*Article* **Small Signal Anti-Jamming Scheme Based on a DMA Linear Array under Strong Jamming**

**Yuankai Wang , Liang Jin \*, Yangming Lou and Yinuo Hao**

Wireless Communication Technology Office, Information Engineering University, Zhengzhou 450002, China

**\*** Correspondence: liangjin@263.net

**Abstract:** Considering the difficulty of receiving small signals under strong electromagnetic jamming, this paper proposes a small-signal anti-jamming scheme based on a single dynamic metamaterial antenna (DMA). Our scheme uses the dynamic-adjustable characteristics of the DMA to perform spatial filtering at the antenna radio frequency (RF) front-end, to suppress strong jamming signals in advance and to improve the receiver's ability to receive and demodulate small signals. Specifically, we take the maximization of signal-to-interference-plus-noise ratio (SINR) as the optimization goal, transform the fractional non-convex objective function model into a quasi-convex semi-definite relaxation (SDR) problem, and use the Charnes-Cooper (CC) transform algorithm to find the optimal DMA arrayelement codeword-state matrix. Simulation results show that DMA has better spatial-beamforming capability than traditional antenna arrays, and the proposed scheme can better resist strong jamming. DMA realizes the effect of digital beamforming at the back end of the traditional communication system, has the advantages of traditional digital-spatial filtering, and further improves the receiver's ability to receive and demodulate small signals.

**Keywords:** dynamic metamaterial antenna; small signal anti-jamming; matched filtering; signal-tointerference-plus-noise ratio maximization

## **1. Introduction**

With the increasingly complex electromagnetic environment on the battlefield, the dynamic range of the signal has deteriorated sharply under battlefield conditions of large broadband and strong jamming. The jamming patterns faced by radar detection are becoming more and more diverse, and it is becoming more and more difficult to receive target signals. In particular, when a strong jamming signal enters the receiver, it will force the analog-to-digital converter (ADC) to work in the nonlinear region, causing a large number of nonlinear spurs in the received signal, resulting in nonlinear distortion and a small signal. It is submerged in the distortion of the large signal, so that the small signal cannot be received and processed normally. In order to correctly identify small signals, it is often necessary to increase the receiving gain. It can be seen that the wireless signal receiver is a key device in the communication link. Its dynamic performance determines the signal range that the receiver can handle. The lower limit of the dynamic range is the sensitivity of the receiver, that is, the smallest signal that the receiver can normally receive. The upper limit of the dynamic range is determined by the maximum acceptable signal power. When the input signal power is too large, distortion occurs due to the nonlinearity of the receiver. Therefore, the dynamic range of the receiver signal becomes an important bottleneck, restricting the receiving performance of the wireless communication system. A typical wideband receiver includes a low-noise amplifier (LNA), mixer, ADC, and filter; a large dynamic receiver requires the above devices to have good linearity and to reduce the generation of nonlinear distortion [1]. Among them, the ADC is required to have a wide spurious-free dynamic range (SFDR). In order to prevent the working range of the ADC from the linear segment into the nonlinear segment, resulting in nonlinear distortion [2],

**Citation:** Wang, Y.; Jin, L.; Lou, Y.; Hao, Y. Small Signal Anti-Jamming Scheme Based on a DMA Linear Array under Strong Jamming. *Electronics* **2023**, *12*, 1389. https:// doi.org/10.3390/electronics12061389

Academic Editors: Naser Ojaroudi Parchin, Mohammad Ojaroudi and Raed A. Abd-Alhameed

Received: 21 February 2023 Revised: 8 March 2023 Accepted: 12 March 2023 Published: 14 March 2023

**Copyright:** © 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

the power back-off method is used in engineering to deal with this problem. It means to adjust the power of the signal by selecting a lower-gain LNA or by adding an attenuator to ensure that the ADC works in a linear state. However, this operation increases the decision threshold for small signals and reduces the dynamic performance of the receiver. However, the traditional self-adaptive zeroing and anti-jamming technology has been unable to achieve signal alignment due to the dynamic deterioration, or even blockage, of the front end [3]. In fact, high-performance ADC devices have not fundamentally reduced the pressure of back-end digital processing. Therefore, under the premise of not reducing the input-signal power, it will have practical engineering significance to ensure the detection and reception of small signals by suppressing the generation of nonlinear distortion.

At present, research on optimizing wireless signal receivers and improving SFDR signal reception is mainly divided into circuit optimization [4] and digital compensation [5]. In [4], in order to reduce the nonlinear error introduced by capacitance mismatch, the residual voltages of the two sampling stages, before and after, were complementary by exchanging the sampling capacitor. However, the complexity of analog-circuit design is high, and it is difficult for the same circuit to meet the needs of different application scenarios. Therefore, digital-compensation technologies having higher flexibility and better performance have gradually become the focus of research. In existing research, the digital-compensation technology is mainly based on the signal-processing technology of the array antenna [6]. Compared with the traditional antenna, the array antenna has the characteristics of high spatial resolution, long detection distance, and anti-jamming. It is widely used in radar, sonar, wireless communication, space telemetry, and other fields [7–14]. For the electromagnetic jamming problem, the classic digital compensation technology is mainly signal separation. One approach is adaptive-beamforming technology, which can automatically adjust the beam pointing and nulling, according to the direction of the target and the jamming with a specific adaptive criterion, so that the signal output of the array antenna forms a main lobe beam in the direction of the target, and forms a null as low as possible in the jamming direction to suppress the influence of the jamming signal. At the same time, the side-lobe level is minimized to reduce the influence of clutter. Existing studies have applied this technology to enhance target signals, suppress jamming signals, and improve the resolution, energy, and channel utilization of airspace detection. The other was based on the jamming-blocking algorithm and its extended application [15–18]. Its core idea is to use the angle information to construct a jamming-blocking matrix to eliminate the jamming part. In addition, there was an extended-noise subspace algorithm [19], which constructs an extended subspace of strong jamming and noise, and performs conventional direction-of-arrival (DOA) estimation on this basis. In [20], the authors proposed an improved jamming-blocking algorithm, and reconstructed the Toeplitz matrix so that the coherent signal is decoherent, and the jamming signal was suppressed by using the jamming blocking matrix method, but the robustness of the algorithm was not strong.

