Robust broadband adaptive beamforming for planar arrays with tunable nulls in high-dynamic scenario

Abstract

Traditional space-time adaptive processing (STAP) is extensively adopted in various fields such as navigation satellite, sonar, radar, etc. It can form the sharp null in the angle domain (azimuth and elevation) to suppress interferences. However, the null cannot continuously match the interference in dynamic interference scenarios. To address the problem, a STAP strategy based on Simpson-statistical constraint for the planar array is proposed, capable of generating wide nulls with flexible width and asymmetry. Firstly, a taper matrix (TM) is calculated, which can achieve asymmetric widening of the null in the angular domain. Asymmetry is achieved by introducing an artificial interference group that satisfies the Simpson-statistical constraint. Then, the eigen-covariance matrix (ECM) is obtained by the eigen-decomposition of the sample covariance matrix (CM). The unequal null width is generated by reconstructing the array CM of the planar array based on the TM and ECM. Finally, the computation of weight is carried out using a linear phase undistorted beamformer. Extensive testing demonstrates the robustness of the proposed solution.

Introduction

Space Time Adaptive Processing (STAP) is an efficacious interference suppression technology which can not only alleviate broadband (or narrowband) interference in both spatial and temporal domains, but also enhance signal reception gain^1,2. It is widely used in navigation satellite, radar, sonar, and other fields^3,4. However, owing to dynamic jammer or platform motion, rapid angular displacement may occur within a short time interval. This dynamic behavior challenges traditional broadband STAP beamforming techniques, where the inherent computational latency in weight vector adaptation causes the null to not continuously match the interference⁵. Consequently, the null cannot promptly follow dynamic interference, leading to degradation of the output signal-to-interference-plus-noise ratio (SINR).

Several suppression strategies have emerged to solve the challenge of the rapid interference movement. The derivative constraint strategy for sidelobe is proposed to widen the anti-interference null in⁶, but it is only applicable to narrowband processing. A strategy for widening the null by imposing a special constraint on the integrated power is proposed⁷. And a convex optimization strategy is proposed to achieve null widening in broadband beamforming⁸. However, in all three strategies, the prior knowledge of the interference direction-of-arrival (DOA) is indispensable which is difficult to obtain in practical scenarios. Recently, the robust broadband beamforming strategies without prior knowledge have been proposed to adaptively widen the null. These strategies are summarized as covariance matrix taper (CMT) technique^9,10,11,12. Nevertheless, the strategies in^9,10,11 are only suitable for linear arrays (LA). Compared to the linear array, the planar array (PA) provides a distinct advantage in signal receiving systems by enabling simultaneous azimuth and elevation measurements. Therefore, robust broadband beamforming of the PA for the rapidly moving interference is becoming increasingly critical. A CMT-based strategy suitable for PA is proposed¹², which is an improvement on the method in⁹. This strategy loses anti-interference degrees of freedom (DOFs) due to symmetrical and equal-width widening nulls. DOF is a momentous resource for beamforming. In most cases, each interference source typically has unique non-stationary characteristics, and a symmetric and equal-width widening strategy is not the optimal solution¹³. To solve this problem, an asymmetric null widening method is proposed in¹⁴. Nevertheless, it is only applicable to scenarios that the interference with movement in the same direction and only derives solutions in discrete form. Furthermore, although we previously established a Simpson-statistical constraint for asymmetric null widening in one-dimensional (1D) LA¹⁵, extending this framework to two-dimensional (2D) PA is not a simple mathematical formality. It requires addressing the complex coupling between azimuth and elevation dynamics. In the 2D PA case, interference dynamics create a volumetric uncertainty region rather than a linear interval. A direct Kronecker product of 1D taper often fails to accurately model the specific asymmetric shapes required in the joint $\left( {\theta ,\varphi } \right)$ domain. The ‘curse of dimensionality’ in PA also demands a more computationally efficient closed-form solution to handle the expanded covariance matrix (CM), a challenge that was not fully addressed in¹⁴.

In this manuscript, a null widening strategy based on Simpson-statistical constraint for the PA is designed, capable of generating wide nulls with tunable width and asymmetry. The innovative aspect and key contribution of this manuscript can be outlined as follows.

• 1. A closed-form 2D Taper Matrix (TM) based on the Simpson-statistical constraint is derived, which enables the flexible construction of wide nulls with independent control over azimuth and elevation widths.

• 2. The artificial interference groups are configured to be independent and asymmetric around each source, and the width and asymmetry of the null can be adjusted. Furthermore, the asymmetry is achieved by introducing an artificial interference group that satisfies the Simpson-statistical constraint. The tunability of the null width is achieved by the eigenvalue decomposition of the sample CM and the CM reconstruction of the PA.

• 3. A rigorous mathematical framework is provided to handle the cross-dimensional coupling in PA, ensuring that the reconstructed CM effectively matches the non-stationarity of interferences.

• 4. A detailed analysis of the computational complexity is conducted, demonstrating that the proposed strategy has effective scalability for 2D arrays without the iterative overhead of convex optimization.

The organization of this manuscript is outlined as follows. The section Ⅱ unveils the STAP receiving model for planar arrays, and the proposed strategy is systematically presented in the section Ⅲ. In the section Ⅳ, strategy performance is assessed through multiple simulated experiments. The fifth section is the conclusion.

The receiving model based on STAP for planar array

As shown in Fig. 1, a PA of N×M antenna elements is considered, where all elements lie on the y-z plane. The DOA of the signal received by PA is characterized by the elevation $\theta$ and the azimuth $\varphi$. One target signal and P broadband interferences are received by the array from directions $\left( {{\theta _p},{\varphi _p}} \right),p=0,1,2, \ldots P$. $p=0$ represents the target signal. The signal model at the $\left( {m,n} \right){\text{-th}}$ element is

$${x_{m,n}}\left( t \right)=\sum\limits_{{p=0}}^{P} {{s_p}\left( {t - {\tau _{m,n}}\left( {{\theta _p},{\varphi _p}} \right)} \right)} +{n_{m,n}}\left( t \right),$$

(1)

where ${x_{m,n}}\left( t \right)$ is the received signal at the $\left( {m,n} \right){\text{-th}}$ element, ${s_p}(\cdot)$ is the target ($p=0$) or the $p{\text{-th}}$ ($p=1,2, \ldots ,P$) interference, ${n_{m,n}}\left( t \right)$ represents the Gaussian white noise at the $\left( {m,n} \right){\text{-th}}$ element. ${\tau _{m,n}}\left( {{\theta _p},{\varphi _p}} \right)$ is the time delay of the $p{\text{-th}}$ signal at the $\left( {m,n} \right){\text{-th}}$ element relative to the reference-element.

