems, radar systems, autonomous vehicles, underwater acoustic communication systems, and so on.

Source localization using DOA estimation is crucial in antenna array signal processing [1]. For far-field sources, the wavefront of signals can be considered plane, and the source position can be characterized by the DOA [2]. It enables accurate source localization and enhances the performance of smart antenna techniques. Ref. [3] demonstrated the benefits of smart antennas in terms of carrier-to-interference ratio, signal-to-noise ratio, and robustness against interference and angular spread. This technology is particularly important for future wireless networks, where smart multiple-input multiple-output (MIMO) antennas with advanced signal processing algorithms are necessary for better network coverage and tracking [4,5].

Seismic exploration relies on DOA estimation to reduce losses caused by earthquakes and identify potential oil and gas reserves [6,7]. Ref. [8] proposed a wireless sensor network based on ultra-wideband (UWB) technology for land seismic exploration for oil and gas reservoirs. By analyzing reflected and refracted seismic waves, geophysicists can accurately identify potential hydrocarbon reservoirs beneath the Earth’s surface, contributing to efficient resource exploration and extraction. Other researchers, such as Chalise et al. [9] and Zhang et al. [10], have also proposed DOA-based algorithms that utilize polarization information and compressive sensing techniques for subsurface imaging, enhancing the capabilities of seismic exploration techniques.

DOA estimation plays an important role in wireless communication systems, which is imperative in remote monitoring and control in critical industries such as manufacturing, energy, utilities, and transportation [11,12]. It improves operational efficiency, reduces downtime, and enhances safety by accurately estimating the direction of the arrival of signals and enabling accurate localization and continuous monitoring of assets. Its applications include remote monitoring of production lines, power generation facilities, water supply networks, and transport vehicles. Dakulagi et al. [13] proposed a DOA algorithm using a single snapshot for solving multi-target localization problems using uniform circular array (UCA). The continued development and implementation of DOA estimation techniques will further advance wireless communication systems, facilitating more intelligent and reliable industrial operations.

In radar systems, DOA estimation is widely used for target detection, tracking, and localization [14]. For instance, ref. [15] introduced a DOA-based technique using multi-monostatic radar for tunnel detection and localization, enabling reliable identification of underground structures. Additionally, ref. [16] proposed a method for high-resolution DOA estimation using MIMO radar in the case of a limited number of snapshots. The simulation results show that this method can significantly improve the DOA estimation accuracy of MIMO radar in the case of a limited number of snapshots. Furthermore, Xu et al. [17] developed a DOA-based algorithm for subsurface imaging using seismic data, allowing precise imaging of underground features. DOA estimation allows radar systems to adapt their sensing parameters based on the environment and target scenario. By estimating the DOA, the radar system can dynamically adjust its beamwidth, transmit power, or waveform properties to optimize detection range, resolution, and sensitivity. These studies demonstrate the significance of DOA estimation in enhancing the performance and capabilities of radar systems across different applications.

In autonomous vehicles, DOA estimation can assist them in localizing and tracking sound sources, such as emergency sirens or approaching vehicles. It enhances situational awareness and contributes to safer navigation. A robust and accurate vehicle localization system consisting of a bistatic passive radar is proposed, which exhibits better performance than the existing sparse signal representation-based algorithms, and performs well in the vehicle localization system [18]. Cantarini et al. [19] introduces a prototype of an emergency vehicle detection system based on audio and video sensors. The author used a four channel microphone array to capture sound and used a spectral correlation-based DOA algorithm to estimate the direction of the sound source. This enables autonomous vehicles to proactively adjust their trajectory and take appropriate actions, ensuring safer navigation and facilitating efficient emergency vehicle passage.

A lot of celebrated algorithms have been put forward to settle the DOA estimation issue, for instance, multiple signal classification (MUSIC) [20], root-MUSIC [21], and ESPRIT [22]. In these classic methods, the eigenvalue decomposition (EVD) or singular value decomposition (SVD) is an indispensable operation to separate the noise subspace and signal subspace. To avoid EVD or SVD, a propagator that constructs the noise subspace from the received signal is also presented [23].