However, the above-mentioned research methods have been widely used in the reception scenarios of homogeneous-array antennas with electromagnetic jamming, such as anti-jamming. Among them, the homogeneous array-antenna adopts the method of enlarging the array aperture to improve the resolution ability of airspace signals, but when the aperture is limited, it is difficult to achieve the desired signal-separation effect simply by increasing the number of antennas. Currently, both 5G massive multiple-input multipleoutput (MIMO) and 6G ultra-massive MIMO face the problem of limited aperture. In fact, the homogeneous array antenna only utilizes the processing gain of the array and ignores the processing gain of the array elements, and the spatial-signal processing-capability of the array antenna is limited [21–23]. At present, an emerging receiver architecture, based on DMA, combines adjustable analog in the hardware level with an array of microstrips, each embedded with configurable radiating-metamaterial elements, the physical properties of which, such as permittivity and magnetic permeability, are dynamically adjustable. By configuring stacks of metamaterials on the surface, they can be tuned to achieve different transformations for transmission, reception, or strike waves. So far, the two main applications of metamaterials have been considered for wireless communications. The first used passive metamaterials as reflectors, also known as reconfigurable intelligent surface (RIS), which generated flexible reflection patterns to change the wireless propagation environment in desired ways [24–26]. Currently, research has been done on holographic imaging [27], intelligent perception [28], wireless communication [29–31], and direction of arrival estimation [32]. The applications are still expanding. Additionally, another emerging application of metamaterials in wireless communications is to exploit their controllable radiation- and reception-patterns as antennas, i.e., the DMA mentioned in this paper. The difference from RIS is that DMA is directly connected to the RF channel, which can change the receiving-response of electromagnetic waves from different directions in real time. It can achieve programmable control of receive-beam patterns through advanced analog signal-processing capabilities. This kind of antenna structure generally consumes less power and costs than traditional array-based architectures. So far, DMA-related research has mainly focused on reducing path loss [33], improving multi-user communication throughput [34–36], and directional beam focusing [37,38], which motivates the study of the effect of DMA on the reception-performance of desired small signals under conditions of strong jamming.

This paper focuses on the matching-reception problem of small signals under the battlefield scenario of strong electromagnetic jamming. Different from the existing antennaarray receiving system, this solution relies on a field-programmable gate array (FPGA) to adjust the DMA array-element codeword-state matrix in real time, which will couple guided waves from in-plane sources into a free space, manipulate the extracted free-space waves on demand, and enable beamforming and steering functions. Each metamaterial element is independently controlled by a positive–intrinsic–negative (PIN) diode to switch the element between the coupled (coded as "1") and uncoupled (coded as "0") states. It can realize the periodic harmonic amplitude and phase-independent adjustment of each metamaterial array element, and can realize the independent adjustment of electromagnetic-wave specific-harmonic pattern and intensity. It is useful to change the electromagnetic-wave response, improve the receiving gain in the direction of the desired small signal, perform spatial-filtering processing in advance at the RF front-end, suppress strong jamming signals, improve the equivalent SFDR of the receiver, and reduce the loss of small-signal information. Considering these advantages of DMA, we innovatively propose a small-signal anti-jamming scheme based on a single DMA, which can perform spatial-filtering and matching reception on multipath channels of multi-stream signals, and improve the receiving gain of desired small signals.

The contributions of the paper are summarized as follows:

First, we established a basic model of small-signal anti-jamming based on single DMA, and the optimal-array element codeword-state matrix of DMA was designed according to the prior direction of arrival and multipath-channel value of the expected small signal and strong jamming, so as to achieve the effect of zero trapping aligned with incoming waves of strong jamming. That is, strong jamming signals will be attenuated, but desired small signals will be boosted at the receiver.

Secondly, in the process of designing the optimal array-element codeword state matrix, our optimization goal is to ensure the maximum SINR of the receiver, so as to maximize the suppression of the impact of strong jamming on the desired small signal. Aiming at the challenging problems caused by non-convex constraints and fractional forms of high-dimensional matrices, we propose a CC transform algorithm, which transforms the fractional-order non-convex objective-optimization model into a quasi-convex SDR problem, to find the optimal DMA array-element codeword-state matrix that maximizes the received SINR to maximize the receive gain of small signals.

Simulation results show that the scheme can achieve higher SINR performance in the face of strong electromagnetic jamming. In addition, the scheme can still improve the effect of small-signal information-loss to a certain extent in the face of strong main lobe jamming. jamming.

