Advancements in Jammer Location Identification and Suppression: Employing a Multi-Target Least Square Constant Modulus Array Approach

Refuting intentional interference or mitigation of jamming signals has continued to be a major challenge for wireless communication and radar researchers for years [1-8]. Jamming remains a considerable threat to applications that go through space-time processing [5-8]. Jammers are used intentionally in radio communication to deliberately block and disrupt the communication channel, typically by reducing the SNR. Consequently, familiarity of specific localization of intentional jammers is of utmost importance to radio engineers and researchers at low SNR values, so as to mitigate and reduce the effect of interfering signals.

In the entire procedure of jamming signal mitigation, the detection of the location of the jammer constitutes an important and primary step. Jammer locations can be precisely identified by estimating the DOA of the jamming signals impinging on an array of antennas or sensors with some array processing capabilities. Traditional strategies for estimating DOA, such as MUSIC [9] or a variant of ESPRIT [10-14], make use of isotropic sensors or omnidirectional antenna elements of dense uniform linear array (ULA) to form and decompose a square array correlation matrix of the impinging signals. Other than settling with a lesser degree of freedom (DOF), a major disadvantage of the conventional DOA estimation methods is their failure to resolve near-correlated jamming sources [11-13, 15-17]. Thus, there always remains a possibility of detecting an incorrect number of jammers whose spatial separation is narrow. Eventually, this affects the mitigation process or the null steering of the antenna array beam.

Various solutions have been proposed by researchers over the years for the detection and mitigation of jamming signals. The authors [1-8] utilized either the time, frequency, or space domains, or a combination of all domains, to exploit the detection and mitigation methods. Striking drawbacks evident in time domain and frequency domain processing are performance deterioration in tracking, inaccurate correlation peaks, and acquisition issues [2-5].

1.1 Related works

In the space domain, as spatial mitigation is employed, the performance gets significantly improved [1, 6]. Space domain processing has the potential to estimate the DOA of the impinging signals, then direct the main lobe towards the location of the signal-of-interest (SOI) (or beam-steering) and place nulls in the estimated direction of jamming signals (or null-steering) [18-21]. A smart antenna system may be used to realize space-domain processing. Typically, a smart antenna system consists of an array of sensors with an adaptive signal processing capability that can successfully accomplish null-steering and beam-steering [12, 14, 16]. Of the exclusive two categories well explored, the blind class of adaptive beam-forming algorithms invariably has certain edge over non-blind classifications, as the obligatory learning signal is not necessary, thus improving the system's spectrum efficiency. The most illustrative of the blind adaptive beam-forming methods is the CMA [22]. It is an iterative method that conserves the envelope of the beam-forming response almost at a constant level so as to separate the SOI from the interference. But at low SNR values, CMA fails to resolve the interference signal with the SOI if the interference signal stringently follows the constant modulus property. This issue can be sorted out to some degree by using the least squares approach, minimizing the cost function of the CMA over one block of data vectors, and then updating block-by-block. The resulting block iterative method is the LSCMA [14, 23, 24], which acquires better global stability and a faster convergence rate as compared to conventional CMA.

Direction finding based on the paradigm of compressive sensing (CS) structure has garnered considerable interest among researchers in the last decade [25-37]. It provides a different perspective to regenerate the original signal with sparsity characteristics by appending a high reconstruction class at the sensing phase and at the receiver phase, respectively. From 2004 onwards, a group of researchers published a succession of research papers in which it was conclusively proved that exact reconstruction of a signal is conceivable with fewer samples than the Nyquist rate necessitates, provided that some statistics of the sparsity characteristics of the signal are known beforehand. This concept led to the development of the theory of CS in signal processing [25-27, 34].

The achievement of CS-based sparse reconstruction is established on two basic properties of the intended signal: compressed or sparse representation and incoherency [25-27]. In source localization problems, in which the DOA of the impinging signals is estimated, the sparsity or compressed representation is guaranteed by making use of a suitable angular transformation. The framework of CS provides a distinguishing edge over the standard models of AOA estimation in reduced computational complexity, reconstruction by single snapshot, and improved degrees of freedom [27-30].