However, most algorithms focus on the DOA estimation exploiting the uniform linear antenna array (ULA) [24]. Compared with the non-uniform linear antenna array (NULA), the structure of the ULA is relatively simple, and the array manifold matrix has Vandermonde structure, which enables the MUSIC algorithm to be applied to the high-resolution estimation of signals. Nevertheless, due to the limitation of the spatial sampling theorem, distance between adjacent elements of the ULA can not be greater than half of the wavelength of the incoming signal. Thus, a sparse NULA with a specific configuration scheme is adopted to break the limitation of the half wavelength.

In addition, under the influence of the Gaussian noise, the estimation accuracy declines due to the noise buried in signal covariance matrix, which can not be eliminated. Making the algorithm more noise-resistant, a high-order cumulant [25,26,27,28,29,30] of the received signal becomes a promising countermeasure, which demonstrates outstanding resilience to Gaussian noise, rather than calculating the second-order covariance matrix directly.

More to the point, in the above mentioned approaches, the source number is required as an a priori condition, thus it needs to be estimated before performing the DOA estimation through these approaches. The source number estimation is another piece of work and it was first proposed by Wu [31]. However, when the source number cannot be estimated precisely, the signal subspace and noise subspace cannot be separated correctly. In reality, the accurate source number is difficult to estimate from the received signal.

To deal with the above problems, a unique approach based on sparse NULA is proposed for the DOA estimation by calculating a propagator with a Hermitian matrix without being aware of the source number. Firstly, a Hermitian matrix is designed with the FOC. The Hermitian matrix is utilized to construct the propagator, which is adopted to avoid the EVD or SVD operation. Then, both the DOA and source number are estimated through the spatial spectrum, which consists of the propagator and steering vector. Finally, with the estimated source number, DOA, and covariance matrix, a distinct method, which projects the covariance matrix of the received signal onto an irregular Toeplitz set, is modified to refine the estimation accuracy.

The article’s remaining sections are arranged as follows. The signal model as well as the questions to be solved in the NULA instance is introduced in Section 2. Section 3 discusses the specific array configuration scheme, which is sufficient for the DOA identifiability. The approach proposed by us is described in depth in Section 4. Section 5 displays several experiment results to attest to the effectiveness of the suggested algorithm. In the end, Section 6 draws a conclusion for the whole article.

2. Signal Model

Figure 1 depicts the signal model of a NULA. If K separate narrow-band signals are impinging on the NULA, their distance from the array is much greater than the wavelength of the signal, then the DOA of the same signal source measured by multiple antennas can be approximately regarded as equal. With M antennas and K sources, the antenna position vector and the angles of sources can be considered as $r={[r_0,...,r_{M-1}]}^T$ and $\theta ={[{\theta }_1,...,{\theta }_K]}^T$, where T is transpose operation. The distance between antennas $d_1,...,d_{M-1}$ are not equal and not greater than the half wavelength of the signal, $\lambda$. Thus, $r_0=0,r_1=d_1,...,r_{M-1}=d_1+d_2+...+d_{M-1}$.

The model for the received signal is

$$\mathbf{y}\left(t\right)=\mathbf{A}(\theta ,r)\mathbf{s}\left(t\right)+\mathbf{n}\left(t\right),t=1,2,3,...L$$

(1)

with $\mathbf{y}\left(t\right)$ being the $M\times 1$ received signal vector:

$$\mathbf{y}\left(t\right)={[y_1\left(t\right),y_2\left(t\right),...,y_M\left(t\right)]}^T$$

(2)

L representing the snapshot number, $\mathbf{A}(\theta ,r)$ being the $M\times K$ steering matrix

$$\mathbf{A}(\theta ,r)=[\mathbf{a}({\theta }_1,r),\mathbf{a}({\theta }_2,r),...,\mathbf{a}({\theta }_K,r)]$$

(3)

$$\mathbf{a}({\theta }_k,r)={[e^{\frac{-j2\pi r_{0}sin{\theta }_k}{\lambda }},e^{\frac{-j2\pi r_{1}sin{\theta }_k}{\lambda }},...,e^{\frac{-j2\pi r_{M-1}sin{\theta }_k}{\lambda }}]}^T$$

(4)

