1. Introduction
Array signal processing has been intensively investigated over the last several decades [
1,
2]. As one of the hotspots in array signal processing, the direction of arrival (DOA) estimation problem of plane waves impinging on an antenna array has drawn widespread attention. Numerous DOA estimation algorithms have been developed to achieve high-resolution performance, e.g., multiple signal classification (MUSIC) [
3], estimating signal parameters via rotational invariance technique (ESPRIT) [
4], maximum likelihood (ML) approach [
5], compressed sensing (CS) method [
6] and sparse Bayesian learning (SBL) [
7]. These estimators work well under the assumption that prior knowledge of the array manifold is available. However, in practical applications, this assumption is unrealistic since a series of array imperfections including gain-phase errors, mutual coupling and sensor location errors, may affect the ideal array manifold and degrade significantly the DOA estimation accuracy [
8,
9,
10,
11,
12]. Therefore, it is highly desirable to estimate and calibrate array imperfections before DOA estimation. We focus the paper on estimating the gain-phase errors of uniform linear arrays (ULAs). Typically, the gain-phase errors are caused by the following factors: gradual changes of the behavior of the sensor itself and the internal circuits (due to thermal effects, aging of components, etc.), changes in location of the array elements induced by the vibrating wing of an aircraft or a hydrophone array behind a ship, and changes to the environment around the sensor array [
13]. Herein, we consider these three main origins of gain-phase errors and assume that the gain-phase error caused by these changes is in the presence of a constant since all the above-mentioned changes are tiny and slow in a short time.
In recent decades, lots of effort has been made to address the gain-phase errors of antenna arrays [
13,
14,
15,
16,
17,
18,
19,
20,
21,
22,
23,
24,
25]. Existing gain-phase error calibration methods can be roughly divided into two categories: self-calibration (without calibration sources) and pre-calibration (with calibration sources). In general, the first type of gain-phase error estimation method, which can jointly estimate the DOA parameters and gain-phase errors, does not require other calibration sources with known directions. Hence, it is more suitable for practice and has evoked much research interest for the past few years [
13,
14,
15,
16,
17,
18]. In [
13,
14], a self-calibration iterative method based on the eigenstructure is proposed to simultaneously estimate the DOAs of source signals and calibrate gain-phase errors. However, the drawback of this method is that it may suffer from suboptimal convergence. The algorithm proposed in [
15] provides a consistent estimate of the gain-phase errors from the ML perspective, but it requires the covariance matrix of the ideal array output to be known. By using the diagonal lines of the covariance matrix, a class of the gain-phase error calibration algorithm is presented for linear equispaced arrays (LEA) in [
16], but the perfect covariance matrix is unavailable when the number of snapshots taken to obtain the covariance matrix is small [
17]. Z. Liu et al. proposed a novel sparse Bayesian array calibration (SBAC) method for compensating for all the typical array imperfections [
18], whereas the online estimation of numerous unknown parameters makes the SBAC more complicated. Many proposals have been conducted on the gain-phase error calibration problem for nonlinear arrays [
19,
20,
21]. In [
19], A. Liu et al. presented an eigenstructure DOA algorithm in the presence of gain-phase errors, which is based on the dot product of the array output and its conjugate. Nevertheless, it requires that at least two signals are spatially far from each other. A Hadamard product-based method [
20], which performs independently of the phase errors and without the requirement of two signals being spatially far separated from each other, achieves better estimation performance with respect to [
19]. However, it suffers from high computational load. In [
21], W. Xie et al. extended the amplitude-only measurements-based technique [
19,
20] into central symmetric arrays (CSAs) and proposed an algorithm for jointly estimating the DOAs of noncircular sources and gain-phase errors. In addition, for underwater sensing tasks, the random acoustic fluctuations in the medium, such as a highly dynamic ocean, may introduce phase errors in the output of the ULA. To address this issue, a novel L1-norm (absolute-error) maximum projection principal component analysis (PCA) method in [
22] is proposed to resist the uncertainty of random acoustic fluctuations. Dubrovinskaya et al. designed acoustic arrays of an arbitrary shape and a practical algorithm to achieve accurate DOA estimates under some array imperfections, such as the spatial ambiguity that must be compensated for [
23].