The rest of the paper is organized as follows: Section 2 introduces the small-signal matching-receiver system model and framework based on a single DMA. Section 3 introduces the small-signal anti-jamming matching-receiving scheme. Section 4 offers the numerical results analysis and performance evaluation. Section 5 draws conclusions. The rest of the paper is organized as follows: Section II introduces the small-signal matching-receiver system model and framework based on a single DMA. Section Ⅲ introduces the small-signal anti-jamming matching-receiving scheme. Section IV offers the numerical results analysis and performance evaluation. Section V draws conclusions.

effect of small-signal information-loss to a certain extent in the face of strong main lobe

#### **2. Multi-Stream Signal Matched Reception Scenario 2. Multi-Stream Signal Matched Reception Scenario**

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In Figure 1, we consider the small-signal matching receiving system scenario under strong jamming conditions based on DMA, where the antenna RF front-end of the receiver Alice is a DMA with metamaterial array elements, and Bob is a small-signal source. Jammer is a strong jamming signal source. In the process of electromagnetic wave propagation, affected by the terrain, the expected signal transmitted by Bob will reach the DMA of Alice through multiple paths. Each element of the DMA array combines the observed signals at the output port of the microstrip, and feeds the RF chain and ADC through Nyquist rate sampling. In Figure 1, we consider the small-signal matching receiving system scenario under strong jamming conditions based on DMA, where the antenna RF front-end of the receiver Alice is a DMA with metamaterial array elements, and Bob is a small-signal source. Jammer is a strong jamming signal source. In the process of electromagnetic wave propagation, affected by the terrain, the expected signal transmitted by Bob will reach the DMA of Alice through multiple paths. Each element of the DMA array combines the observed signals at the output port of the microstrip, and feeds the RF chain and ADC through Nyquist rate sampling.

**Figure 1.** Multi-stream signal matched reception scenario based on DMA. **Figure 1.** Multi-stream signal matched reception scenario based on DMA.

In this paper, we assume that the directions of arrival of Bob and Jammer are known, and the corresponding multipath channel values have been obtained from the channel estimation results. Then in a single symbol period, Bob's multipath channel estimate can be expressed as: In this paper, we assume that the directions of arrival of Bob and Jammer are known, and the corresponding multipath channel values have been obtained from the channel estimation results. Then in a single symbol period, Bob's multipath channel estimate can be expressed as:

$$h(\theta\_B) \stackrel{\Delta}{=} \left[ h(\theta\_{B,1})\_{\prime} \dots \iota\_{\prime} h(\theta\_{B,L\_1}) \right]^T \tag{1}$$

<sup>1</sup> ,1 , ( ) [ ( ), , ( )]*<sup>T</sup> B B BL hh h* , (1) where θ <sup>×</sup> ∈ <sup>1</sup> <sup>1</sup> ( ) *<sup>L</sup> <sup>B</sup> h* is the multipath channel estimate between Bob and Alice, the multipath number is 1 *L* , and θθ θ <sup>1</sup> ,1 , [ ,, ]*<sup>T</sup> B B BL* represents Bob's multipath angle of arrival where *<sup>h</sup>*(*θB*) <sup>∈</sup> <sup>C</sup>*L*1×<sup>1</sup> is the multipath channel estimate between Bob and Alice, the multipath number is *L*<sup>1</sup> and *θ<sup>B</sup>* , [*θB*,1, . . . , *θB*,*L*<sup>1</sup> ] *T* represents Bob's multipath angle of arrival (AOA) at Alice.

(AOA) at Alice. Meanwhile, Jammer's multipath channel estimate can be expressed as:

$$\lg(\theta\_{\mathcal{I}}) \triangleq \left[ \lg(\theta\_{\mathcal{I},1}), \dots, \lg(\theta\_{\mathcal{I},L\_2}) \right]^T,\tag{2}$$

θθ θ <sup>2</sup> ,1 , ( ) [ ( ), , ( )]*<sup>T</sup> J J JL gg g* , (2) where θ <sup>×</sup> ∈ <sup>2</sup> <sup>1</sup> ( ) *<sup>L</sup> <sup>J</sup> g* is the multipath channel estimate between Jammer and Alice, the where *<sup>g</sup>*(*θJ*) <sup>∈</sup> <sup>C</sup>*L*2×<sup>1</sup> is the multipath channel estimate between Jammer and Alice, the multipath number is *L*2, and *θ<sup>J</sup>* , [*θJ*,1, . . . , *θJ*,*L*<sup>1</sup> ] *T* represents Jammer's multipath AOA at Alice.

multipath number is 2 *L* , and θθ θ <sup>1</sup> ,1 , [ ,, ]*<sup>T</sup> J J JL* represents Jammer's multipath AOA at Alice. Then the wireless channels between Bob, Jammer, and Alice, respectively, can be expressed as:

$$\hbar\_{\rm B} = \phi^T(\theta\_{\rm B})\hbar(\theta\_{\rm B})\_\prime \tag{3}$$

$$\mathbf{g}\_I = \boldsymbol{\phi}^T(\boldsymbol{\theta}\_I)\mathbf{g}(\boldsymbol{\theta}\_I),\tag{4}$$

where

 

 1 1

( ) *<sup>L</sup> B*

where *<sup>φ</sup>*(*θB*) <sup>∈</sup> <sup>C</sup>*L*1×<sup>1</sup> is the patten response of DMA at *<sup>θ</sup>B*, *<sup>φ</sup>*(*θJ*) <sup>∈</sup> <sup>C</sup>*L*2×<sup>1</sup> is the patten response of DMA at *θ<sup>J</sup>* and *h<sup>B</sup>* is the equivalent channel between Bob and Alice, *g<sup>J</sup>* is the equivalent channel between Jammer and Alice. response of DMA at *J* , and *B h* is the equivalent channel between Bob and Alice, *J g* is the equivalent channel between Jammer and Alice.