$\mathbf{s}\left(t\right)$ being the $K\times 1$ source signal vector

$$\mathbf{s}\left(t\right)={[s_1\left(t\right),s_2\left(t\right),...,s_K\left(t\right)]}^T$$

(5)

and $\mathbf{n}\left(t\right)$ being the $M\times 1$ noise vector

$$\mathbf{n}\left(t\right)={[n_1\left(t\right),n_1\left(t\right)...,n_M\left(t\right)]}^T$$

(6)

3. A Specific Configuration Scheme for Inter-Element Spacing

Although the ULA is a mature and stable array configuration, it is also limited by factors such as the limited inter-element spacing. Additionally, its manufacturing process requires high precision, which will result in higher costs. In reality, the difficulty in controlling the inter-element spacing is completely equal in practical application. In this sense, the sparse NULA is more convenient since it allows inter-element spacing greater than half of the wavelength, it also brings great flexibility to the antenna layout with the same antenna number.

As we all know, the resolution of a linear array is directly proportional to the size of its aperture [32,33,34,35]. Adopting a sparse NULA with every inter-element spacing greater than half a wavelength can provide a maximum aperture. The NULA based on this approach offers excellent performance for the DOA estimation compared to the ULA.

However, when the inter-element spacing exceeds $\lambda /2$, the issue of ambiguous angles rises, which can be resolved only in specific array layouts. Now we adopt a specific configuration scheme for inter-element spacing, which is applicable when the antenna number is not less than both 3 and the source number. As shown in Figure 2, the distance d can be any arbitrary value, including the one greater than $\lambda /2$. The spacing between each element forms an arithmetic progression with a common difference of $▵$, which is restricted to the range of $0<▵\le \lambda /2$.

For a single snapshot and $\theta$, the received signal vector at instant $t_0$ with a size of $M\times 1$, is given by:

$$\mathbf{y}\left(t_0\right)=\mathbf{a}\left(\theta \right)\mathbf{s}\left(t_0\right)+\mathbf{n}\left(t_0\right)$$

(7)

where

$$\mathbf{a}\left(\theta \right)={[1,e^{-j\pi dsin\left(\theta \right)},e^{-j\pi (2d+▵)sin\left(\theta \right)},...,e^{-j\pi [\frac{M^2-3M+2}{2}▵+(M-1)d]sin\left(\theta \right)}]}^T$$

(8)

Both d and $▵$ have been measured in units of half wavelength. $\mathbf{s}\left(t_0\right)$ is the signal sample and $\mathbf{n}\left(t_0\right)$ is the noise at instant $t_0$. In order to guarantee the identifiability of the DOA, one needs to ensure that each value of $\theta$ generates a distinct $\mathbf{a}\left(\theta \right)$. To put it cleverly, $\mathbf{a}\left(\theta \right)$ can be written as:

$$\mathbf{a}\left(\theta \right)={\mathbf{a}}_{ULA}\left(\theta \right)\circ {\mathbf{a}}_{▵}\left(\theta \right)$$

(9)

where

$${\mathbf{a}}_{ULA}\left(\theta \right)={[1,e^{-j\pi dsin\left(\theta \right)},e^{-j\pi 2dsin\left(\theta \right)},...,e^{-j\pi (M-1)dsin\left(\theta \right)}]}^T$$

(10)

$${\mathbf{a}}_{▵}\left(\theta \right)={[1,1,e^{-j\pi ▵sin\left(\theta \right)},...,e^{-j\pi \frac{M^2-3M+2}{2}▵sin\left(\theta \right)}]}^T$$

(11)

and ∘ represents the Hadamard product operation. It is obvious that ${\mathbf{a}}_{ULA}\left(\theta \right)$ is the ULA steering vector. To ensure the DOA to be identifiable, either ${\mathbf{a}}_{ULA}\left(\theta \right)$ or ${\mathbf{a}}_{▵}\left(\theta \right)$ is unique of each value of $\theta$. For ${\mathbf{a}}_{ULA}\left(\theta \right)$, as we all know, it is certainly achievable when d is not greater than $\lambda /2$. However, we hope to surpass the limit of $\lambda /2$ for d. Therefore, ${\mathbf{a}}_{ULA}\left(\theta \right)$ cannot be the exploited component to achieve the goal. Instead, we should be concerned with the ${\mathbf{a}}_{▵}\left(\theta \right)$. Note that, when $M\ge 3$, the minimum value of $\frac{M^2-3M+2}{2}$ is 1 and can only be achieved when M equals 3, and it is the unique minimum value in this case.