Unlike the self-calibration method, the second type of method, named the pre-calibration method, requires calibration sources with known directions. The eigenstructure-based pre-calibration method with two calibration sources with known directions was developed in [
24], which depends on the covariance differencing technique and iterative method. However, since the gain-phase errors are estimated by using the phase of each entry in the steering vector, the possible phase ambiguities need to be checked. Based on the elegant analysis of the Cramer–Rao bounds on calibration and source location accuracies under three different sensor location situations, the authors of [
26] indicate that two calibration sources are needed to calibrate the planar array, and the gain-phase estimation errors tend to decrease as the calibrating signal strength or power increases. By utilizing a set of calibration sources in known locations, a ML calibration algorithm is presented in [
27] and the array perturbation parameters can be solved by means of the least square method. Nonetheless, the unique determination (identifiability) of the perturbation parameters is not satisfied in this method. Based on the null characteristic of the MUSIC spectrum, the array sensor gain-phase error calibration problem can be formulated as a series of linear equations and solved by a constrained optimization [
25]. When the number of the linear equation is larger than that of the gain-phase errors to be estimated, a unique solution can be obtained. However, plenty of calibration sources with known DOAs are needed to solve for the uncertainties of array gain-phase errors. Under the existence of array phase errors, a non-coherent DOA estimation method is proposed in [
28,
29] based on the modified version of the greedy sparse phase retrieval (GESPR) framework [
30]. Nevertheless, multiple calibration sources are required to cope with the phase ambiguity problem, and the global minima cannot be guaranteed [
31]. Compared with the self-calibration method, although the pre-calibration method with calibration sources is somewhat limited practically when numerous calibration sources are required, there are two reasons why we take it to be once again in the spotlight. First, the pre-calibration one, in general, is more computationally efficient than the self-calibration since it avoids the alternation between the DOA estimation and array calibration, and the uncertainty calibration procedures are accomplished one at a time. Second, the pre-calibration can provide more satisfactory calibrating accuracies than the self-calibration due to some prior knowledge of the calibration sources [
27]. Therefore, to obtain both satisfactory calibrating accuracy and low-cost implementation, we intend to explore an accurate, low-cost, broadly applicable gain-phase error calibrating scheme, which can be available for both small-scale and large-scale ULAs.
The adaptive antenna nulling technique, also known as power-inversion adaptive array [
31], was firstly proposed by Appelbaum [
32]. Its primitive application was to reduce sidelobe levels in the unknown direction of interferences by weighting the received signal vectors to minimize the interference powers. Alternatively, it is only necessary to have the steering vectors in the directions of interest to be zeros, i.e., null steering. In [
33], several constrained null steering algorithms based on adaptive antenna nulling array are introduced, including mainly constrained least-mean-square (CLMS) and QR-recursive least-squares (QR-RLS) algorithms, and some convergence behaviors of the corresponding algorithms are also provided. The infinite impulse response (IIR) adaptive nulling array structure is designed in [
34], which consists of a number of shift-invariant subarrays and takes the outputs of previous subarrays as spatial feedback. Moreover, the null steering scheme has been applied in combating spatial acoustic feedback between the hearing aid loudspeaker and microphone(s) [
35,
36], in which the calculation of null steering beamformer coefficients can be formulated as a least-square (LS) optimization or a min-max optimization so as to be null in the direction of the acoustic feedback. As an extension of [
36], the same authors present a soft constraint on designing the null steering beamformer coefficients [
37] in order to cope with the incoming acoustic signal preservation and feedback cancellation simultaneously.