The CM of the received signal can be given by.

$${{\varvec{R}}_{xx}}=E\left\{ {{\varvec{x}}\left( t \right){{\varvec{x}}^H}\left( t \right)} \right\},$$

(2)

where.

$$\begin{gathered} {\varvec{x}}\left( t \right)=[{x_{1,1}}\left( t \right), \ldots ,{x_{M,1}}\left( t \right),{x_{1,2}}\left( t \right), \ldots ,{x_{M,2}}\left( t \right), \ldots ,{x_{M,N}}\left( t \right), \hfill \\ {\text{ }}{x_{1,1}}\left( {t - {T_s}} \right), \ldots ,{x_{M,1}}\left( {t - {T_s}} \right),{x_{1,2}}\left( {t - {T_s}} \right), \ldots ,{x_{M,2}}\left( {t - {T_s}} \right), \ldots ,{x_{M,N}}\left( {t - {T_s}} \right), \hfill \\ {\text{ }}{x_{1,1}}\left( {t - 2{T_s}} \right), \ldots ,{x_{M,N}}\left( {t - 2{T_s}} \right), \ldots ,{x_{M,N}}\left( {t - \left( {D - 1} \right){T_s}} \right){]^T}, \hfill \\ \end{gathered}$$

where D is the amount of filter taps connected in series for apiece element, and the structure as shown in Fig. 2. ${T_s}$ is the sampling interval, ${\left( \cdot \right)^H}$ and ${\left( \cdot \right)^T}$ are respectively conjugate transpose and transpose.

Fig. 1

The element layout diagram of planar array.

Fig. 2

The element layout diagram of planar array.

The frequency-domain representation of ${\varvec{x}}\left( t \right)$ is

$$\begin{gathered} {\varvec{x}}\left( {{f_l}} \right)={\varvec{A}}\left( {{f_l}} \right) \cdot {\varvec{S}}\left( {{f_l}} \right)+{\varvec{N}}\left( {{f_l}} \right),{\text{ }}l=1,2, \ldots L \hfill \\ {\varvec{A}}\left( {{f_l}} \right)=\left[ {{\varvec{a}}\left( {{f_l},{\theta _0},{\varphi _0}} \right),{\varvec{a}}\left( {{f_l},{\theta _1},{\varphi _1}} \right),...,{\varvec{a}}\left( {{f_l},{\theta _p},{\varphi _p}} \right)} \right], \hfill \\ \end{gathered}$$

(3)

where ${\varvec{S}}\left( {{f_l}} \right)={\left[ {{s_0}\left( {{f_l}} \right),{s_1}\left( {{f_l}} \right),...,{s_P}\left( {{f_l}} \right)} \right]^T}$ is the complex envelope of the target and interferences in the frequency domain, which is obtained by applying the Fourier Transform (FT), ${\varvec{N}}\left( {{f_l}} \right)$ is the FT of the Gaussian white noise, ${\varvec{a}}\left( {{f_l},{\theta _p},{\varphi _p}} \right)$ is the space-time steering vector, L is the amount of discrete frequency bins.

$${\varvec{a}}\left( {{f_l},{\theta _p},{\varphi _p}} \right)={{\varvec{a}}_t}\left( {{f_l}} \right) \otimes {{\varvec{a}}_\tau }\left( {{\theta _p},{\varphi _p}} \right),$$

(4)

where.

$${{\varvec{a}}_t}\left( {{f_l}} \right)={\left[ {1,{e^{ - j2\pi {f_l}{T_s}}}, \ldots ,{e^{ - j2\pi {f_l}\left( {D - 1} \right){T_s}}}} \right]^T},$$

$${{\varvec{a}}_\tau }\left( {{\theta _p},{\varphi _p}} \right)={[1,{e^{ - j\frac{{2\pi {f_l}}}{c}{d_y}{u_p}}}, \cdots ,{e^{ - j\frac{{2\pi {f_l}}}{c}(M - 1){d_y}{u_p}}},{e^{ - j\frac{{2\pi {f_l}}}{c}{d_z}{v_p}}}, \cdots ,{e^{ - j\frac{{2\pi {f_l}}}{c}((M - 1){d_y}{u_p}+(N - 1){d_z}{v_p})}}]^T},$$

where $\otimes$ and c represent the kronecker product and the speed of electromagnetic propagation, ${d_y}$ and ${d_z}$ are the interval between elements along the horizontal and vertical axes of the antenna array. ${u_p}=\cos {\theta _p}\sin {\varphi _p}$, ${\nu _p}=\sin {\theta _p}$. ${f_0}$ is the center-frequency of the received signal. ${{\varvec{a}}_\tau }$ and ${{\varvec{a}}_t}$ are the spatial steering vector and the temporal steering vector, respectively.

The response of the array system can be written as.

$$H\left( {{f_l},\theta ,\varphi } \right)=\sum\limits_{{m=1}}^{M} {\sum\limits_{{n=1}}^{N} {\sum\limits_{{d=1}}^{D} {w_{{m,n,d}}^{ * }{e^{\frac{{ - j2\pi {f_l}}}{c}((m - 1){d_y}u+(n - 1){d_z}v)}}{e^{ - j2\pi {f_l}(d - 1){T_s}}}} } } ,$$

(5)

where $\theta ,\varphi \in {\Theta _s}$, ${f_l} \in {\Theta _f}$, ${\Theta _s}$ and ${\Theta _f}$ represent the entire angular domain of the spatial region and the in-band frequency range. $u=\cos \theta \sin \varphi$, $v=\sin \theta$. $\left( \cdot \right)_{{}}^{ * }$ represents conjugate operation. $w_{{m,n,d}}^{{}}$ is the weight at the $\left( {m,n} \right){\text{-th}}$ element and the $d{\text{-th}}$ tap. Equation (5) can be further written as

$$H\left( {{f_l},\theta ,\varphi } \right)=\sum\limits_{{l=1}}^{L} {{{\varvec{w}}^H}{\varvec{a}}\left( {{f_l},\theta ,\varphi } \right)} ,$$

(6)

where ${\varvec{w}}$ is the weight.

Proposed beamforming strategy

Rapidly-moving interference is a hazardous interference pattern that significantly degrades the performance of communication systems. The conventional broadband beamformer is limited to generating sharp nulls that suppress static interference, which is ineffective for interference with high angular mobility. In addition, the existing null widening strategy in broadband beamforming sacrifices the DOF due to the symmetry widening of the null. To solve the rapidly-moving interference received by the PA, a beamforming method based on Simpson statistical prior for the PA is proposed, capable of generating wide nulls with flexible width and asymmetry. Firstly, a TM is calculated, which can achieve asymmetric widening of the null in the angular domain. Then, the unequal null width is effected by reconstructing the array CM of PA. Finally, the weight is computed by the linear phase undistorted beamformer.