4. Proposed DOA Estimator

To better explain the proposed solution, we first introduce the FOC model. The DOA estimators in two scenarios, with and without knowing the source number, are elaborated considering the FOC. Then, the refined solution using projection on an irregular Toeplitz set is proposed.

4.1. Fourth-Order Cumulant

Firstly, the influence of the Gaussian noise is removed by using the FOC. In [36], it is mentioned that if the random variables ${\mathbf{x}}_i$ are independent of the random variables ${\mathbf{y}}_i$, $i=1,2,...,k$, then

$$cum\{{\mathbf{x}}_1+{\mathbf{y}}_1,...,{\mathbf{x}}_k+{\mathbf{y}}_k\}=cum\{{\mathbf{x}}_1,...,{\mathbf{x}}_k\}+cum\{{\mathbf{y}}_1,...,{\mathbf{y}}_k\}$$

(12)

where cum{·} is the cumulant operation.

Suppose $\mathbf{y}\left(t\right)=\mathbf{x}\left(t\right)+\mathbf{n}\left(t\right)$, where $\mathbf{x}\left(t\right)$ and $\mathbf{n}\left(t\right)$ are independent. Then the cumulant of $\mathbf{y}\left(t\right)$ can be written as:

$$cum\{{\mathbf{y}}_1,...,{\mathbf{y}}_k\}=cum\{{\mathbf{x}}_1,...,{\mathbf{x}}_k\}+cum\{{\mathbf{n}}_1,...,{\mathbf{n}}_k\}$$

(13)

On the one hand, according to the probability theory, if a subset of the k random variables ${\mathbf{n}}_i$ is independent of the rest, and the mean of at least one variable is zero, then we have

$$cum\{{\mathbf{n}}_1,...,{\mathbf{n}}_k\}=0$$

(14)

On the other hand, the FOC of source signal can be given by [26,27,37]:

$$cum\{{\mathbf{s}}_a,{\mathbf{s}}_b^{*},{\mathbf{s}}_c,{\mathbf{s}}_d^{*}\}=\left\{\begin{array}{c}{\mathbf{C}}_{4_{{\mathbf{s}}_k}},(k=a=b=c=d)\hfill \\ 0,\left(others\right)\hfill \end{array}\right.$$

(15)

where ${\mathbf{C}}_{4_{s_k}}$ is the FOC of ${\mathbf{s}}_k$.

Ultimately, the FOC of received signal is written as:

$$cum\{{\mathbf{y}}_a,{\mathbf{y}}_b^{*},{\mathbf{y}}_c,{\mathbf{y}}_d^{*}\}=\left\{\begin{array}{c}\sum _{k=1}^K{\mathbf{C}}_{4_{{\mathbf{s}}_k}}{e}^{\frac{-j2\pi r_{g}sin{\theta }_k}{\lambda }},g\ge 0\hfill \\ \sum _{k=1}^K{\mathbf{C}}_{4_{{\mathbf{s}}_k}}{e}^{\frac{-j2\pi r_{-g}sin{\theta }_k}{\lambda }},g<0\hfill \end{array}\right.$$

(16)

where $g=a-b+c-d$.

4.2. Estimator with Source Number

Assuming the source number K can be determined, a method without EVD or SVD operation is developed to localize the source. Define $\mathbf{C}$ as an FOC matrix of the received signal with size $M\times M$. The calculation formula of the $(a,b)$th element of $\mathbf{C}$ is written as:

$$\begin{array}{cc}\hfill & \mathbf{C}(a,b)\hfill \\ \hfill & =cum\{{\mathbf{y}}_a,{\mathbf{y}}_0^{*},{\mathbf{y}}_0,{\mathbf{y}}_b^{*}\}\hfill \\ \hfill & =\left\{\begin{array}{c}\sum _{k=1}^K{\mathbf{C}}_{4_{s_k}}{e}^{\frac{-j2\pi r_{u}sin{\theta }_k}{\lambda }},u\ge 0\hfill \\ \sum _{k=1}^K{\mathbf{C}}_{4_{s_k}}{e}^{\frac{-j2\pi r_{-u}sin{\theta }_k}{\lambda }},u<0\hfill \end{array}\right.\hfill \end{array}$$

(17)

where $u=a-b$, and $\mathbf{C}$ can also be expressed as [38]:

$$\mathbf{C}={\mathbf{AC}}_{4_{s_k}}{\mathbf{A}}^H$$

(18)

where H stands for the complex conjugate operation.