In this paper, unlike using nulls of the MUSIC spectrum, we present a new gain-phase error pre-calibration method for ULAs by exploiting the adaptive antenna nulling technique [
33,
34] and null steering algorithm [
35,
36,
37]. Only one known-DOA calibration source is required. We divide a ULA with
M array elements into
$M-1$ sub-arrays, and the gain-phase errors of each sub-array can be uniquely extracted one by one. To be specific, for the
mth sub-array, it is able to derive the gain-phase errors as the reciprocal of the cumulative product of a series of complex coefficients, which relate to the sub-array unperturbed null steering vector (SAUNSV). In order to reliably estimate the SAUNSV,
${\overline{\mathbf{w}}}_{m}$, we formulate the received data structure of ULAs as an errors-in-variables (EIV) model, based on which a weighted total least-squares (WTLS) algorithm is presented. Furthermore, we show that (1) the optimal location of the unique calibration source is the normal direction of the ULA. (2) When the calibration signal-to-source signal power ratio (CSR) increases, the gain-phase error estimation performance improves. As a consequence, it requires the calibration signal power to be much larger than the source signal. Our main contributions are therefore as follows: (1) By using the adaptive antenna nulling technique, a new gain-phase error pre-calibration method with only one calibration source is presented. (2) We formulate an errors-in-variables (EIV) model and propose a WTLS algorithm in order to estimate the SAUNSV, which is a crucial factor for the gain-phase error estimation. (3) Some comparative statistical analyses on the solution to the WTLS problem are derived.
The paper is organized as follows. In
Section 2, the signal model for gain-phase error estimation is established and the adaptive antenna nulling technique is reviewed. The proposed method is derived in
Section 3. In
Section 4, we formulate an EIV model, propose a WTLS algorithm for estimating the SAUNSV and give the statistical analyses on the solution to the WTLS.
Section 5 describes the simulation results, and the conclusion is given in
Section 6.
Notation: Superscripts ${(\xb7)}^{H}$, ${(\xb7)}^{T}$ and ${(\xb7)}^{*}$ stand for conjugate transpose, transpose and complex conjugate, respectively. Matrices and vectors are represented by bold upper-case and bold lower-case characters, respectively. The notation $\mathrm{diag}\left(\mathbf{a}\right)$ means forming a diagonal matrix by using the vector $\mathbf{a}$ as its main diagonal entries, while $\mathrm{diag}(\Lambda )$ denotes the vector formed by the diagonal of $\Lambda $ if $\Lambda $ is a diagonal matrix. The notation $E(\xb7)$ denotes the mathematical expectation, $\mathrm{tr}(\xb7)$ stands for the trace operator, $\left|\right|\xb7\left|\right|$ and $|\xb7|$ represent the Euclidean norm and absolute value, respectively. ⊗ denotes the Kronecker product. ${\mathbf{I}}_{M}$ denotes a $M\times M$ identity matrix, ${\mathbf{0}}_{M}$ is a $M\times 1$ column vector with all zero entries and ${0}_{MM}$ is a $M\times M$ matrix with all zero entries. $vec(\xb7)$ denotes the vectorization operator, which stacks all columns of a matrix one below the other to form a column vector.
5. Simulation Results and Discussion
In this section, several simulations are conducted to verify the validity and effectiveness of the proposed gain-phase error estimation method. We first consider the gain-phase error estimation performance of the proposed method for both small-scale and large-scale ULAs in
Section 5.1. Then, in
Section 5.2, the proposed method is compared with the eigenstructure method [
13], diagonal line method [
17], and MUSIC nulling method [
25]. At last, in
Section 5.3, the DOA estimation performance results are provided, which depend on the aforementioned gain-phase error estimation methods. All the results are obtained by averaging over 500 Monte Carlo trials. Two far-field uncorrelated source signals, with equal power
${\sigma}_{s}^{2}$, impinge on the ULA from directions
${5}^{\circ}$ and
${30}^{\circ}$. The power of the unique calibration signal,
${\sigma}_{c}^{2}$ is set to be 1, which is located at
${\gamma}_{0}={0}^{\circ}$. The signal-to-noise ratio (SNR) is defined as
$\mathrm{SNR}\left(\mathrm{dB}\right)=10{log}_{10}\left({\sigma}_{s}^{2}/\phantom{{\sigma}_{s}^{2}{\sigma}_{n}^{2}}\phantom{\rule{0.0pt}{0ex}}{\sigma}_{n}^{2}\right)$. In addition, since one calibration signal coexists with the source signals in the gain-phase error estimation stage, we define the calibration signal-to-source signal ratio (CSR) as
$\mathrm{CSR}\left(\mathrm{dB}\right)=10{log}_{10}\left({\sigma}_{c}^{2}/\phantom{{\sigma}_{c}^{2}{\sigma}_{s}^{2}}\phantom{\rule{0.0pt}{0ex}}{\sigma}_{s}^{2}\right)$.