 ( ) ( ) *<sup>T</sup>*

 ( ) ( ) *<sup>T</sup>*

*B B B h h* , (3)

*J J J g g* , (4)

is the patten

 2 1

( ) *<sup>L</sup> J*

 *B* , 

= 

= 

is the patten response of DMA at

#### **3. Small-Signal Anti-Jamming Model Based on Single DMA 3. Small-Signal Anti-Jamming Model Based on Single DMA**

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DMA can adjust the response of each metamaterial to incident electromagnetic waves in real time to change the mode, as shown in Figure 2. We consider DMA to be a ULA composed of *N* × 1 metamaterial elements. DMA can adjust the response of each metamaterial to incident electromagnetic waves in real time to change the mode, as shown in Figure 2. We consider DMA to be a ULA composed of *N*1 metamaterial elements.

**Figure 2.** Small-signal anti-jamming model based on single DMA. **Figure 2.** Small-signal anti-jamming model based on single DMA.

The metamaterial array-element pattern of DMA can be written as The metamaterial array-element pattern of DMA can be written as

$$\boldsymbol{\phi}^{T}(\boldsymbol{\theta}) = \begin{bmatrix} 1 & \cdots & 1 \end{bmatrix} [\boldsymbol{\Psi}(\boldsymbol{\omega}) \odot \boldsymbol{A}(\boldsymbol{\theta})], \tag{5}$$

$$\begin{split} \boldsymbol{\Psi}(\boldsymbol{\omega}) &= \begin{bmatrix} \boldsymbol{\psi}(\omega\_{1}, \theta\_{1}) & \cdots & \boldsymbol{\psi}(\omega\_{1}, \theta\_{L}) \\ \vdots & \ddots & \vdots \\ \boldsymbol{\psi}(\omega\_{N}, \theta\_{1}) & \cdots & \boldsymbol{\psi}(\omega\_{N}, \theta\_{L}) \end{bmatrix} \\ &= \begin{bmatrix} \boldsymbol{\psi}^{j\boldsymbol{\varphi}(\omega\_{1}, \theta\_{1})} & \cdots & \boldsymbol{\psi}^{j\boldsymbol{\varphi}(\omega\_{1}, \theta\_{L})} \\ \vdots & \ddots & \vdots \\ \boldsymbol{\psi}^{j\boldsymbol{\varphi}(\omega\_{N}, \theta\_{1})} & \cdots & \boldsymbol{\psi}^{j\boldsymbol{\varphi}(\omega\_{N}, \theta\_{L})} \end{bmatrix} \end{split} \tag{6}$$

1 ( , ) ( , ) *N N L j j e e* where represent the direction of arrival AOA of the signal, is for Hadamard product. *A*( ) represents the manifold matrix, where 1 ( ) [ ( ), , ( )] *<sup>L</sup> A a a* , and 2 ( 1)sin / ( ) [1, , ] *j d N T a e* − = . ( , ) *n l* represents the amplitude and phase of the inciwhere *θ* represent the direction of arrival AOA of the signal,  is for Hadamard product. *A*(*θ*) represents the manifold matrix, where *A*(*θ*) , [*a*(*θ*1), . . . , *a*(*θL*)], and *a*(*θ*) = [1, . . . ,*e j*2*πd*(*N*−1) sin *θ*/*λ* ] *T* . *ψ*(*ωn*, *θ<sup>l</sup>* ) represents the amplitude and phase of the incident electromagnetic wave changed by metamaterial elements. Then (5) can be further derived as follows:

$$\boldsymbol{\phi}^T(\boldsymbol{\theta}) = \boldsymbol{\Omega}^T(\boldsymbol{\omega})\boldsymbol{A}(\boldsymbol{\theta}),\tag{7}$$

derived as follows: where Ω(*ω*) , [*e jϕ*(*ω*<sup>1</sup> ) , . . . ,*e jϕ*(*ωN*) ] *T* , *ϕ*(*ωn*) represent the DMA codeword-state parameter used in the *k*-th symbol period. The mode of DMA can be customized by adjusting the value of the codeword. Therefore, during the *k*-th symbol period, assuming the channel remains constant during estimation, the channels of Bob and Jammer can be re-expressed as

$$\begin{array}{l} h\_B = \Omega^T(\omega) A(\theta\_B) h(\theta\_B) \\ g\_I = \Omega^T(\omega) A(\theta\_I) g(\theta\_I) \end{array} \tag{8}$$

#### **4. Small Signal Anti-Jamming Scheme**

To simplify the analysis, we assume that the scheme is in a quasi-static wireless communication, and the channel is a fast-fading channel, so the channel remains unchanged during signal-reception and subsequent digital processing.