The general development of sparse signal classification for DOA estimation in the CS domain is well studied in several literatures [26-30, 34]. The single snapshot recovery issue is effectively addressed in several literatures [26, 27, 30, 35], while multiple-time snapshot-based DOA is being estimated and studied in the literature [28], exhibiting promising resolution characteristics. In the literature [29], various optimization algorithms based on $\ell_0, \ell_1$ and $\ell_2$ norms for direction finding are examined, and performances are being compared in terms of computational complexity, mean-square error, resolution, and recovery time. The problem of random selection of sensors and the pair-matching issue in 2-D DOA estimation by an L-shaped array are well studied in several literatures [31-33]. Instead of a ULA, a co-prime array structure is effectively used in the CS paradigm for estimating the DOA of the receiving signals in several literatures [36, 37]. One of the major advantages of the co-prime array structure is that it provides a larger array aperture as compared to ULA, which enables better resolution in DOA estimation problems.

In recent years, a deep learning framework has been introduced by researchers for identifying a specific signal direction from multiple signals impinging on an array of antennas [38-41]. Both 1-D and 2-D DOA estimation parameters are studied in ULA, uniform circular array (UCA), and retro-directive arrays (RDA), respectively. In deep learning models, improved resolution is obtained as the network does not depend on the statistical properties of the signal.

Contemporary developments in DOA estimation, as discussed, have revealed striking advantages over standard models, particularly in obtaining enhanced degrees of freedom, lower computational complexity, a single snapshot instance, and more accurate resolution for near-coherent targets. But the requirements of the training data and the decision and construction of the sensing matrices remain prominent hindrances in both deep learning and sparse reconstruction methods. Also, finding the location of an interferer or jammer and being able to steer nulls of the antenna array beam, when the SNR is low have not been studied or talked about much in recent research.

The main aim of this paper is to introduce a novel high-resolution DOA estimation method for the jammer signals at low SNR estimates that are nearly correlated with each other and null-steer the smart antenna system so as to suppress or mitigate the jamming directions effectively. The MT-LSCMA, which is a variant of the LSCMA algorithm, is a robust adaptive beam-forming method that is well exploited in separating the multipath signals from the SOI [42]. The jamming and the SOI sources or targets are assumed to be located in the far-field region with respect to the ULA-based smart antenna receiver, and the narrowband signals impinge on the ULA indiscriminately from all directions. The intrinsic characteristics of the MT-LSCMA are effectively used to create two groups of adaptive weights to distinguish between the jamming signals and the SOI, so as to beam-steer the main lobe on the way to the direction of the SOI and null-steer by inserting nulls in the direction of the jamming signals. The weights of the steering vector are calculated through a novel iterative methodology that results in highly precise DOA estimation. The suggested method can clearly resolve closely spaced DOAs of jamming signals as well as DOAs of signals of interest with the DOAs of the jammers, which makes the antijamming effect stronger. Furthermore, it is capable of improving the null-steering process and avoiding SOI signal level drops while rejecting jamming signals.

The effectiveness of the proposed approach is evaluated and compared with the conventional MUSIC [9] algorithm, the recently developed modified MUSIC (M-MUSIC) [43] algorithm, and CS-based DOA estimation for randomly selected sensors of a ULA [35] and a sparse coprime array [37] in terms of RMSE, resolution probability with varying SNR, and computational complexity. The M-MUSIC algorithm employs the block adaptation method, similar to the proposed technique here. Consequently, the suggested methodology has a faster convergence speed as compared to the MUSIC algorithm. Investigations are accomplished for single and multiple highly correlated jamming signals. Simulation findings show that the proposed approach can determine the jammer locations more accurately at low values of SNR and performs better in terms of probability of detection, failure rate, RMSE, and resolution probability, therefore enhancing the general system performance.

This is how the rest of the paper is structured. Section 2 introduces the system modeling of the DOA estimation problem, while Section 3 provides a brief description of the conventional MUSIC and the M-MUSIC algorithms. In Section 4, the formation of CS-based array pattern reconstruction for DOA estimation from randomly selected sensors of a ULA and a co-prime array structure is discussed. The proposed jammer mitigation algorithm based on MT-LSCMA is developed in Section 5. Section 6 accords with the simulation results and performance comparisons with respect to RMSE, computational complexity analysis, probability of resolution, etc. The paper is completed by the conclusion in Section 7.