The random gain-phase errors, which are proved to be feasible in [
51], can be assumed to be Gaussian distribution, i.e.,
${g}_{m}\sim N(0,{\sigma}_{g}^{2})$, where
${\sigma}_{g}^{2}$ is the variance of gain errors. In addition, for the phase error, we also have
${\gamma}_{m}\sim N(0,{\sigma}_{\gamma}^{2})$, where
${\sigma}_{\gamma}^{2}$ is the variance of phase errors. In simulations,
${\sigma}_{g}^{2}$ and
${\sigma}_{\gamma}^{2}$ are set to be 0.1 and 36, respectively. The deterministic gain-phase error is also considered in
Section 5.2 and
Section 5.3 for a comparison with the cited algorithms. In order to evaluate the estimation precision of the proposed method, the root-mean-square error (RMSE) of the gain-phase error estimation is used, which is defined as
where
K denotes the number of trials and
${\widehat{\mathbf{\Gamma}}}^{\left(k\right)}$ is the estimated gain-phase error in the
kth trial.
Similarly, the RMSE for testing the DOA estimation performance is defined as shown below:
where
${\widehat{\theta}}_{l,k}$ is the estimated DOA of the
lth source signal in the
kth trial.
5.1. Gain-Phase Error Estimation Performance
In this subsection, we study the gain-phase error estimation performance results of the proposed method in terms of CSRs, snapshots, number of array elements, spatial location of the calibration signal, and correlation between the source signals. The simulated RMSEs are calculated depending on the proposed method, while the theoretical RMSE values are derived by using (47). Unless otherwise stated, the total number of snapshot $T=80$ and $N=40$. Accordingly, we can see that $T/\phantom{TN}\phantom{\rule{0.0pt}{0ex}}N=2$ iterations in WTLS are carried out.
Figure 3 shows the gain-phase error estimation RMSE for the proposed method versus CSRs. The numbers of array elements are set to
$M=6,8,12,16$ for
Figure 3a–d, respectively. Similar cases for large-scale ULAs, i.e.,
$M=64,128,256,512$ are illustrated in
Figure 4. We can observe that the RMSEs decrease as CSRs increase, and more satisfactory gain-phase error estimation performance results can be achieved when the power of the calibration source is far larger than that of the signal source. This is due to statistical perturbation of the SAUNSVs according to (46), which depends on CSRs. Furthermore, it is evident that the simulation results closely agree with theoretical predictions when the CSR is larger than 20 dB.
In
Figure 5, the RMSE of the gain-phase estimation using the proposed method is plotted versus numbers of snapshots
T for different array elements
$M=1,050,100$. Two iterations are carried out for the proposed method with
$N=T/2$. It can clearly be seen that the RMSE decreases as
T increases and increases by increasing
M, which coincides with the explicit expression of the gain-phase error estimate (8). The more estimated coefficients
${\widehat{\overline{w}}}_{m}$ involve, the larger the estimated error becomes. Consequently, the gain-phase error estimation accuracy and RMSE degrade with the increase in the number of array elements.
The gain-phase error estimation RMSE for different spatial locations of the calibration source,
${\gamma}_{0}$, is shown in
Figure 6a. We observe that the RMSE performance of the proposed method degrades with the increase in
${\gamma}_{0}$, which attributes to the null position bias of the practical null spectrum in (51). Furthermore, the corresponding null position bias,
$\Delta {\gamma}_{0}$, is plotted in
Figure 6b, from which the simulation results are nearly close in agreement with the theoretical values calculated by (54). The minimum null position bias is achieved in the case of
${\gamma}_{0}={0}^{\circ}$.