Taper matrix construction for planar array

According to Eq. (3), the CM of the PA in the frequency domain is represented as.

$${\varvec{R}_{xx}} = \sum\limits_{l = 1}^L {\sum\limits_{p = 0}^P {{Q_p} \varvec{a}\left( {{f_l},{\theta _p},{\varphi _p}} \right){ \varvec{a}^H}\left( {{f_l},{\theta _p},{\varphi _p}} \right)} + \sum\limits_{l = 1}^L {\sigma _n^2\varvec{I}} } ,$$

(7)

where ${Q_p}$ is the power, and $\sigma _{n}^{2}$ is the noise power. ${\varvec{I}}$ is an identity matrix of $MND \times MND$ dimension. Moreover, ${{\varvec{R}}_{xx}}$ is transformed into

$${\varvec{R}_{xx}} = \varvec{B}{\varvec{B}^H} + \sigma _n^2 \varvec{I},$$

(8)

where.

$${\varvec{B}}=\left[ {\begin{array}{*{20}{c}} {\sqrt {{Q_1}} }& \cdots &{\sqrt {{Q_P}} } \\ {\sqrt {{Q_1}} {e^{ - j{{\phi ^{\prime}}_1}}}}& \cdots &{\sqrt {{Q_P}} {e^{ - j{{\phi ^{\prime}}_P}}}} \\ \vdots & \ddots & \vdots \\ {\sqrt {{Q_1}} {e^{ - j\left( {M - 1} \right){{\phi ^{\prime}}_1}}}}& \cdots &{\sqrt {{Q_P}} {e^{ - j\left( {M - 1} \right){{\phi ^{\prime}}_P}}}} \\ {\sqrt {{Q_1}} {e^{ - j{{\phi ^{\prime\prime}}_1}}}}& \cdots &{\sqrt {{Q_P}} {e^{ - j{{\phi ^{\prime\prime}}_P}}}} \\ \vdots & \ddots & \vdots \\ {\sqrt {{Q_1}} {e^{ - j(M - 1){{\phi ^{\prime}}_1} - j(N - 1){{\phi ^{\prime\prime}}_1}}}}& \cdots &{\sqrt {{Q_P}} {e^{ - j(M - 1){{\phi ^{\prime}}_P} - j(N - 1){{\phi ^{\prime\prime}}_P}}}} \\ \vdots & \ddots & \vdots \\ {\sqrt {{Q_1}} {e^{ - j(D - 1)\xi }}}& \cdots &{\sqrt {{Q_P}} {e^{ - j(D - 1)\xi }}} \\ \vdots & \ddots & \vdots \\ {\sqrt {{Q_1}} {e^{ - j(D - 1)\xi - j(M - 1){{\phi ^{\prime}}_1} - j(N - 1){{\phi ^{\prime\prime}}_1}}}}& \cdots &{\sqrt {{Q_P}} {e^{ - j(D - 1)\xi - j(M - 1){{\phi ^{\prime}}_P} - j(N - 1){{\phi ^{\prime\prime}}_P}}}} \end{array}} \right],$$

$$\begin{gathered} \phi _{p}^{\prime }=2\pi {f_l}{d_y}{u_p}/c,\phi _{p}^{{\prime \prime }}=2\pi {f_l}{d_z}{v_p}/c,\xi =2\pi {f_l}{T_s}, \hfill \\ 1 \leqslant p \leqslant P,1 \leqslant l \leqslant L. \hfill \\ \end{gathered}$$

The element of ${{\varvec{R}}_{xx}}$ can be formulated as

$${\left( {{\varvec{R}_{xx}}} \right)_{h,g}} = \sigma _n^2{I_{h,g}} + \mathop \sum \limits_{p = 1}^P {Q_p}{e^{j2\pi {f_l}{u_p}{d_y}{N_1}/c}}{e^{j2\pi {f_l}{v_p}{d_z}{N_2}/c}}{e^{j2\pi {f_l}{T_s}{N_3}}},$$

(9)

where.

$$\begin{gathered} {N_1}={\text{rem}}\left( {g - 1,M} \right) - {\text{rem}}\left( {h - 1,M} \right), \hfill \\ {N_2}={\text{int}}\left( {{\text{rem}}\left( {g - 1,MN} \right),M} \right) - {\text{int}}\left( {{\text{rem}}\left( {h - 1,MN} \right),M} \right), \hfill \\ {N_3}={\text{int}}\left( {g - 1,MN} \right) - {\text{int}}\left( {h - 1,MN} \right), \hfill \\ 1 \leqslant h,g \leqslant M \times N \times D, \hfill \\ \end{gathered}$$

${\text{rem}}\left( {a,b} \right)$ denotes the remainder when a is divided by b, $\operatorname{int} \left( {a,b} \right)$ denotes the integer quotient of a divided by b. h and g represent rows and columns, respectively.

To effectively mitigate rapidly-moving interference, the artificial interference group is added around the initial direction of interference to flexibly widen the null. Given that the PA is two-dimensional unlike the linear counterpart, artificial interference group must be deployed along the azimuth u and elevation v.

Fig. 3

The probability density of Simpson-distribution.

To effectively mitigate rapidly-moving interference, the artificial interference group is added around the initial direction of interference to flexibly widen the null. In previous work, a null widening strategy based on the Simpson-statistical constraint for LA¹⁵ is proposed. While that method effectively handles 1D high-dynamic interference, it cannot be directly applied to PA due to the decoupling limitation. Unlike LA, the PA involve simultaneous constraints on the artificial interference group in both azimuth and elevation domains. The extension to 2D is not merely a mathematical formality; it requires addressing the cross-dimensional coupling problem where interference dynamics create a volumetric uncertainty region rather than a linear interval. A direct Kronecker product of 1D tapers often fails to accurately model the specific asymmetric shapes required in the joint $\left( {\theta ,\varphi } \right)$ domain. Therefore, a new derivation of the 2D TM that maps the tensor product of array dimensions to the 2D spatial domain is essential. The artificial interference group must be deployed along the azimuth u and elevation v, which can solve the coupling problem between angles $\theta$ and $\varphi$.

To obtain tunable wide nulls, the perturbation sector of interference along the azimuth u is limited to $- {{\text{I}}_1}$ to ${{\text{I}}_2}$, where $- {{\text{I}}_1} \ne {{\text{I}}_2}$. Similarly, along the direction v, the perturbation sector is limited to $- {{\text{I}}_3}$ to ${{\text{I}}_4}$, where $- {{\text{I}}_3} \ne {{\text{I}}_4}$. In practical engineering applications, the determination of the widening parameters (${{\text{I}}_1}$, ${{\text{I}}_2}$, ${{\text{I}}_3}$ and ${{\text{I}}_4}$) depends on the dynamic characteristics of the interference and the system update rate. The dynamic characteristics of the interference can be estimated by source matching and tracking algorithms^16,17. Postulating that the perturbation ${{{\Delta}}}{u_q}$ of the p-th interference along the direction ${u_q}$ follows the Simpson-statistical in the range $\left[ { - {{\text{I}}_1},{{\text{I}}_2}} \right]$, the probability density function is the

$$f\left( {{{{\Delta}}}{u_p}} \right)=\left\{ {\begin{array}{*{20}{c}} {\frac{{2\left({{{{\Delta}}}{u_p}+{{\text{I}}_1}} \right)}}{{{{\text{I}}_1}\left( {{{\text{I}}_2}+{{\text{I}}_1}} \right)}}, - {{\text{I}}_1} \leqslant {{{\Delta}}}{u_p} \leqslant 0} \\ {\frac{{2\left( {{{\text{I}}_2} - {{{\Delta}}}{u_p}} \right)}}{{{{\text{I}}_2}\left( {{{\text{I}}_2}+{{\text{I}}_1}} \right)}},0<{{{\Delta}}}{u_p} \leqslant {{\text{I}}_2},} \end{array}} \right.$$

(10)

where $- {{\text{I}}_1}$ is the lower limit, the mode is 0, ${{\text{I}}_2}$ is the upper limit, and the Eq. (10) is a continuous function. Setting the mode to 0 indicates that the space-location of the original-interference is taken as a fixed reference. Figure 3 is the probability density.