#### *4.1. Signal Processing*

During the *k*-th symbol period, we set the number of samples at *N<sup>s</sup>* = 512, *k* ∈ [1, *K*]. When the *k*-th pilot symbol is sent, the signal received by DMA can be expressed as

$$y\_k = h\_{B}s\_{B,k} + g\_{J}s\_{J,k} + n\_{k\prime} \tag{9}$$

where *sB*,*<sup>k</sup>* ∈ C is the pilot signal sent by Bob, *sJ*,*<sup>k</sup>* ∈ C is the pilot signal sent by Jammer. Also *n<sup>k</sup>* , [*nk*,1, . . . , *nk*,*N*] *T* is the noise received by the *N* metamaterial elements of the DMA, following the independent and identical distributed (i.i.d.) CN 0, *σ* 2 *n* . Arranging the signals received by *N* metamaterial elements into a column vector, Formula (10) can be re-expressed as

$$y\_k = \boldsymbol{\Omega}^T(\omega)(\boldsymbol{A}(\theta\_B)h(\theta\_B)\boldsymbol{s}\_{B,k} + \boldsymbol{A}(\theta\_I)\boldsymbol{g}(\theta\_I)\boldsymbol{s}\_{I,k}) + \boldsymbol{n}\_{k\prime} \tag{10}$$

where Y , [*y*1, . . . , *yK*], *S<sup>B</sup>* , [*sB*, . . . ,*sB*] and *S<sup>J</sup>* , [*sJ*,1, . . . ,*sJ*,*K*] represents the pilot sequence respectively. Therefore, in the *k*-th symbol period, the complex baseband received signal power output by the DMA array can be expressed as

$$P\_{out} = \mathbb{E}[\left|y\_k y\_k^H\right|]\_\prime \tag{11}$$

According to (10), the complex baseband received signal power output by the DMA array can be re-expressed as

$$P\_{\rm out} \triangleq \Omega \hat{A}(\theta\_{\rm B}) \hat{P}\_{\rm B} \hat{A}^{H}(\theta\_{\rm B}) \Omega^{H} + \Omega \hat{A}(\theta\_{\rm I}) \hat{P}\_{\rm I} \hat{A}^{H}(\theta\_{\rm I}) \Omega^{H} + \sigma^{2} \,\,\,\,\,\,\tag{12}$$

where *σ* <sup>2</sup> <sup>∈</sup> <sup>C</sup> is the noise power received by DMA. To simplify the analysis, we normalize the signal powers sent by Bob and Jammer to be 1. Then the receiving *SINR* of Alice at the receiving terminal can be written as

$$SINR = \frac{\Omega \hat{A}(\theta\_B) \hat{P}\_B \hat{A}^H(\theta\_B) \Omega^H}{\Omega \hat{A}(\theta\_I) \hat{P}\_I \hat{A}^H(\theta\_I) \Omega^H + \sigma^2} \tag{13}$$

#### *4.2. Problem Formulation*

In order to ensure the reliability of small-signal transmission, we take the maximization of the *SINR* of the received signal as the optimization goal. Therefore, the problem is transformed into optimizing the codeword corresponding to each metamaterial array element of DMA, that is, optimizing Ω(*ω*) to maximize the *SINR*. Therefore, the optimization objective function can be written as

$$P1: \begin{array}{l} \max\limits\_{\Omega} \frac{\Omega \Phi \Omega^{H}}{\Omega \mathcal{R} \Omega^{H} + \sigma^{2}}\\ \text{s.t.} \ |\Omega\_{\text{n}}|^{2} = 1, \ n = 1, \dots, N\end{array} \tag{14}$$

where (*P*1) is actually a problem of maximizing a fractional non-convex objective function [39]. For such non-convex problems, we can transform it into another more feasible form.

That is ΩΦΩ*<sup>H</sup>* = *Tr*(ΦΩ*H*Ω) and Ω**R**Ω*H*(*ω*) = *Tr*(**R**Ω*H*Ω), where we define *<sup>V</sup>* <sup>=</sup> <sup>Ω</sup>*H*<sup>Ω</sup> in which *<sup>V</sup>* <sup>≥</sup> 0 and *rank*(*V*) = 1, (*P*1) is equivalent to

$$\begin{array}{ll} P\mathbf{2}: & \max\_{V} \frac{Tr(\Phi V)}{Tr(\mathbf{R}V) + \sigma^{2}} \\ & s.t. \ V\_{\mathbf{n},\mathbf{n}} = 1 \; \mathbf{n} = 1, \dots, N \; . \\ & V \ge 0 \\ & rank(V) = 1 \end{array} . \tag{15}$$

We will discuss the optimal solution to (*P*2) in the next section.

#### *4.3. Optimization Algorithm for P2*

For this kind of problem, this paper uses SDR to solve it. The term (*P*2) can be used for the problem of maximizing a fractional non-convex objective function. This usually requires solving using a convex optimization tool. Firstly, we need to remove the *rank*(*V*) = 1 constraint, so (*P*2) is simplified as the following SDR models as

$$\begin{array}{ll} P3: & \max\_{V} \frac{\text{Tr}(\Phi V)}{\text{Tr}(\mathbf{R}V) + \sigma^{2}} \\ & s.t. \ V\_{n,n} = 1, \ n = 1, \dots, N \ \cdot \\ & V \ge 0 \end{array} \tag{16}$$