The main aim of the DOA estimation methodology is to determine a function that provides an implication of the impinging DOA on an array of sensors based on a plot between the maxima and angular values. This functional plot is known as the pseudospectrum P(θ) is dependent on the angle of arrival. One of the best approaches to defining pseudospectrum is by minimizing the mean-squared error of the correlation matrix, formed by the array output, by eigenvalue decomposition. This gives rise to the MUSIC algorithm, developed by Schmidt [9]. It is a subspace-based method, as it specifically exploits the noise subspace. This approach has been found to work effectively for uncorrelated impinging signals as well as noise and has established itself as one of the most popular solutions to the DOA estimation problem. However, for highly correlated impinging signals, MUSIC fails to separate noise as the source correlation matrix becomes singular, and the estimation method breaks down considerably. Besides, due to eigenvalue decomposition, the computational complexity is high and becomes exorbitant for a large sensor or antenna array.

This weakness of the MUSIC algorithm is tackled and substantially reduced in the M-MUSIC technique by exploiting the Nyström method to approximate the noise subspace [43]. A computationally more efficient variant of MUSIC, M-MUSIC is founded on the randomly selected sensor array output. The eigenvalue decomposition of the array correlation matrix is obtained by the Nyström approximation for reconstructing the corresponding noise subspace. Finally, the modified pseudospectrum function is modeled, from which the DOAs are estimated. Consequently, the computational efficiency of M-MUSIC is well established due to the use of smaller array correlation matrices (for randomly chosen small arrays), which reduces eigenvalue decomposition complexities. However, if the original sensor array itself is small-scale, the computational efficiency of M-MUSIC reduces.

2.png

Figure 2. N- element array with impinging signals. Example of spatial filtering

Figure 2 shows a spatial filter with an antenna array of N isotropic components with N prospective weights. D narrowband signals (SOI and jamming signals), originating from the far-field region, impinge on the array from D directions as shown. It is assumed that D<N. Each received signal x_i(k) consists of additive white Gaussian noise (AWGN) with a zero mean. The array output at the k-th snapshot can be written as:

$y(k)=\boldsymbol{w}^T . \boldsymbol{x}(k)$          (1)

where, w=[w₁ w₂ w₃……w_N]^T is the array weight vector.

$\boldsymbol{x}(k)=\left[\boldsymbol{a}\left(\theta_1\right) \boldsymbol{a}\left(\theta_2\right) \boldsymbol{a}\left(\theta_3\right) \ldots .. \boldsymbol{a}\left(\theta_D\right)\right] .\left[\begin{array}{c}s_1(k) \\ s_2(k) \\ \vdots \\ s_D(k)\end{array}\right]+\boldsymbol{n}(k) \Rightarrow \boldsymbol{x}(k)=\boldsymbol{A}(\theta) . \boldsymbol{s}(k)+\boldsymbol{n}(k)$               (2)

where, s(k) is the impinging narrowband signal vector, n(k) is the zero mean, and $\sigma_n^2$ variance AWGN vector, a(θ_i) is the N-element array steering vector for DOA θ_i, and A(θ)=[a(θ₁) a(θ₂) a(θ₃) …a(θ_D)]_N×D is the array manifold matrix of the steering vectors a(θ_i).

For the conventional MUSIC algorithm [9, 14], the correlation matrix is formed by using Eq. (2) as:

$\boldsymbol{R}_{x x}=\boldsymbol{A}(\theta) . \boldsymbol{R}_{s s} . \boldsymbol{A}(\theta)^H+\boldsymbol{R}_{n n}$            (3)

where, $\boldsymbol{R}_{x x}=E\left[\boldsymbol{x}(k) . \boldsymbol{x}(k)^H\right]$ is the N×N array correlation matrix, $\boldsymbol{R}_{s s}=E\left[\boldsymbol{s}(k). \boldsymbol{s}(k)^H\right]$ is the D×D source correlation matrix and $\boldsymbol{R}_{n n}=\sigma_n^2 \boldsymbol{I}$ is the N×N noise correlation matrix (I is the identity matrix). Due to the fact that MUSIC is a subspace-based approach, it estimates the signal and noise subspaces (E_S and E_N respectively) by eigenvalue decomposition of $\widehat{\boldsymbol{R}}_{x x}$.

The MUSIC pseudospectrum is given by [9, 42] as:

$P(\theta)=\frac{1}{\left|\boldsymbol{a}(\theta)^H \boldsymbol{E}_N \boldsymbol{E}_N^H \boldsymbol{a}(\theta)\right|}$               (4)

The plot of Eq. (4) generates sharp peaks at the angles of arrival of the impinging signals (D).