Obviously, the values of $- {{\text{I}}_1}$ and ${{\text{I}}_2}$ determine the sector of the widened null because the function describes the space-location of artificial interference groups. Thus, a tunable wide null can be created. Similarly, the perturbation ${{{\Delta}}}{v_q}$ of the p-th interference along the direction ${v_q}$ follows the Simpson-statistical in the range $\left[ { - {{\text{I}}_3},{{\text{I}}_4}} \right]$, the probability density function is

$$f\left( {{{{\Delta}}}{v_q}} \right)=\left\{ {\begin{array}{*{20}{c}} {\frac{{2\left( {{{{\Delta}}}{v_q}+{{\text{I}}_3}} \right)}}{{{{\text{I}}_3}\left( {{{\text{I}}_4}+{{\text{I}}_3}} \right)}}, - {{\text{I}}_3} \leqslant {{{\Delta}}}{v_q} \leqslant 0} \\ {\frac{{2\left( {{{\text{I}}_4} - {{{\Delta}}}{v_q}} \right)}}{{{{\text{I}}_4}\left( {{{\text{I}}_4}+{{\text{I}}_3}} \right)}},0<{{{\Delta}}}{v_q} \leqslant {{\text{I}}_4}.} \end{array}} \right.$$

(11)

Based on Eqs. (9)-(11), the CM with artificial interference groups can be derived, whose element is.

$$\begin{aligned} {\left( {\tilde {\varvec{R}}} \right)_{h,g}} &= \sigma _n^2{ {\varvec{I}}_{h,g}} + \sum\limits_{p = 0}^P {\int\limits_{ - {I_1}}^{{I_2}} {\int\limits_{ - {I_3}}^{{I_4}} {{Q_p} f\left( {\Delta {u_p}} \right){e^{j\frac{{2\pi {f_l}({u_p} + \Delta {u_p}){d_y}{N_1}}}{c}}}f\left( {\Delta {v_p}} \right){e^{j\frac{{2\pi {f_l}({v_p} + \Delta {v_p}){d_z}{N_2}}}{c}}}{e^{j2\pi {f_l}{T_s}{N_3}}}d\Delta {u_p}d\Delta {v_p}} } } \\ & = \sigma _n^2{I_{h,g}} + \sum\limits_{p = 0}^P {{Q_p}{e^{j2\pi {f_l}{T_s}{N_3}}}{e^{j\frac{{2\pi {f_l}{u_p}{d_y}{N_1}}}{c}}}{e^{j\frac{{2\pi {f_l}{v_p}{d_z}{N_2}}}{c}}}\int\limits_{ - {I_1}}^{{I_2}} {f\left( {\Delta {u_p}} \right){e^{j\frac{{2\pi {f_l}\Delta {u_p}{d_y}{N_1}}}{c}}}d\Delta {u_p}\int\limits_{ - {I_3}}^{{I_4}} {f\left( {\Delta {v_p}} \right){e^{j\frac{{2\pi {f_l}\Delta {v_p}{d_z}{N_2}}}{c}}}d\Delta {v_p}} } } \\ & = \sigma _n^2{I_{h,g}} + \sum\limits_{p = 0}^P {{Q_p}{e^{j2\pi {f_l}{T_s}{N_3}}}{e^{j\frac{{2\pi {f_l}{u_p}{d_y}{N_1}}}{c}}}{e^{j\frac{{2\pi {f_l}{v_p}{d_z}{N_2}}}{c}}}} \\ & {\text{ }} \cdot \frac{{2\left( {{I_1} - {I_1}{e^{j \cdot {I_2} \cdot \hat y}} + {I_2} - {I_2}{e^{ - j \cdot {I_1} \cdot \hat y}}} \right)}}{{{I_1} \cdot {I_2} \cdot \left( {{I_1} + {I_2}} \right) \cdot {{\hat y}^2}}}\frac{{2\left( {{I_3} - {I_3}{e^{j \cdot {I_4} \cdot \hat z}} + {I_4} - {I_4}{e^{ - j \cdot {I_3} \cdot \hat z}}} \right)}}{{{I_3} \cdot {I_4} \cdot \left( {{I_3} + {I_4}} \right) \cdot {{\hat z}^2}}} \\ & = \sigma _n^2{I_{h,g}} + \sum\limits_{p = 0}^P {{Q_p}{e^{j2\pi {f_l}{T_s}{N_3}}}{e^{j\frac{{2\pi {f_l}{u_p}{d_y}{N_1}}}{c}}}{e^{j\frac{{2\pi {f_l}{v_p}{d_z}{N_2}}}{c}}} \cdot {{\varvec{T}}_{h,g}}} , \end{aligned}$$

(12)

where ${\varvec{T}}$ is the TM for PA, and

$${{\varvec{T}}_{h,g}}=\frac{{2\left( {{{\text{I}}_{\text{1}}} - {{\text{I}}_{\text{1}}}{e^{j \cdot {{\text{I}}_2} \cdot \hat {y}}}+{{\text{I}}_2} - {{\text{I}}_2}{e^{ - j \cdot {{\text{I}}_1} \cdot \hat {y}}}} \right)}}{{{{\text{I}}_{\text{1}}} \cdot {{\text{I}}_2} \cdot \left( {{{\text{I}}_{\text{1}}}+{{\text{I}}_2}} \right) \cdot {{\hat {y}}^2}}}\frac{{2\left( {{{\text{I}}_3} - {{\text{I}}_3}{e^{j \cdot {{\text{I}}_4} \cdot \hat {z}}}+{{\text{I}}_4} - {{\text{I}}_4}{e^{ - j \cdot {{\text{I}}_3} \cdot \hat {z}}}} \right)}}{{{{\text{I}}_3} \cdot {{\text{I}}_4} \cdot \left( {{{\text{I}}_3}+{{\text{I}}_4}} \right) \cdot {{\hat {z}}^2}}},$$

(13)

where.

$$\begin{gathered} \hat {y}=2\pi {f_l}{d_y}{N_1}/c, \hfill \\ \hat {z}=2\pi {f_l}{d_z}{N_2}/c. \hfill \\ \end{gathered}$$

The non-diagonal elements of $\tilde {{\varvec{R}}}$ are the elements of ${{\varvec{R}}_{xx}}$ multiplied by two fractions. Ordinarily, the interference power is substantially greater than that of noise., so $\tilde {{\varvec{R}}}$ is approximately

$$\tilde {{\varvec{R}}}_{h,g} = {\left( {{\varvec{R}_{xx}}} \right)_{h,g}} \cdot {\varvec{T}_{h,g}}$$

(14)