In the situation of knowing K, a new matrix can be created using the first K columns of $\mathbf{C}$:

$${\mathbf{U}}_s=[{\mathbf{c}}_1,{\mathbf{c}}_2,...{\mathbf{c}}_k]$$

(19)

Define a propagator:

$${\mathbf{U}}_n={\mathbf{E}}_M-{\mathbf{U}}_s{\left({\mathbf{U}}_s^H{\mathbf{U}}_s\right)}^{-1}{\mathbf{U}}_s^H$$

(20)

where ${\mathbf{E}}_M$ is an $M\times M$ identity matrix and it is obviously that:

$${\mathbf{U}}_s^H{\mathbf{U}}_n={\mathbf{0}}_{K\times M}$$

(21)

where ${\mathbf{0}}_{K\times M}$ is a $K\times M$ zero matrix. Thus, ${\mathbf{U}}_s$ is orthogonal to ${\mathbf{U}}_n$, and it corresponds to:

$${\mathbf{A}}^H{\mathbf{U}}_n={\mathbf{0}}_{K\times M}$$

(22)

Then, by constructing the spatial spectrum based on this formula, the DOA is available. It can be seen from the above steps that the EVD or SVD operation is avoided.

4.3. Propagator Based on Elementary Transformation without Source Number

Traditionally, before performing the DOA estimation, signal detection methods are required [39]. Then, a source number estimator will be utilized, for instance, CorrM [40], BIC [41], and AIC [42]. Actually, source number estimation can be achieved through elementary column transformation, without any source number estimator. We apply iterative transformation to $\mathbf{C}$:

$$\frac{{\mathbf{C}}_{i,(M+1-i)}}{{\mathbf{C}}_{(i+j),(M+1-i)}}{\mathbf{C}}_{i+j}-{\mathbf{C}}_i\stackrel{}{\to }{\mathbf{C}}_{i+j}$$

(23)

where i traverses from 1 to M and remains constant through each repetition. Meanwhile, j traverses 1 to $M+1-i$. We repeat this operation until an upper triangular-like matrix is acquired. The maximal linearly independent subsystem (i.e., the nonzero column) of $\mathbf{C}$ is represented by ${\mathbf{F}}_c$. Thus, the propagator can be reconstructed with ${\mathbf{F}}_c$, which is of course a full column-rank matrix, serving as a substitute for ${\mathbf{U}}_s$:

$${\mathbf{U}}_n={\mathbf{E}}_M-{\mathbf{F}}_c{\left({\mathbf{F}}_c^H{\mathbf{F}}_c\right)}^{-1}{\mathbf{F}}_c^H$$

(24)

4.4. Estimator Based on Improved Propagator without Source Number

Notwithstanding the achievement of DOA estimation based on the above propagator without the priori K, the transformation of (21) is a K estimator in nature, which poses great uncertainty as well as extra computation. That is to say, the propagator depends on whether K is correct or not. If K is estimated incorrectly, the spatial spectrum cannot be constructed correctly, which causes the error of the DOA estimation. Therefore, there is a need to propose another method that does not require K at all.

Now, assuming that K can no longer be obtained, and the first $\widehat{K}$ columns are extracted from $\mathbf{C}$ to calculate the propagator ${\widehat{\mathbf{U}}}_n$:

$${\widehat{\mathbf{U}}}_s=[{\mathbf{c}}_1,{\mathbf{c}}_2,...,{\mathbf{c}}_{\widehat{K}}]$$

(25)

$${\widehat{\mathbf{U}}}_n={\mathbf{E}}_M-{\widehat{\mathbf{U}}}_s{\left({\widehat{\mathbf{U}}}_s^H{\widehat{\mathbf{U}}}_S\right)}^{-1}{\widehat{\mathbf{U}}}_s^H$$