The angle estimation of Eq. (4) is recognized to achieve a splendid balance between the complexity of computation and the performance of DOA estimation. Subsequently, the MUSIC algorithm has remained a generally accepted method in practice as well as in the literature as a standard proposition [9-10, 43]. But the MUSIC algorithm is very sensitive to near-correlated targets and model mismatches. Nevertheless, the MUSIC algorithm has stimulated the array processing research community to seek better performance in terms of locating correlated targets with reduced computational complexity.

In M-MUSIC algorithm [43], the Nyström approximation is used to construct a new array correlation matrix $\boldsymbol{R}_{x x N^{\prime}}$ by selecting N' elements at random from N for N'<N. The new correlation matrix is designed as:

$\boldsymbol{R}_{x x N^{\prime}}=E\left[\boldsymbol{x}(k) . \boldsymbol{y}(k)^H\right]=\boldsymbol{A}(\theta) . \boldsymbol{R}_{s s} . \boldsymbol{A}_y(\theta)^H+\sigma^2 \boldsymbol{I}_{N \times N^{\prime}}$              (5)

where, $\boldsymbol{y}(k)=\left[\begin{array}{lllll}y_1 & y_2 & \ldots & \ldots & y_{N^{\prime}}\end{array}\right]$ is the output vector of the randomly chosen N' array elements from x(k), A_y(θ) is the direction matrix or manifold matrix of the observation vector y(k) and $\boldsymbol{I}_{N \times N^{\prime}}$ is the N×N' dimensional matrix, in which the diagonal elements are one and other elements are zero.

Applying Nyström approximation to estimating the noise subspace, $E_{N^{\prime}}$, the eigenvalue decomposition yields: $\boldsymbol{R}_{x x N^{\prime}} . \boldsymbol{e}_y=\lambda_y . \boldsymbol{e}_n$, where e_n is the approximate principle eigenvector. Following this, the process continues using an approach akin to the MUSIC algorithm in estimating the DOA angle θ.

Eventually, the spectral function in Eq. (4) can be reformed as:

$P(\theta)=\frac{1}{\left|\boldsymbol{a}(\theta)^H\left(\boldsymbol{I}_N-\boldsymbol{E}_{N^{\prime}} \boldsymbol{E}_{N^{\prime}}^H\right) \boldsymbol{a}(\theta)\right|}$             (6)

The DOA of the far-field targets is estimated from the angles related to the plot of D spectral peaks in Eq. (6).

5.1 Signal model of LSCM algorithm

4.png

Figure 4. Blind adaptive beamforming method. Example: CMA and its variants like LSCMA and MT-LSCMA

Figure 4 shows a schematic of an adaptive weight updating beamformer with a blind algorithm that generates the error function without the requirement of the desired signal. Blind adaptive spatial filtering is the mathematical basis for the family of CMA and the LSCMA [14, 42]. The cost function of LSCMA using non-linear Gauss’s method is given as [14, 23, 24]:

$J(\boldsymbol{w})=\sum_{k=1}^K|| y(k)|-| \alpha||^2$            (11)

where, α is the array output amplitude. A cost function is an optimizing criterion or performance surface defined based on requirements. In the family of CMA, the minimization of the weighted-sum-of-error-squares criterion is adhered to as an optimizing requirement [23]. The J(w) in Eq. (11) is also known as the dispersion function. This dispersion function (or cost function) can be minimized iteratively by associating the slope of the function with zero. Assuming the narrowband impinging signals (SOI and jammers) are of constant amplitude, Eq. (11) can be re-written as:

$J(\boldsymbol{w})=\sum_{k=1}^K|| y(k)|-1|^2=\left.|| g_k(\boldsymbol{w})\right||_2 ^2=\left.|| g(\boldsymbol{w})\right||^2$                 (12)

where, |g_k(w)| is the non-linear function of the k^th signal and $|.|_2$ is the $\ell_2$ norm. In vector form, |g_k(w)| can be expressed as $\boldsymbol{g}(\boldsymbol{w})=\left[g_1(w) g_2(w) \ldots \ldots g_K(w)\right]^T$. Using the offset vector δ, the cost function of Eq. (11) can be expanded by Taylor series as $J(\boldsymbol{w}+\boldsymbol{\delta})=\left\|g(\boldsymbol{w})+\boldsymbol{D}^H(\boldsymbol{w}) \boldsymbol{\delta}\right\|_2^2$, where $\boldsymbol{D}(\boldsymbol{w})=\boldsymbol{\nabla} g_k(\boldsymbol{w})=\left[\nabla\left(g_1(\boldsymbol{w})\right) \nabla\left(g_2(\boldsymbol{w})\right) \ldots \ldots \nabla\left(g_k(\boldsymbol{w})\right]\right.$. Taking the gradient of the expanded cost function and equating it to zero (to achieve the global minima of the error surface), we get:

$\boldsymbol{\delta}=-\left[\boldsymbol{D}(\boldsymbol{w}) \boldsymbol{D}^H(\boldsymbol{w})\right]^{-1} . \boldsymbol{D}(\boldsymbol{w}) \boldsymbol{g}(\boldsymbol{w})$             (13)

If n denotes the iteration number, then the updated weight vector w(n+1) can be generated from Eq. (9) as:

$\begin{gathered}\boldsymbol{w}(n+1)=\boldsymbol{w}(n)-\boldsymbol{D}(\boldsymbol{w}(n)) \boldsymbol{D}^H(\boldsymbol{w}(n))^{-1} \boldsymbol{D}(\boldsymbol{w}(n)) \boldsymbol{g}(\boldsymbol{w}(n)) \\ \Rightarrow \boldsymbol{w}(n)-\left(\boldsymbol{Z} \boldsymbol{Z}^H\right)^{-1} \boldsymbol{Z} \boldsymbol{Z}^H \boldsymbol{w}(n)-\left(\boldsymbol{Z} \boldsymbol{Z}^H\right)^{-1} \boldsymbol{Z} r^*(n) \\ \Rightarrow\left(\boldsymbol{Z Z}^H\right)^{-1} \boldsymbol{Z} r^*(n)\end{gathered}$              (14)

where,

$\boldsymbol{Z}=[z(1) z(2) \ldots \ldots z(K)]^T$              (15)

and r(n) indicates a hard limiter for the operation of y, such that:

$y(n)=\left[\boldsymbol{w}(n)^H \boldsymbol{Z}\right]^T$          (16)

and

$\boldsymbol{r}(n)=\left[\frac{y(1)}{|y(1)|} \frac{y(2)}{|y(2)|} \ldots \ldots \frac{y(K)}{|y(K)|}\right]^T=\boldsymbol{L}(y)$              (17)

The LSCM algorithm is represented by the weight update Eq. (14) and Eqs. (15)-(17). It uses the data block Z(K), and iteration is performed within the block of data to estimate the updated weight vector w(n+1). From the new r(n+1) value, the output y is then calculated. The iteration continues until the weight vector converges.

5.2 MT-LSCM algorithm for location identification and beam-forming

One of the major drawbacks of conventional CMA in direction finding is that their convergence characteristics are highly dependent on the initial values of the weight vector [22, 23, 42]. Thus, there remains a high possibility of convergence at the local minima rather than the true minima point if the initial weight vectors are not chosen appropriately. Consequently, achieving proper minima and convergence criteria is highly sensitive to the initial values of the weight vector.

Also, as the number of array elements or sensors is higher than the number of sources to be detected, it gives rise to different independent beam-forming vectors for an identical output signal [22, 23]. Hence, it is not sufficient to necessitate the independence of the weight vector w. A solution, as proposed [44, 45], supplements the cost function with a term exhibiting independence, but in concurrence with a slower and unpredictable convergence rate [45].

To separate the jamming signal direction from the SOI, the proposed MT-LSCMA resolves two sets of adaptive weights, consequently to null-steer by producing nulls towards the interfering signals and beam-steer the main lobe in the direction of the SOI. The null-steering weights are optimized by MT-LSCMA by $\underset{\boldsymbol{w}}{\operatorname{argmin}}\left\|\boldsymbol{w}^* \widetilde{\boldsymbol{x}}(k)-\boldsymbol{s}(k)\right\|_F$.

From Eq. (1), beam-forming solutions are desirable for known values of either A(θ) or s(k). If A(θ) is known, then the weight vector is set as $\boldsymbol{w}^*=\boldsymbol{A}(\theta)^{\dagger}$, and thus s_i(k)=w^*.x(k). Now, considering that if s_i(k) is known by apriori, then we set w^*=s_i(k).x(k), with $\boldsymbol{A}(\theta)=\left(\boldsymbol{w}^*\right)^{\dagger}$, where ϯ denotes the Moore-Penrose pseudo inverse. As w^*.A(θ)=I attained in both the above cases, the achieved beam-forming exactly cancels all interference.