Then, $\tilde {{\varvec{R}}}$ is rewritten as

$$\tilde {{\varvec{R}}}={\varvec{T}} \odot {{\varvec{R}}_{xx}},$$

(15)

where, $\odot$ represents Hadamard product. In addition, rem(h-1, M), rem(g-1, M), int(rem (h-1, MN), M), and int(rem (g-1, MN), M) are periodic functions of MN, so the TM ${\varvec{T}}$ can be expressed as

$${\varvec{T}}=1 \otimes \tilde {{\varvec{T}}},$$

(16)

where 1 is a matrix with $D \times D$ dimensions and all 1s, $\tilde {{\varvec{T}}}$ can be recorded as

$$\tilde {{\varvec{T}}}={\tilde {{\varvec{T}}}_N} \otimes {\tilde {{\varvec{T}}}_M},$$

(17)

where.

$$\begin{gathered} {\left[ {{{\tilde {{\varvec{T}}}}_M}} \right]_{k,l}}=\frac{{2\left( {{{\text{I}}_{\text{1}}} - {{\text{I}}_{\text{1}}}{e^{j \cdot {{\text{I}}_2} \cdot \hat {y}}}+{{\text{I}}_2} - {{\text{I}}_2}{e^{ - j \cdot {{\text{I}}_1} \cdot \hat {y}}}} \right)}}{{{{\text{I}}_{\text{1}}} \cdot {{\text{I}}_2} \cdot \left( {{{\text{I}}_{\text{1}}}+{{\text{I}}_2}} \right) \cdot {{\hat {y}}^2}}}, \hfill \\ {\left[ {{{\tilde {{\varvec{T}}}}_N}} \right]_{a,b}}=\frac{{2\left( {{{\text{I}}_3} - {{\text{I}}_3}{e^{j \cdot {{\text{I}}_4} \cdot \tilde {z}}}+{{\text{I}}_4} - {{\text{I}}_4}{e^{ - j \cdot {{\text{I}}_3} \cdot \tilde {z}}}} \right)}}{{{{\text{I}}_3} \cdot {{\text{I}}_4} \cdot \left( {{{\text{I}}_3}+{{\text{I}}_4}} \right) \cdot {{\tilde {z}}^2}}}, \hfill \\ \tilde {z}=2\pi {f_r}{d_z}{N_3}/c, \hfill \\ {N_3}={\text{rem}}(l - 1,M) - {\text{rem}}(k - 1,M), \hfill \\ {N_4}={\text{rem}}(b - 1,N) - {\text{rem}}(a - 1,N), \hfill \\ 1 \leqslant k,l \leqslant M,1 \leqslant a,b \leqslant N. \hfill \\ \end{gathered}$$

It should be pointed out that for systems like radar, which process pulse echoes, the frequency ${f_r}$ in the taper matrix should be set to the highest frequency of the processing band¹⁸. In contrast, for signals such as navigation satellite, it should be designated as the in-band center-frequency, due to the applicability of the narrowband assumption.

Interference matching and tunable widening

Based on the theoretical derivation in the previous section, the TM ${\varvec{T}}$ applied to the PA can be obtained, and it successfully achieves asymmetric widening of nulls. In the proposed TM, artificial interference groups can be adaptively added, and since ${\varvec{T}}$ does not contain the quantity parameter of artificial interference, it is more robust than the methods in references^12,18,19. This fulfills a portion of our goal. A processing is presented in this section to widen different nulls by applying distinct sector limits. The goal of flexibility and tunability will be fully achieved.

The CMT-based method utilizes the spatial-temporal (frequency) correlation between sensors. From Eq. (12), it is clear that taper processing is applied to each individual interference. However, the non-stationarity of each interference is unique, so it is necessary to widen each null separately to make certain that the interference and the null are matched. The individual widening of each null necessitates the identification and segregation of distinct interferences present in the snapshot. Indeed, eigen-decomposition constitutes an outstanding solution to this requirement. All first-order and second-order eigen of interference are included in the covariance, and the eigen-decomposition is the optimal second-order decomposition¹³. Eigen-decomposition formulated as.

$$\varvec{R} = \sum\limits_{i = 1}^{N \times M \times D} {{\lambda _i}{{\mathbf{u}}_i}{\mathbf{u}}_i^H} ,$$

(18)

Here, ${\lambda _i}$ is the i-th eigenvalue, and ${{\mathbf{u}}_i}$ is the i-th eigenvector. ${\lambda _1}>{\lambda _2}> \ldots >{\lambda _{M \times N \times D}}$. Based on the prolate spherical wave function, the eigenvector can be linearly represented by steering vectors of different interferences²⁰.

$${{\mathbf{u}}_r}={c_{1r}}{\mathbf{a}}\left( {{f_l},{\theta _1},{\varphi _1}} \right)+{c_{2r}}{\mathbf{a}}\left( {{f_l},{\theta _2},{\varphi _2}} \right)+ \cdots +{c_{pr}}{\mathbf{a}}\left( {{f_l},{\theta _p},{\varphi _p}} \right) \ldots +{c_{Pr}}{\mathbf{a}}\left( {{f_l},{\theta _P},{\varphi _P}} \right),$$

(19)

where $r=1,2, \ldots ,P$, ${c_{pr}}$ is the cross-correlation coefficient between interferences in both the temporal and spatial domains. These eigenvectors are collectively referred to as the “signal-subspace,” and the remaining $M \times N \times D - P$ eigenvectors span the complementary “noise-subspace”. Separation of the two subspaces is based on the trend exhibited by their respective eigenvalues²¹. More significantly, as a result of the statistical independence of the interferences, each eigenvector within the signal subspace is predominantly associated with a particular steering vector. For instance, in the case of two uncorrelated sources possessing equal power, the two eigenvectors composing the subspace are

$${\mathbf{u}_{1(2)}}=\frac{{a\left( {{f_l},{\theta _{1(2)}},{\varphi _{1(2)}}} \right) \pm {e^{ - j\phi (\rho _{c}^{{12}})}}a\left( {{f_l},{\theta _{2(1)}},{\varphi _{2(1)}}} \right)}}{{\sqrt {2\left( {M \cdot N \cdot D} \right)\left( {1 \pm \left| {\rho _{c}^{{12}}} \right|} \right)} }}$$

(20)

where $\rho _{c}^{{12}}$ denotes the spatial correlation between the two interferences, and $\phi \left( \cdot \right)$ is the operation of extracting phase components. It is evident from Eq. (20) that the eigenvectors are a composition of the two steering vectors. A greater separation between the sources leads to a diminished $\rho _{c}^{{12}}$, resulting in each eigenvector approximating its respective steering vector. This correspondence remains valid in the general case of more than two interfering sources. An eigenvector for a respective source can still be found, albeit it is difficult to express in a simple form like Eq. (20).