(26)

When $\widehat{K}<K$, we have $rank\left({\widehat{\mathbf{U}}}_s\right)\le \widehat{K}<K=rank\left({\mathbf{A}}_1\right)$, which would destroy the orthogonality between ${\mathbf{A}}_1$ and ${\widehat{\mathbf{U}}}_n$. When $\widehat{K}>K$, the size of ${\widehat{\mathbf{U}}}_s$ is $M\times K$, so ${\widehat{\mathbf{U}}}_s$ is not full column rank, which makes the inverse operation:${\left({\widehat{\mathbf{U}}}_s^H{\widehat{\mathbf{U}}}_S\right)}^{-1}$ impossible. But the real K is unknown, so taking the whole $\mathbf{C}$ as ${\widehat{\mathbf{U}}}_s$ is a feasible way to ensure that the propagator can be calculated. Since ${\mathbf{C}}^H\mathbf{C}$ is positive semi-definite, introducing an identity matrix can make ${\mathbf{C}}^H\mathbf{C}$ a positive definite. Then, the improved propagator is written by:

$${\mathbf{U}}_i={\mathbf{E}}_M-\mathbf{C}{({\mathbf{C}}^H\mathbf{C}+\mu {\mathbf{E}}_M)}^{-1}{\mathbf{C}}^H$$

(27)

where $\mu$ is a hyperparameter that relates to both calculation efficiency and estimation accuracy. When it is too small, the term $\mu {\mathbf{E}}_M$ may not work, which causes the failure of the construction of the propagator. When it is too large, the estimation accuracy would be significantly affected, which is similar to the affection by noise. Thus, it is necessary to carefully consider the value of $\mu$ in the actual scenario.

Finally, the spatial spectrum can be given by:

$$f(\theta ,r)=\frac{1}{\left|\right|diag\left(\mathbf{a}{(\theta ,r)}^{*}{\mathbf{U}}_i{\mathbf{U}}_i^H\mathbf{a}{(\theta ,r)}^T\right)\left|\right|}$$

(28)

Finding all the peaks that are higher than the preset threshold, the DOA and source number can be obtained.

4.5. Refine Accuracy Based on the Projection onto an Irregular Toeplitz Set

According to the matrix theory, any Toeplitz matrix, $\mathbf{T}$, can be decomposed into Vandermonde components:

$$\mathbf{T}=\mathbf{VD}{\mathbf{V}}^H$$

(29)

where D is a diagonal matrix with positive entries.

As shown in (4) and (5), $\mathbf{A}$ is not a Vandermonde matrix, but it has similar structure to the Vandermonde matrix. Hence, the irregular Vandermonde matrix is deemed as a generalization. Similarly, a Toeplitz-like matrix created from an irregular Vandermonde matrix can be defined as an irregular Toeplitz matrix.

With the estimated source number and DOA, for the sake of improvement of the estimation precision, we provide a modified technique that projects the covariance matrix of the received signal onto an irregular Toeplitz set.

Firstly, we define another irregular Vandermonde matrix, which is similar to $\mathbf{A}$: $\mathbf{Q}(p,\varphi )=\mathbf{A}(\theta ,r)$, introducing the mapping [43]:

$$\varphi =r,p_k=e^{-j\pi sin{\theta }_k},{\theta }_k=-{sin}^{-1}\left(\frac{\angle p_k}{\pi }\right)$$

(30)

where ∠ represents the operation of finding the amplitude of a complex number. From the mapping based on root-MUSIC, we can see that with the estimated $\theta$, the roots on the unit circle p can be calculated.