However, since the objective function of (*P*3) has a linear fractional structure, an SDR solution cannot be obtained. A classic solution is a binary search algorithm with high computational complexity, which searches for the global optimum by solving a series of SDR solutions. We use the CC transform algorithm to effectively solve this kind of fractional objective function problem, and get the optimal solution of (*P*3). By introducing a variable *µ* ≥ 0, we transform *V* as

$$
\overline{V} = \mu V.\tag{17}
$$

Thus (*P*3) can be re-expressed as

$$\begin{array}{ll} P4: & \max\_{\overline{V}} Tr(\Phi \overline{V}) \frac{Tr(\Phi V)}{Tr(\mathbf{R} V) + \sigma^2} \\ & s.t. \ Tr(\mathbf{R} \overline{V}) + \mu \sigma^2 = \zeta \\ & \overline{V}\_{n,n} = \mu, \ n = 1, \ldots, N \\ & \overline{V} \ge 0, \quad \mu \ge 0 \end{array} \tag{18}$$

where *ζ* is a constant that *ζ* 6= 0 according to [25]. To solve (*P*4). We use the CVX toolbox to solve the problem.

#### **5. Numerical Results**

In order to establish its input-output relationship, the DMA used in this paper is a single-antenna uniform linear array composed of eight metamaterial array elements. In this paper, we assume that the directions of arrival of Bob and Jammer are known, and the corresponding multipath-channel values have been obtained from the channel estimation results.

#### *5.1. Comparison of Strong Jamming Suppression Effects*

In order to verify the performance of the scheme against a strong jamming signal, we first needed to simulate the beam pattern of a single DMA. We set the number of DMA's metamaterial array element bits to *bits* = 2. The metamaterial array-element spacing of the DMA was set to 0.25, the number of snapshots is *N<sup>s</sup>* = 512, and the number of metamaterial array elements was *N* = 8. Bob uses a conventional omnidirectional antenna. The simulation scenario architecture is shown in Figure 1. Without loss of generality, the scheme assumes that the channel remains unchanged within a frame period, and each path follows CN (0, 1). The signal-to-noise ratio is *SNR* = 10 dB, and the jamming signal-to-

noise ratio is *JNR* = 30 dB. The multipath numbers of the small signal and the strong jamming signal were *<sup>L</sup>* <sup>=</sup> 3. The multipath angle distribution was *<sup>θ</sup>* = [−*<sup>π</sup>* 2 , *π* 2 ]. Finally, the single DMA and the homogeneous array antenna have the anti-jamming reception effect on small signals, and the solved pattern is shown in Figure 3. Finally, the single DMA and the homogeneous array antenna have the anti-jamming reception effect on small signals, and the solved pattern is shown in Figure 3.

*bits* = 2.

*N* = 8.

In order to establish its input-output relationship, the DMA used in this paper is a single-antenna uniform linear array composed of eight metamaterial array elements. In this paper, we assume that the directions of arrival of Bob and Jammer are known, and the corresponding multipath-channel values have been obtained from the channel estima-

In order to verify the performance of the scheme against a strong jamming signal, we first needed to simulate the beam pattern of a single DMA. We set the number of DMA's

The simulation scenario architecture is shown in Figure 1. Without loss of generality, the scheme assumes that the channel remains unchanged within a frame period, and each

The metamaterial array-element spacing of

Bob uses a conventional omnidirectional antenna.

, and the number of met-

, and the jamming sig-

 

.

 = −[ , ] 2 2

= 512 *N s*

*SNR dB* = 10

*L* = 3 . The multipath angle distribution was

. The multipath numbers of the small signal and the

*Electronics* **2023**, *12*, x FOR PEER REVIEW 8 of 14

*5.1. Comparison of Strong Jamming Suppression Effects*

the DMA was set to 0.25, the number of snapshots is

path follows (0,1). The signal-to-noise ratio is

*JNR dB* = 30

metamaterial array element bits to

amaterial array elements was

strong jamming signal were

nal-to-noise ratio is

**5. Numerical Results**

tion results.

**Figure 3.** Pattern comparison of the CC algorithm based on DMA and the same-dimensional homogeneous array antenna. **Figure 3.** Pattern comparison of the CC algorithm based on DMA and the same-dimensional homogeneous array antenna.

In Figure 3, the green vertical line represents the incoming wave direction of the expected small signal. The red vertical line represents the strong jamming signal. The blue solid line represents the DMA reception pattern of the CC algorithm of this scheme. The orange dotted line represents the receiving pattern of the same-dimension homogeneous array antenna ( *PHI* = [1 1] ). It can be seen that the scheme forms a null in the incoming wave direction of strong jamming, and effectively suppresses strong jamming. Compared with the homogeneous array antenna, the array pattern of DMA is aligned with the direction of arrival of the expected small signal, and the small signal is effectively received, In Figure 3, the green vertical line represents the incoming wave direction of the expected small signal. The red vertical line represents the strong jamming signal. The blue solid line represents the DMA reception pattern of the CC algorithm of this scheme. The orange dotted line represents the receiving pattern of the same-dimension homogeneous array antenna (*PH I* = [1 · · · 1]). It can be seen that the scheme forms a null in the incoming wave direction of strong jamming, and effectively suppresses strong jamming. Compared with the homogeneous array antenna, the array pattern of DMA is aligned with the direction of arrival of the expected small signal, and the small signal is effectively received, which proves the effectiveness of the scheme.