Hence, the association between eigenvectors and steering vectors can be utilized for interference matching. Owing to the delayed operation inherent in space-time filtering, the ${{\mathbf{u}}_r}$ is formed by $\hat {g}$(2≤$\hat {g}$≤D) adjacent eigenvectors. ${\hat {g}_p}$ is the number of eigenvectors for the p-th interference, and it is related to the signal bandwidth. ${\hat {g}_p}=2+\left( {D - 2} \right) \cdot \hbox{min} \left( {1,{B_p}\left( {D - 1} \right)/B} \right)$,$D \geqslant 2$, where ${B_p}$ is the interference bandwidth and B is the system processing bandwidth. Critically, the eigen-covariance matrix (ECM) for each interference, built from eigenvalues and eigenvectors, includes all its statistical knowledge. Thus, with this paramount solution, the tunability of null width can be enabled by applying a taper to the ECM corresponding to each interference. Moreover, ECM reconstruction can mask the target signal and solve the null of the target signal in beamforming.

$${\varvec{R}_T} = \sum\limits_{p = 1}^P {{\varvec{R}_p} \odot {\varvec{T}_p}} + \sum\limits_{i = \varepsilon + 1}^{M \times N \times D} {{\lambda _i}{{\mathbf{u}}_i}{\mathbf{u}}_i^H + } \gamma {I_{M \times N \times D}},$$

(21)

where.

$${\varvec{R}_p} = \sum\limits_{i = \varsigma + 1}^{{{\hat g}_p}} {{\lambda _i}{{\mathbf{u}}_i}{\mathbf{u}}_i^H,}$$

$$\varepsilon = \sum\limits_{p = 1}^P {{{\hat g}_p},} \;\;\varsigma = \sum\limits_{q = 1}^{p - 1} {{{\hat g}_q},}$$

where ${{\varvec{R}}_p}$ and ${{\varvec{T}}_p}$ are the ECM and corresponding TM of the p-th interference. $\gamma {{\text{I}}_{M \times N \times D}}$ is the diagonal loading term employed to stabilize the sidelobes. The application of this diagonal loading is essential in most scenarios to guarantee non-singularity since flexible null-widening slightly disturbs the noise subspace²². Here, some analysis will be provided²³. It is evident from Eq. (15) that the general TM is applied to the complete CM. However, flexible widening operations based on ECM can alter the distribution of signal and noise subspaces, resulting in singularity. The impact of the proposed strategy on the noise subspace can be illustrated by a widening process with an identical TM.

$$\eqalign{ {\varvec{R}_T} & = \mathop \sum \limits_{i = 1}^\varepsilon {\lambda _i}{\mathbf{u}_i}\mathbf{u}_i^H \odot \varvec{T} + \mathop \sum \limits_{i = \varepsilon + 1}^{M \times N \times D} {\lambda _i}{\mathbf{u}_i}\mathbf{u}_i^H \cr & {\text{ }} = \left( {\mathop \sum \limits_{i = 1}^\varepsilon {\lambda _i}{\mathbf{u}_i}\mathbf{u}_i^H + \mathop \sum \limits_{i = \varepsilon + 1}^{M \times N \times D} {\lambda _i}{\mathbf{u}_i}\mathbf{u}_i^H \odot \varvec{T}'} \right) \odot \varvec{T} \cr}$$

(22)

Based on Eq. (22), the first term consists of the interference-plus-noise, and the second term is attributable to noise. ${\varvec{T}}^{\prime} \odot {\varvec{T}}=1$. 1 is the $MND \times MND$ all-1s matrix. Whereas the conventional CMT applies tapering to all interference components within the CM, the proposed strategy restricts the tapering to the selected principal component. As demonstrated by the final expression in Eq. (22), the proposed strategy can be viewed as an implementation of the conventional CMT on a modified CM that the noise eigen is tapered. The modified noise subspace inevitably results in sidelobe fluctuations. This interpretation also holds when applying tapers to different ECM. This is reason why diagonal loading is essential to achieve a stable beam pattern. Diagonal loading can stabilize the perturbed noise subspace and retain the taper information.

Nevertheless, applying an excessively large loading distorts the nulls. A minimal loading factor $\gamma$ on the order of ${10^{ - 2}}\sigma _{n}^{2}$ is adequate to solve the matrix singularity. Diagonal loading constitutes a routine procedure in actual systems.

Weight solution for beamforming

It should be noted that traditional STAP-MVDR becomes inapplicable when the received signal is generated by the Direct Sequence Spread Spectrum (DSSS) technique. This is because it fails to consider the particularity that the amplitude response of received signal should be 1 in the entire frequency band²⁴. To further explain, the coherent integration function²⁵ of the received signal is analyzed in the frequency domain following STAP and matched filtering.

$${\varvec{R}}\left( {\tau ,u,v} \right)=A\int_{{ - \infty }}^{\infty } {\Phi \left( {{f_l}} \right)H\left( {{f_l},\theta ,\varphi } \right){e^{j{\psi _0}}}{e^{j2\pi f\left( {\tau - {\tau _0}} \right)}}df} ,$$

(23)

where $\tau$, A, $\Phi \left( f \right)$, ${\psi _0}$ and ${\tau _0}$ are the delay of the spread-spectrum code, the amplitude, the power spectral density, the true carrier phase and the true code phase, respectively. $H\left( {{f_l},\theta ,\varphi } \right)$ denotes the STAP system function, which is dependent on the DOA of signal and received frequency f. When $H\left( {{f_l},\theta ,\varphi } \right)=1$, meaning the amplitude response is 1 in the entire frequency band, it logically follows that the measured code phase and carrier phase are both true values and the received signal is undistorted.

However, in traditional STAP technology, only the center frequency ${f_0}$ is constrained, which makes it difficult to ensure $H\left( {{f_l},\theta ,\varphi } \right)=1$. Furthermore, $H\left( {{f_l},\theta ,\varphi } \right)$ varies with the changing DOA of the received signal. Consequently, its response is non-linear, which can introduce a bias in the phase measurement.

For the purpose of distortionless reception and real phase measurement, the beamformer is developed based on the theoretical principle²⁶ that the coefficients of the linear phase finite impulse response (FIR) filter are complex-conjugate symmetry. The optimization function of beamformer is formulated as.

$$\left\{ \begin{gathered} \mathop {\hbox{min} }\limits_{{\varvec{w}}} {{\varvec{w}}^H}{{\varvec{R}}_T}{\varvec{w}} \hfill \\ {{\varvec{w}}^H}{\left[ {\begin{array}{*{20}{c}} 0&{{{\varvec{a}}_{s0}}^{T}}&0 \end{array}} \right]^T}=1 \hfill \\ D{\text{ is odd}} \hfill \\ \end{gathered} \right.,$$

(24)

where ${\varvec{w}}$and ${{\varvec{a}}_{s0}}$ are the STAP weight and the spatial steering vector of the target. 0 represents an all-zero vector with the dimension 1×MN(D-1)/2. The first sub-constraint ensures that the filtering coefficients of the beamformer satisfy the complex-conjugate symmetry²⁷, and a solution to the optimization problem can be obtained by Lagrange multipliers. The weight can be solved as

$${\varvec{w}}=\mu {\varvec{R}}_{T}^{{ - 1}}{{\varvec{a}}_{s0}},$$

(25)

where $\mu$ is a real constant, typically set to $1/{\varvec{a}}_{{s0}}^{H}{\varvec{R}}_{T}^{{ - 1}}{{\varvec{a}}_{s0}}$, which is used to normalize the weight.