Secondly, we define the source power:

$$\mathbf{c}={[c_1,...,c_k]}^T$$

(31)

$$c_k=\left|\right|{\mathbf{s}}_k{\left|\right|}_2$$

(32)

which can also be expressed as [43]:

$$\mathbf{c}=diag\left({\mathbf{Q}}^{†}\mathbf{R}{\left({\mathbf{Q}}^{†}\right)}^H\right)$$

(33)

where $\mathbf{R}$ is the covariance matrix of the received signal and

$${\mathbf{Q}}^{†}={\left({\mathbf{Q}}^H\mathbf{Q}\right)}^{-1}{\mathbf{Q}}^H$$

(34)

In the ULA case, $\mathbf{R}$ is calculated by the following procedure:

$$\mathbf{R}=\mathbf{A}diag\left(\mathbf{c}\right){\mathbf{A}}^H$$

(35)

Obviously, $\mathbf{R}$ is a Toeplitz matrix. Thus, as a generalization of the covariance matrix from the ULA to NULA, the projection $\mathbf{R}\to \tilde{\mathbf{R}}$ is given by:

$$\tilde{\mathbf{R}}=\mathit{P}\left(\mathbf{R}\right)=\mathbf{Q}diag\left(\mathbf{c}\right){\mathbf{Q}}^H$$

(36)

In conclusion, the optimization process can be summarized as a function: $\left[\tilde{\mathbf{R}}\right]=optimization(r,\mathbf{R},K,\theta )$, where r is the antenna position vector, $\mathbf{R}$ the covariance matrix of the received signal, and K and $\theta$ the estimated source number and DOA.

Noise subspace ${\mathbf{V}}_n$ can be obtained from $\tilde{\mathbf{R}}$ through the SVD operation, then a new spatial spectrum can be constructed by:

$$\tilde{f}(\theta ,r)=\frac{1}{{\left|\right|\mathbf{a}(\theta ,r)}^H{\mathbf{V}}_n{\mathbf{V}}_n^H\mathbf{a}(\theta ,r)\left|\right|}$$

(37)

The estimation accuracy has been further improved by using this spatial spectrum.

5. Experimental Results

To validate the feasibility of our proposed method, we conducted four experiments to examine it from different perspectives and the results are shown in this section.

Supposing that the electromagnetic wave of a signal source far away from the antennas hits the antennas, the incidence angle $\theta$ is set to −45° and 60°. It should be emphasized that our proposed method does not require the source number K as a prior knowledge in the following experiments, whereas other comparison algorithms rely on it.

Experiment 1. 

The root mean square error (RMSE) is investigated for different combinations of d and $▵$ using our proposed algorithm with a specific array configuration scheme under the following conditions: signal-to-noise ratio (SNR) = 10, antenna number = 5, and the snapshot number of 200. RMSE is a metric that can directly reflect the estimation accuracy, which is measured in degrees, given by:

$$RMSE=\sqrt{\frac{1}{D}\sum _{d=1}^D{({\widehat{\theta }}_d-\theta )}^2}$$

(38)

where D is the trial number in each SNR value, which is set to 500. $\theta$ is the true DOA and $\widehat{\theta }$ is the estimated DOA.

As far as we are concerned, there is no need to display the RMSE of every incident angle. Only the average RMSE of all incident angles needs to be evaluated. So the average RMSE for the two angles, −45° and 60°, is shown in

Table 1. RMSE under various schemes.

	d	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9	1
$▵$
1		0.6781	0.3770	0.3668	0.3161	0.4226	0.2484	0.2718	0.2082	0.2678	0.1886
2		0.2206	0.2423	0.2019	0.1841	0.1753	0.1835	0.1744	0.1531	0.1553	0.1536
3		0.1557	0.1693	0.1642	0.1438	0.1399	0.1402	0.1511	0.1187	0.1235	0.1197
4		0.1177	0.1413	0.1117	0.1632	0.1176	0.1232	0.1069	0.1427	0.1204	0.1067
5		32.9941	0.1225	0.1110	0.1347	0.0940	0.1166	0.0869	0.1271	0.0753	0.1067
6		0.1211	3.2585	0.0847	0.0727	0.0962	0.0899	0.0722	0.0680	0.0789	0.0872
7		0.0670	0.0733	0.0733	0.0566	0.0660	0.0626	0.0657	0.0551	0.0536	0.0602
8		0.0539	0.0501	0.0870	0.0481	0.0575	0.0447	0.0766	0.0507	0.0497	0.0360
9		0.0575	0.0549	0.0542	0.0480	0.0667	0.0373	0.0647	0.0332	0.0762	0.0298
10		63.3342	0.0547	0.0306	0.0703	0.0419	0.0467	0.0346	0.0471	0.0423	0.0281

. d as well as $▵$ are measured in units of $\lambda /2$.