#### which proves the effectiveness of the scheme. *5.2. Comparison of Small Signal Anti-Jamming Performance*

In order to further verify the feasibility of the proposed algorithm, the simulation parameters are basically the same as in Section 5.1, only the value ranges of SNR and JNR are changed respectively. We compared the small-signal anti-jamming performance between the traversal search algorithm (*PH I* = *phiTra*) and the traditional adaptive beamforming MVDR algorithm (*PH I* = *phiMVDR*), and the CC algorithm of this scheme, as shown in Figure 4. It can be seen that the overall range of the SINR of the received signal of the CC algorithm of the scheme is better than that of the other two algorithms, and the SINR of the homogeneous array antenna is the worst. In particular, when the JNR of the strong jamming signal gradually increases, the algorithm can still effectively receive the small signal, and the anti-jamming performance of the small signal is strong, as shown in Figure 4a. This further verifies the effectiveness of this scheme for the strong jamming suppression effect.

In order to further verify the feasibility of the proposed algorithm, the simulation parameters are basically the same as in Section 5.1, only the value ranges of SNR and JNR are changed respectively. We compared the small-signal anti-jamming performance be-

) and the traditional adaptive beam-

), and the CC algorithm of this scheme, as

= *Tra PHI phi*

shown in Figure 4. It can be seen that the overall range of the SINR of the received signal of the CC algorithm of the scheme is better than that of the other two algorithms, and the SINR of the homogeneous array antenna is the worst. In particular, when the JNR of the strong jamming signal gradually increases, the algorithm can still effectively receive the small signal, and the anti-jamming performance of the small signal is strong, as shown in Figure 4a. This further verifies the effectiveness of this scheme for the strong jamming

**Figure 4.** The anti-jamming performance of the CC algorithm compared with the traversal search algorithm, MVDR, and homogeneous array antennas. (**a**) SNR. (**b**) JNR. **Figure 4.** The anti-jamming performance of the CC algorithm compared with the traversal search algorithm, MVDR, and homogeneous array antennas. (**a**) SNR. (**b**) JNR.

#### *5.3. Single DMA Performance Analysis 5.3. Single DMA Performance Analysis*

*5.2. Comparison of Small Signal Anti-Jamming Performance*

= *MVDR PHI phi*

tween the traversal search algorithm (

forming MVDR algorithm (

suppression effect.

In order to further verify the influence of DMA on the small signal anti-jamming performance, we changed the multipath value in the simulation parameters, the number of bits of the metamaterial element, and the metamaterial-element spacing from the perspective of the array antenna itself. The simulation results are shown in the following chapter. In order to further verify the influence of DMA on the small signal anti-jamming performance, we changed the multipath value in the simulation parameters, the number of bits of the metamaterial element, and the metamaterial-element spacing from the perspective of the array antenna itself. The simulation results are shown in the following chapter.

#### 5.3.1. Effect of the Number of Paths 5.3.1. Effect of the Number of Paths

Under the same simulation conditions, we set the multipath number of small signal and strong jamming signal to *L*[3,5,7] . The SINR of the received signal after processing is shown in Figure 5. It can be seen that the overall SINR of the signal received by this algorithm is better than that of the traversal search algorithm, and the small-signal receiving effect of the homogeneous array antenna is the worst. From only the three curves of this algorithm, it can be seen that the anti-jamming performance of the number *L*=3 is the best, followed by *L*=5 , and finally *L*=7 . This shows that the sparsity of the channel in the space will affect the receiving effect of the array antenna. With the increase of the multi-channel number L, the difficulty of aligning the beam of the array antenna to the small signal increases, and only a few of the channels can be matched, thereby reducing Under the same simulation conditions, we set the multipath number of small signal and strong jamming signal to *L* ∈ [3, 5, 7]. The SINR of the received signal after processing is shown in Figure 5. It can be seen that the overall SINR of the signal received by this algorithm is better than that of the traversal search algorithm, and the small-signal receiving effect of the homogeneous array antenna is the worst. From only the three curves of this algorithm, it can be seen that the anti-jamming performance of the number *L* = 3 is the best, followed by *L* = 5, and finally *L* = 7. This shows that the sparsity of the channel in the space will affect the receiving effect of the array antenna. With the increase of the multi-channel number *L*, the difficulty of aligning the beam of the array antenna to the small signal increases, and only a few of the channels can be matched, thereby reducing the reception performance of the signal DMA. *Electronics* **2023**, *12*, x FOR PEER REVIEW 10 of 14

**Figure 5.** Comparison of the number of paths to the anti-jamming reception performance of different algorithms. **Figure 5.** Comparison of the number of paths to the anti-jamming reception performance of different algorithms.

adaptive beamforming MVDR algorithm, and the homogeneous-array antenna. The num-

rameters are the same as in Section 5.1. The SINR of the received signal after processing is shown in Figure 6. It can be seen that by increasing the number of bits of a single metamaterial element, this solution can slightly improve the receiving SINR of small signals, but the trend is not obvious. This shows that although the increase in the number of bits can enable metamaterial elements to have more receiving states, the signal-processing capability of a single DMA line array is limited, while the shape and parameters of the DMA should be configured according to specific scenarios. Even so, this solution still has good anti-jamming performance against strong jamming, and the method of increasing the signal-processing gain by increasing the number of bits cannot be denied, just like the suboptimal traversal search algorithm in Figure 6. However, the homogeneous array antenna only utilizes the processing gain of a single array and ignores the processing gain of the array element itself. Therefore, the method of increasing the number of bits does not essentially improve the processing performance of the homogeneous array antenna for air-

*bits* [1, 2, 3]

, and the other pa-

ber of bits of each metamaterial element is adjusted to

space signals.