Equation (24) shows that merely one DOF is expended. Moreover, limiting the quantity of taps to an odd number dose not increase computational complexity, which remains the same as the conventional STAP-MPDR method²⁷.

It is worth noting that the beamformer has a constant shift of $\left( {D - 1} \right){T_s}/2$ in code phase measurement. It is easy to solve it by modifying the STAP tap structure to have a symmetrical delay around the center tap.

Subsequent supplements and methodological processes

Based on the above principles and steps, the main work of the proposed strategy has been completed. Based on the above principles and steps, the core framework of the proposed strategy is now completed. However, certain essential procedures cannot be overlooked, including non-stationarity estimation and eigenvector matching.

The non-stationarity is a crucial parameter that makes the proposed strategy effective. In practical operation, the DOA estimation method is a feasible way. Capon or MUSIC method can be used periodically to track the non-stationary change of DOA. The estimation results of the method reveal the non-stationarity per adaptive interval. Furthermore, specific techniques for non-stationary DOA estimation have been proposed in²², which can estimate the time-varying DOA. It is critical to note that platform motion is typically predictable. When DOA estimation is unavailable, this motion provides a valuable non-stationarity indicator, as seen in mechanically rotating of the radar.

The matching of eigenvectors is an important supplementary procedure. As established in the analysis of Sect. 3.2, each eigenvector corresponds to a particular steering vector in a statistically independent interference scenario. The matching procedure exploits the orthogonality of the steering vectors.

$$\hbox{max} \left| {\hat {{\varvec{a}}}{{\left( {{f_l},{u_p},{v_p}} \right)}^H}{{\varvec{u}}_i}} \right|$$

(26)

where $i=1,2, \ldots ,P$, and $\hat {{\varvec{a}}}\left( {{f_l},{u_p},{v_p}} \right)$ is the estimated space-time steering vector. The search process matches each interference to its eigenvector. Once determined, these matching results are substituted into Eq. (21).

It is pertinent to discuss why the proposed strategy utilizes eigenvectors, rather than steering vectors, to achieve separate widening operations. In practical systems, antenna arrays invariably contain inherent errors. The reconstruction of the ECM based on steering vectors would require an ideal array manifold. In contrast, eigenvectors are derived directly from the sampled snapshots, thereby encapsulating the true characteristics of the actual array. Consequently, employing eigenvectors for ECM reconstruction makes the process inherently robust against manifold errors.

With all requisite steps now determined, the complete strategy with tunable null sector is summarized as follows (Table 1).

Table 1 Algorithm flow.

Simulation results

The effectiveness of the proposed strategy is validated through some experimental results. A planar array of 15 × 10 elements is considered, with 7 delay taps connected in series behind each element. The spacing between elements in rows and columns is half the wavelength of the received signal. The target direction comes from $\left( {{\theta _0},{\varphi _0}} \right)=\left( {{0^ \circ },{0^ \circ }} \right)$ ($\left( {u,v} \right)=\left( {0,0} \right)$), and the center frequency of the down converter is 6.12 MHz, with a bandwidth of 2 MHz. Interference 1 comes from direction $\left( {{u_1},{v_1}} \right)=\left( {0.6,0.4} \right)$ with an interference-to-noise ratio (INR) of 70dB, while interference 2 comes from direction $\left( {{u_2},{v_2}} \right)=\left( { - 0.4, - 0.4} \right)$ with an INR of 50dB. For comparison, the conventional Extended-MVDR (E-MVDR) method²⁸ and the CMT method¹¹ are simulated. In the following experiment, 500 Monte Carlo trials were conducted.

Flexible widening of an individual null

This section investigates the flexible widening of an individual null. For the proposed strategy, parameters: I₁ = 0.1, I₂ = 0.05, I₃ = 0.05, I₄ = 0.1. To cover the worst-case scenario of interference DOA variation, the widening parameter $\Delta =0.2$ for the conventional CMT is set, and the number of discrete artificial interferences is $K=5$. A comparison of the three-dimensional beam patterns with the E-MVDR and conventional CMT methods is illustrated in Figs. 4, 5 and 6.

As demonstrated in Fig. 4, the E-MVDR strategy can form sharp nulls at the initial direction of interference. Given the limitations in hardware resources and update rates, it is evident that a sharp null cannot match and suppress the rapidly-moving interference. For the conventional CMT strategy, two wide nulls with symmetric and equal-width are formed. Symmetric widening is superfluous. It squanders the system’s DOF, given that a non-stationary interference exhibits a constant direction of movement within a short time interval. In order to highlight the flexible selectivity of the proposed null-widening strategy, Interference 1 is chosen to match the asymmetrically extended null. The pattern is shown in Fig. 6, where the asymmetric null widening is oriented towards the negative axis in the u direction and towards the positive axis in the v direction. On the other hand, a sharp null is maintained at the location of Interference 2. To facilitate a more pronounced comparison, we provide a two-dimensional contrast plot in Fig. 7. Note that the v-axis is normalized. This beam pattern clearly demonstrates the effectiveness of the proposed strategy for the flexible widening of an individual null.

Fig. 4

The beam pattern of E-MVDR.

Fig. 5

The beam pattern of the conventional CMT.

Fig. 6

The beam pattern of the proposed strategy.

Fig. 7

The two-dimensional contrast plot for three methods.

Tunability of wide nulls

This section is dedicated to further verification of the null widening tunability. The tunability encompasses both the tunability of the interference sector and the null width. Interference 1 and Interference 2 are selected to match different asymmetrically widened nulls. To validate the tunability of the interference sector, the widening parameters for Interference 1 are configured as I₁ = 0.1, I₂ = 0.05, I₃ = 0.05, I₄ = 0.1. Similarly, the parameters for Interference 2 are set to I₅ = 0.05, I₆ = 0.1, I₇ = 0.1, I₈ = 0.05. It is evident from the three-dimensional beam pattern in Fig. 8 that the proposed strategy successfully accomplishes distinct asymmetric widening of the interference sectors for different interferences.

Fig. 8

Flexible null widening with different asymmetry.

To verify the tunability of the null width, a more complex scenario is considered. The non-stationarity of interference 1 and interference 2 is different, that is, the sectors and widths of the two widened nulls are different. The widening parameters for Interference 1 are set to I₁ = 0.1, I₂ = 0.05, I₃ = 0.05, I₄ = 0.1. For Interference 2, the parameters are set to I₅ = 0.02, I₆ = 0.08, I₇ = 0.08, I₈ = 0.02. The three-dimensional beam pattern is pictured in Fig. 9. It is obvious that the proposed strategy can successfully achieve distinct adjustments of the null width for different interferences.

Fig. 9

Flexible null widening with different widths.

DOF expenditure

To verify the reduction in DOF expenditure achieved by the proposed strategy, a DOF comparative simulation is shown in Fig. 10. The proposed strategy widens nulls precisely as required, whereas the conventional CMT must cover the worst case of interference DOA variation. The parameters are as in experiment A, and in addition, the widening parameter for interference 2 is set to I₅ = 0.05, I₆ = 0.1, I₇ = 0.1, I₈ = 0.05。.