From Table 1, we can conclude that as $▵$ and d increase, the array performance improves and becomes more stable. Overall, the proposed method performs well, even though the estimation process did not use the source number. It can be seen that when $d=1$ and $▵=10$, the average RMSE value is the smallest. In addition, greater $▵$ brings a larger aperture, which benefits estimation accuracy. Eventually, we decided to choose this scheme to construct the antenna position vector for our proposed algorithm in the next two experiments, which is given by: $r={[0,10,21,33,46]}^T$.

Experiment 2. 

The effectiveness of our proposed estimators using the above r is considered by calculating the change of RMSE with SNR. The experiment condition is set to: antenna number = 5, incident angles of −45° and 60°, and the snapshot number of 100.

Figure 3 demonstrates that the proposed method achieves the highest accuracy, although the source number is unknown. As comparison, the second-order cumulant (SOC) [44] and the fourth-order cumulant (FOC) [45] methods as well as the classic MUSIC algorithm are introduced in the experiment. Obviously, regardless of the signal-to-noise ratio, the performance of FOC and SOC always falls short of the proposed method. MUSIC, on the other hand, performs slightly worse than FOC. We infer that this is likely due to the fact that MUSIC is only applicable to arrays with inter-element spacing not greater than $\lambda /2$. Consequently, with the same antenna number, the array aperture is the smallest, which is the main reason for its slightly lower estimation accuracy.

Experiment 3. 

In addition to the relationship between the SNR and RMSE, the one between the snapshot number and RMSE is also crucial for considering the performance of our proposed method. In this experiment, it is explored under the condition: antenna number = 5, incident angles of −45° and 60°, and the SNR of 10. Again, the SOC and FOC methods as well as the classic MUSIC algorithm are introduced. The simulation results of the experiment are shown in Figure 4, from which we can draw a conclusion that the snapshot number also plays an important role in decreasing RMSE. As the snapshot number increases, the performance of all four algorithms improves. It also shows that the proposed method consistently performs the best without prior knowledge of the source number.

Experiment 4. 

We depicted the relationship between the antenna number M and the estimation success rate P, using our estimator without prior knowledge of the antenna number, which is detailed in Chapter 4, Section 4. In this experiment, we recorded the P as M increased from 3 to 10, while keeping $d=\mathit{3}$, $▵=\mathit{1}$, SNR = 5, and snapshot number of 200. P represents the percentage of successful estimation instances out of the total number of estimation attempts: $P=\frac{S}{D}\times 100\%$, where S is the number of estimated successful attempts and D is the total number of estimation attempts.

Table 2. P increases as M increases.

M	3	4	5	6	7	8	9	10
$P(\%)$	58.4	85.2	88.6	91	94.8	96.8	98	98.4

shows the simulation results and it is evident that as M increases, the estimation success rate P also increases for the reason that an array with more antennas has a larger array aperture, which improves the performance of the array. Experiment 1 and Experiment 4 have demonstrated that increasing d, $▵$, and M can all improve the performance of the array.

To some extent, these experimental results demonstrate that our estimator achieves comparable or better estimation accuracy compared to other methods. The most important point is that the DOA estimation is achieved by our estimator without prior knowledge of the source number, which confirms that the proposed algorithm eliminates the dependency on the source number.

6. Conclusions

Sparse non-uniform linear array-based direction of arrival estimation using spatial spectrum searching without prior knowledge of the source number is considered. To achieve this goal, we adopted the fourth cumulant to eliminate the impact imposed by Gaussian noise, so as to minimize noise interference in the estimation process. Next, we constructed a Hermitian matrix that avoids the need for eigenvalue decomposition, which requires knowledge of the source number. Based on the proposed propagator, a spatial spectrum is constructed to estimate the direction of arrival. To further improve the estimation accuracy, we introduced the projection onto an irregular Toeplitz set for optimization. In addtion, a specific array configuration scheme is utilized to ensure the identifiability of the direction of arrival. Eventually, in comparison to previous methods, we have maintained a good level of estimation accuracy without prior knowledge of the source number. In summary, we have proposed a sparse non-uniform linear array-based propagator method that ensures the identifiability of the direction of arrival without prior knowledge of the source number.