Now, in the presence of AWGN, the formulation of two sets of linear least-squares minimization models can be put forward. The problems can be either based on minimizing the output error, as:

$\begin{aligned} & \qquad \min _{\boldsymbol{w}, \boldsymbol{s}}\left\|\boldsymbol{w}^* \boldsymbol{x}(k)-\boldsymbol{s}(k)\right\|_F^2 \\ & { such \,\, that \,\, conditions \,\, on }(\boldsymbol{w}, \boldsymbol{s}(k))\end{aligned}$            (18)

or based on minimizing the modelling error, as:

$\begin{gathered}\min _{\boldsymbol{A}, \boldsymbol{s}}\|\boldsymbol{x}(k)-\boldsymbol{A}(\theta) \boldsymbol{s}(k)\|_F^2 \\  { \,\,such \,\,that \,\,conditions \,\,on }(\boldsymbol{A}(\theta), \boldsymbol{s}(k))\end{gathered}$              (19)

For the blind beamforming problem, the signature of the impinging signal is |s_ij|=1, the constant modulus (CM) condition. Now considering instantanous solution for x(k) in Eq. (2), $\widetilde{\boldsymbol{x}}(k)=\boldsymbol{A}(\theta) . \boldsymbol{s}(k)+\boldsymbol{n}(k)$, the minimization solution of Eq. (15) can be formulated as (if s(k) is known):

$\widehat{\boldsymbol{A}}(\theta)=\underset{\boldsymbol{A}}{\operatorname{argmin}}\|\widetilde{\boldsymbol{x}}(k)-\boldsymbol{A}(\theta) \boldsymbol{s}(k)\|_F^2=\widetilde{\boldsymbol{x}}(k) \boldsymbol{s}(k)^{\dagger}$               (20)

For known values of A(θ), s(k) can be estimated as:

$\boldsymbol{s}(k)=\underset{\boldsymbol{s}}{\operatorname{argmin}}\|\widehat{\boldsymbol{x}}(k)-\boldsymbol{A}(\theta) \boldsymbol{s}(k)\|_{\boldsymbol{F}}^2$               (21)

with the corresponding beam-forming weights as $\boldsymbol{w}=$ $\boldsymbol{A}(\theta)^{\dagger} . \boldsymbol{x}(k)$. Estimated $\widehat{\boldsymbol{A}}(\theta)$ tends to converge to actual values of $\boldsymbol{A}(\theta)$ for noise vector independent of the sources and having zero mean value. The optimization problem of Eq. (14) minimizes the difference of the output signals as:

$\boldsymbol{w}^*=\underset{\boldsymbol{w}}{\operatorname{argmin}}\left\|\boldsymbol{w}^* \widetilde{\boldsymbol{x}}(k)-\boldsymbol{s}(k)\right\|_F^2=\boldsymbol{s}(k) \widetilde{\boldsymbol{x}}(k)$                 (22)

Using the identity, $\tilde{\boldsymbol{x}}^{\dagger}=\tilde{\boldsymbol{x}}^*\left(\widetilde{\boldsymbol{x}} \tilde{\boldsymbol{x}}^*\right)^{-1}$, we get:

$\boldsymbol{w}^*=\frac{1}{n} \boldsymbol{s}(k) \widetilde{\boldsymbol{x}}(k)^*\left[\frac{1}{n} \widetilde{\boldsymbol{x}}(k) \widetilde{\boldsymbol{x}}(k)^*\right]^{-1}=\widetilde{\mathcal{R}}_{x x}^{-1} . \widetilde{\mathcal{R}}_{s x}^*$                 (23)

where, $\widetilde{\mathcal{R}}_{x x}=\frac{1}{n} \widetilde{\boldsymbol{x}}(k) \widetilde{\boldsymbol{x}}(k)^*$ is the array correlation matrix and $\widetilde{\mathcal{R}}_{s x}=\frac{1}{n} \boldsymbol{s}(k) \widetilde{\boldsymbol{x}}(k)^*$ source-array cross correlation matrix.