It can be observed that the proposed strategy saves approximately 100 DOFs compared to conventional CMT. The reason is that the proposed strategy can better match DOA changes by adjusting the sector and width of the widened null, while conventional CMT needs to cover the worst case of interference DOA variation by widening each null with the maximum and equal width.

Fig. 10

The comparison chart of DOF expenditure.

Anti-interference capability

Here, the anti-interference effectiveness of the proposed strategy is verified. Simulation experiments are conducted from two perspectives: general performance metric and signal processing performance. For a clearer demonstration, the scenario is simplified to include only Interference 1. Throughout the weight interval, the DOA offset of Interference 1 shifts linearly from (0.6, 0.4) to (0.5, 0.5). A total of 2048 snapshots are selected to train the beamformer for null generation, and the DOA of Interference 1 in these snapshots is approximately (0.6, 0.4).

Fig. 11

The comparison chart of output SINR at different input SNRs.

In terms of general performance metric, output SINR is a commonly used evaluation metric. The input signal-to-noise ratio (SNR) is set to -30dB to 0dB. The SINR comparison experiment between E-MVDR, conventional CMT and the proposed strategy is shown in Fig. 11. It is apparent that the E-MVDR method suffers from degraded performance. This is attributed to the misalignment between sharp null and the dynamic interference. Conversely, the conventional CMT and the proposed strategy achieve excellent and nearly identical output SINR. The reason is that the wide null of the two strategies can cover the worst case of interference DOA variation, so that the interference can be suppressed.

Fig. 12

Time domain and frequency spectrum data before anti-interference filtering. (a) Time domain sampling data; (b) Frequency spectrum data.

In terms of signal processing performance, a software-defined receiver^28,29,30 is employed to validate the anti-interference capability of the proposed strategy. The time-domain sampled data before STAP anti-interference filtering and its power spectrum are depicted in Fig. 12(a) and 12(b), respectively. A signal resembling the target is selected as Interference 1. Consequently, the target and the interference are distinguishable only by their power levels in the spectrum. The target energy is -15dB. The time-domain sampled data after the proposed strategy filtering is displayed in Fig. 13(a), and the power spectrum is exhibited in Fig. 13(b). It is evident that the amplitude is substantially suppressed relative to the originaltime-domain data, and the resulting power spectrum closely matches the target energy level, thereby unambiguously verifying the effectiveness of the proposed anti-interference strategy.

Fig. 13

Time domain and frequency spectrum data after anti-interference filtering. (a) Time domain sampling data; (b) Frequency spectrum data.

Computational complexity

PA with $N \times M$ elements suffer from the dimensionality, leading to significantly higher computational costs compared to LA. The most computationally intensive step in STAP is the inversion of the CM, with a complexity of$O\left( {{{\left( {NMD} \right)}^3}} \right)$, where D is the temporal tap length.

A critical advantage of the proposed Simpson-constraint strategy is its computational efficiency. Unlike convex optimization-based widening methods that require iterative solutions to find the optimal weight vector, the proposed strategy constructs the TM using a closed-form solution (as derived in Eq. 16). The calculation of the TM has a complexity of$O\left( {{{\left( {NMD} \right)}^2}} \right)$, which is negligible compared to the matrix inversion step. In addition, the computational complexity of TM for conventional CMT is also $O\left( {{{\left( {NMD} \right)}^2}} \right)$, and the computational complexity of CM reconstruction for the proposed strategy is $O\left( {{{\left( {NMD} \right)}^3}} \right)$ which is the same as inversion. Therefore, the proposed strategy achieves robust 2D null widening with virtually no additional computational burden compared to the E-MVDR, making it highly scalable for PA. The comparison of computational complexity is as follows (Table 2).

Table 2 The comparison of computational complexity.

Steering vector mismatch

In a dense 2D element grid, the strong electromagnetic coupling between adjacent elements distorts the array manifold, causing a mismatch between the theoretical steering vector and the actual one. To demonstrate the performance advantage of the proposed strategy in the array manifold error scenario, the simulation parameters are set as follows. Without loss of generality, the v of the target is fixed to 0, the u of the target varies within [-1.5, 1.5], and the beam direction remains at (0, 0). Only Interference 1 is received and it is stationary. The widening parameter is set to I1 = 0.3, I2 = 0.05, I3 = 0.05, I4 = 0.3. The performance comparison between the proposed strategy and traditional E-MVDR in array manifold error scenarios is shown in Fig. 14. It can be clearly observed that E-MVDR is more sensitive to array manifold errors than the proposed strategy. This is because the CMT-based method broadens the beamwidth while widening the null. The target group split by equal power of the target is also added. Therefore, the output SINR of the strategy proposed for array manifold matching is not as good as E-MVDR, but the difference is very small and acceptable.

Fig. 14

The performance comparison in array manifold error scenarios.

Target preservation and gain loss

Verifying the preservation of the target signal, especially when the interference is spatially close to the target, is as important as verifying interference suppression. A target signal at $\left( {u,v} \right)=\left( {0,0} \right)$ is introduced in the simulation. Then, a strong interference source is placed at various angular distances from the target, ranging from the far sidelobe region to the first sidelobe (very close to the main beam). Define $\Delta d$ as the distance from the initial position of the interference to the beam steering $\left( {u,v} \right)=\left( {0,0} \right)$. Other parameter settings are the same as in Sect. 4.4. The comparison between the proposed strategy and traditional CMT in terms of beam gain loss at beam steering is shown in Fig. 15. The first sidelobe of the beam is located at $\Delta d=0.28$.

Fig. 15

The gain loss comparison between the proposed strategy and traditional CMT.

The simulation results demonstrate that when the interference is at the first sidelobe, the proposed strategy has a gain loss of approximately 3dB, whereas the conventional CMT suffers a loss of over 9 dB due to main lobe erosion. This confirms that the proposed asymmetric widening is not only more flexible but also superior in preserving the integrity of the target signal in interference scenarios. When the interference is close to the target (e.g., on the right side), the conventional-CMT widens the null symmetrically to both left and right. The “left” side of the widened null inevitably extends into the target’s main lobe, causing significant gain degradation. The widening parameters (I1, I2, I3, I4) can be configured to extend the null broadness away from the target (outward) while keeping a sharp transition on the side facing the target.

Conclusions

A robust broadband adaptive beamformer of planar arrays for anti-dynamic interference is proposed in this paper. The proposed strategy is capable of generating wide nulls with diverse widths and asymmetries. Asymmetry is implemented by introducing an artificial interference group that satisfies the Simpson-statistical constraint. The various width is obtained by reconstructing the array CM based on the TM and EM. In addition, the proposed strategy offers the advantage of reduced DOF expenditure. The validation results reveal that the proposed strategy not only effectively suppresses rapidly-moving interference but also provides tunable-null flexibility, thereby offering enhanced practicality over the conventional CMT approach. In the future, we will further explore source tracking to estimate widened parameters.