Figure 7 shows the gain-phase error estimation performance for different correlation factors of two source signals. In this case, the source covariance matrix can be expressed as
where
$\xi $ is the correlation factor. We can note that the RMSE slightly increases by increasing the correlation factor; however, this adverse impact can be reduced by raising the CSR.
5.2. Comparison to Other Calibration Methods
In this subsection, both deterministic and random gain-phase errors are considered. The deterministic gain-phase error is modeled as
$\Gamma =\mathrm{diag}\left(\left[1{e}^{j{0}^{\circ}}2{e}^{j8.{45}^{\circ}}5{e}^{j12.{75}^{\circ}}3.2{e}^{-j{15}^{\circ}}0.2{e}^{j6.{2}^{\circ}}\right]\right)$ with
$M=5$, and the random gain-phase error is also subject to the Gaussian distribution with zero mean and variances given in
Section 5.1. For a fair comparision, we also apply only one calibration signal located at
${0}^{\circ}$ for the MUSIC nulling method [
25].
Figure 8a,b compare the gain-phase error estimation performance results of the corresponding methods for different SNRs in 5-element ULA with deterministic gain-phase errors and 16-element ULA with random gain-phase errors, respectively. The eigenstructure method [
13] and diagonal line method [
17] belong to the method of the self-calibration type, whereas the MUSIC nulling method [
25] is a pre-calibration method like our proposed one. Generally, the pre-calibration methods are superior to the self-calibration type because of the application of calibration sources with known directions. The eigenstructure method [
13] fails to work no matter how high the SNR is due to its suboptimal solutions and lack of unique properties. For the diagonal line method [
17], an insufficient number of snapshots limits the estimation of the covariance matrix, resulting in a weak gain-phase error estimation performance. The proposed method behaves better than other competitive methods. The RMSE performance comparison on the corresponding methods with a different number of array element is shown in
Figure 9. It is observed that the RMSEs of the MUSIC nulling and proposed methods increase as the number of array element increases because more parameters are needed to be estimated for larger arrays which causes larger estimating errors.
Figure 10 presents the performance comparison for different numbers of snapshot
T in 5-element ULA with deterministic gain-phase errors. For the eigenstructure method [
13] and proposed method, two iterations are carried out with
$N=T/2$. We can see that the proposed one performs better than others.
5.3. DOA Estimation Performance
In this subsection, we compare the DOA estimation performance results of the corresponding methods. The CRB on DOA estimation is also given, which is calculated by Equation (4.6) in ref. [
52]. All the simulation scenarios are identical to
Section 5.2. For estimating DOA of source signals, another
${T}_{0}=40$ signal snapshots are needed to be collected. For effective evaluation and comparison of the proposed method, the DOA estimation performance of the MUSIC algorithm in the absence of gain-phase errors is taken as the benchmark. When applying the eigenstructure method [
13], the DOA can be jointly estimated with the gain-phase errors, while the MUSIC estimator is used to obtain the DOA results after the gain-phase errors are calibrated by the diagonal line method [
17], MUSIC nulling method [
25] and proposed method. Once the gain-phase error estimate
$\widehat{\Gamma}$ is available, the MUSIC spatial spectrum can be obtained by
where
${\widehat{U}}_{e}$ is the gain-phase error-free noise subspace with dimension
$M\times (M-L)$. It can be obtained after the gain-phase error calibration and extracted by using the eigenvalue decomposition (EVD) of the signal covariance matrix estimate, which is calculated by
${\widehat{\mathbf{R}}}_{e}=(1/\phantom{(1{T}_{0}}\phantom{\rule{0.0pt}{0ex}}{T}_{0}){\displaystyle \sum _{t=T+1}^{T+{T}_{0}}}\widehat{\Gamma}\mathbf{e}\left(t\right){\mathbf{e}}^{H}\left(t\right){\widehat{\Gamma}}^{H}.$Finally, we can find the DOA estimate
$\widehat{\theta}$ by using the peak searching of
$f\left(\theta \right)$. In order to reduce the grid mismatch, we herein make use of a recursive grid refinement searching scheme, which contains three steps: (1) Create the maximum and minimum angular grids, whose grid step sizes are
${\theta}_{s,max}$ and
${\theta}_{s,min}$, respectively, and obtain the rough DOA estimate by using
${\theta}_{s,max}$. (2) Decrease the grid step size,
${\theta}_{s,max}={\theta}_{s,max}/\phantom{{\theta}_{s,max}I}\phantom{\rule{0.0pt}{0ex}}I$ and
$I>1$, and obtain a refined DOA estimate by using the updated
${\theta}_{s,max}$ at a local range around the rough DOA estimate in the last step. (3) Return to step (2) until
${\theta}_{s,max}\le {\theta}_{s,min}$. In our simulations,
${\theta}_{s,max}={1}^{\circ}$,
${\theta}_{s,min}=0.{01}^{\circ}$ and
$I=10$.