Hence,

$\boldsymbol{w} \approx \widetilde{\mathcal{R}}_{x x}^{-1} \cdot \boldsymbol{A}(\theta)$             (24)

Eq. (20) above provides the null-steering weight update solution for MT-LSCMA. With similar considerations, the beam-steering weight update result is derived in the literature [42] as:

$\boldsymbol{w} \approx \boldsymbol{X}^{\dagger}(\boldsymbol{\theta}) . \boldsymbol{S}(\theta)$               (25)

Using Eqs. (22), (23) and (24), the block iterative optimization for null-steering capability of MT-LSCMA can be formulated as:

$\begin{gathered}S(k)^{\prime}=A(k)^{\dagger} \tilde{X} \\ S(k+1)=\boldsymbol{r}_n\left(S(k)^{\prime}\right) \\ A(k+1)=\tilde{X} S(k+1)^{\dagger}\end{gathered}$               (26)

which can be simplified to,

$\begin{gathered}S(k)^{\prime}=w(k)^* \tilde{X} \\ S(k+1)=\boldsymbol{r}_n\left(S(k)^{\prime}\right) \\ w(k+1)^*=s(k+1) \tilde{X}^{\dagger}\end{gathered}$             (27)

with suitable initial values of A(0) and w(0), Eqs. (24), (26) and (27), converge to Wiener solution with descent computational complexity.

In most practical cases, the total number of jammers or interferers to be estimated is less than the number of sensors in the array (N). This leads to a rank deficiency in X=AS. Hence, the beam-forming solution would not be unique. To ensure that the solution would lie in the column span of A, a dimension-reducing prefiltering is performed such that span (F) =span (A), where F is any N×D matrix. Then in the column span of A, all the beam-forming matrices are given by W=FT_D×D, where T_D×D  is a non-singular square matrix for linearly independent beamformers. The noisy pre-filtered data matrix is given by:

$\underline{\tilde{X}}=F^* \tilde{X}$             (28)

where, $\underline{\tilde{X}}=\underline{A} S+\underline{N}, \underline{A}=F^* A$, and $\underline{N}=F^* N$. The underscored terms denote the pre-filtered variables. Figure 5 depicts a schematic of the MT-LSCMA pre-filtering structure. Prefiltering is an extension of collaborative filtering, which aims at dimensional reduction. The first block, $R_x^{-1}$ performs the whitening of the input coloured noise, and then carry on with the process for the white noise instance. The dimensionality reduction is generally realized by projecting the predictor onto a low-dimensional subspace. The subspace estimator, together with the subspace filter blocks, accomplishes the projection. Finally, blind signal separation with interference is carried out by the proposed method. Commonly, the subspace filter and estimator represent the prefiltering stage.

5.png

Figure 5. MT-LSCMA beamforming pre-filtering structure

Table 1 provides a summary of the proposed null-steering based on MT-LSCMA method, as below:

Table 1. Proposed method of jammer direction identification

A new blind adaptive beamforming method is suggested in this paper. It guesses the direction and null-steers towards intentional interferers or jammers in the far-field region. The proposed MT-LSCMA is more noteworthy than its earlier family variants (CMA and others) as it is a block iterative adaptation method and resolves two sets of weights: one for beamforming towards the SOI and the other for null-steering towards the signal-not-of-interest (SNOI) or jammer directions. The null-steering is achieved by optimizing the CMA least squares equation, and the weight update solution is determined by pseudo-inverting the N×N array correlation matrix. Comparing with conventional and popular DOA estimation methods, including the CS paradigm, the proposed method shows more promising results in terms of RMSE, probability of resolution, detection probability, and failure rate, even at lower SNR levels. Higher computational complexity as compared to CS methods is unavoidable due to the requirement of inverting the array correlation matrix. DOA estimation in a CS framework has some clear advantages, such as increased DOFs, single snapshot instances, lower computational complexity, etc. But at lower SNR values, resolution becomes a predominant issue. Simulation results clearly depict that the proposed method works well in terms of probability of resolution, detection probability, failure rate, and RMSE at reduced SNR values as well.

However, all the methods discussed in this paper (proposed MT-LSCMA, conventional MUSIC, M-MUSIC, and CS framework-based) tend to depend highly on the number of array elements while measuring the complexity of computation. The deep learning approach for DOA estimation seems to reduce computational complexity (as the algorithm performance is independent of the physical array structure and array aperture length), which is much lower than the CS paradigm methods but shows inferior resolution at low SNR levels.

The proposed method would find applicability in modern battlefield scenarios, where the noise power is very high, in identifying intentional interferers or jammers in the far-field region with moderate computational complexity. Further, the proposed method can be extended to estimate two-dimensional (2D) DOA estimations of jammer locations comprising both elevation and azimuthal angles. A two-dimensional or planar array would be required for precise estimation of the angle of arrival [31-33].