Figure 11a,b show the DOA estimation performance results of the proposed method and MUSIC algorithm versus CSR in 5-element ULA and 16-element ULA, respectively. It can be observed that the proposed method can achieve satisfactory DOA performance, like MUSIC, without gain-phase errors at high CSR.
Figure 12 and
Figure 13 show the DOA estimation performance comparison among the corresponding methods in 5-element ULA and 16-element ULA, respectively. It is found that our proposed method offers similar DOA estimation performance results to MUSIC in the absence of gain-phase errors, owing to the fact that the gain-phase errors are correctly estimated and calibrated.
Figure 14 provides the DOA estimation performance comparison for different numbers of the array element; we also can see the superiority of the proposed one.
Figure 15 offers the DOA estimation performance comparison of different numbers of snapshots in the 5-element array. It can be noted that the RMSE decreases when increasing the number of snapshot
T.
5.4. Discussion
The simulation results presented in
Section 5.2 and
Section 5.3 show that our method attains more satisfactory gain-phase error and DOA estimation performance with respect to the eigenstructure method [
13], diagonal line method [
17] and MUSIC nulling method [
25]. The simulation was carried out in a MATLAB 2020a environment using an Intel 3.1-GHz processor with 8 GB of RAM and under a Windows 10 operating system. All the simulation results were obtained by averaging over 500 independent Monte Carlo trials. The eigenstructure method requires a two-step iteration, the first one involving the gain-phase error estimation and the second, calculating the DOA estimates. As such, a convergence to a global minimum and a unique solution may not been guaranteed, which typically leads to the performance degradation of the parameter estimation. The diagonal line method makes use of the different diagonal lines of the data covariance matrix to estimate the gain-phase errors. Nevertheless, it is known that the diagonal elements of the covariance matrix are mostly contaminated by the environment noises. As shown in
Figure 8, in contrast to the Diagonal line method, the proposed method is not sensitive to the environment noises. It obtains a satisfactory estimation performance of the gain-phase error, even when the SNR is 0 dB, due to one high-power calibration source. The MUSIC nulling method requires multiple calibration sources and needs to solve a set of linear equations. Due to the adjustable CSR and the optimal spatial location of the calibration source, our proposed method can offer relatively better estimates than the MUSIC nulling method, which does not provide results about the adjustable CSR and optimal spatial location. However, the proposed method requires a high CSR (larger than 20 dB), and the estimation of covariance matrix
${\mathbf{Q}}_{e}$ in (33) leads to the increase in the number of signal snapshots as compared with these three competing methods.
Our future work is to validate the effectiveness and efficiency of the proposed method by using NI software-defined radio radio testbed. The testbed will work with real Wi-Fi data under IEEE 802.11az for indoor localization and positioning. In this prototype, the gain and phase errors are mainly caused by imperfections in antennas, feedlines, and RF chains. Thus, integrating the proposed calibration procedure in the testbed is helpful in the application for the DOA estimation problem.