5.3.2. Effect of the Number of Bits in a Single Metamaterial Element

### 5.3.2. Effect of the Number of Bits in a Single Metamaterial Element

In order to illustrate the effect of the algorithm in this paper on improving small-signal information loss, it is compared with the traversal search algorithm, the traditional adaptive beamforming MVDR algorithm, and the homogeneous-array antenna. The number of bits of each metamaterial element is adjusted to *bits* ∈ [1, 2, 3], and the other parameters are the same as in Section 5.1. The SINR of the received signal after processing is shown in Figure 6. It can be seen that by increasing the number of bits of a single metamaterial element, this solution can slightly improve the receiving SINR of small signals, but the trend is not obvious. This shows that although the increase in the number of bits can enable metamaterial elements to have more receiving states, the signal-processing capability of a single DMA line array is limited, while the shape and parameters of the DMA should be configured according to specific scenarios. Even so, this solution still has good anti-jamming performance against strong jamming, and the method of increasing the signal-processing gain by increasing the number of bits cannot be denied, just like the suboptimal traversal search algorithm in Figure 6. However, the homogeneous array antenna only utilizes the processing gain of a single array and ignores the processing gain of the array element itself. Therefore, the method of increasing the number of bits does not essentially improve the processing performance of the homogeneous array antenna for airspace signals. *Electronics* **2023**, *12*, x FOR PEER REVIEW 11 of 14

**Figure 6.** Comparison of the anti-jamming performance of different algorithms based on the number of bits in a single metamaterial element. **Figure 6.** Comparison of the anti-jamming performance of different algorithms based on the number of bits in a single metamaterial element.

#### 5.3.3. Effect of Metamaterial Element Spacing

signal.

element spacing.

5.3.4. Effect of Strong Main Lobe Jamming

5.3.3. Effect of Metamaterial Element Spacing Under the same simulation conditions, we set the multipath number of small signal and strong jamming signal to *<sup>d</sup>* / 1 / 2 1 / 3 1 / 4 [ , , ] . The SINR of the received signal after processing is shown in Figure 7. It can be seen from Figure 7 that the element spacing can also affect the SINR of the array antenna to the received signal. On the whole, when the metamaterial element spacing is further reduced, the SINR of DMA for small signal reception shows a downward trend, which indicates that in the strong jamming scenario, the metamaterial element spacing should not be too small, and the spacing should be ad-Under the same simulation conditions, we set the multipath number of small signal and strong jamming signal to *d*/*λ* ∈ [1/2, 1/3, 1/4]. The SINR of the received signal after processing is shown in Figure 7. It can be seen from Figure 7 that the element spacing can also affect the SINR of the array antenna to the received signal. On the whole, when the metamaterial element spacing is further reduced, the SINR of DMA for small signal reception shows a downward trend, which indicates that in the strong jamming scenario, the metamaterial element spacing should not be too small, and the spacing should be adjusted according to the actual scene to improve each metamaterial element's perception and observation ability. The signal can further improve the receiving-gain of the small signal.

justed according to the actual scene to improve each metamaterial element's perception and observation ability. The signal can further improve the receiving-gain of the small

**Figure 7.** Comparison of anti-jamming performance of different algorithms based on metamaterial

In order to further analyze the performance of this scheme against strong main-lobe jamming, we set strong jamming signal sources in the range of 2° deviation of small signal

**Figure 7.** Comparison of anti-jamming performance of different algorithms based on metamaterial element spacing. **Figure 7.** Comparison of anti-jamming performance of different algorithms based on metamaterial element spacing. *Electronics* **2023**, *12*, x FOR PEER REVIEW 12 of 14

**Figure 6.** Comparison of the anti-jamming performance of different algorithms based on the number

*<sup>d</sup>* / 1 / 2 1 / 3 1 / 4

Under the same simulation conditions, we set the multipath number of small signal

. The SINR of the received signal

[ , , ]

after processing is shown in Figure 7. It can be seen from Figure 7 that the element spacing can also affect the SINR of the array antenna to the received signal. On the whole, when the metamaterial element spacing is further reduced, the SINR of DMA for small signal reception shows a downward trend, which indicates that in the strong jamming scenario, the metamaterial element spacing should not be too small, and the spacing should be adjusted according to the actual scene to improve each metamaterial element's perception and observation ability. The signal can further improve the receiving-gain of the small

#### 5.3.4. Effect of Strong Main Lobe Jamming

of bits in a single metamaterial element.

and strong jamming signal to

signal.

5.3.3. Effect of Metamaterial Element Spacing

5.3.4. Effect of Strong Main Lobe Jamming In order to further analyze the performance of this scheme against strong main-lobe jamming, we set strong jamming signal sources in the range of 2° deviation of small signal In order to further analyze the performance of this scheme against strong main-lobe jamming, we set strong jamming signal sources in the range of 2◦ deviation of small signal incoming direction. We confirmed that the jamming signal enters the main lobe range, and that the other simulation conditions were the same as Section 5.1. At this time, the SINR of the received signal is shown in Figure 8. It can be seen that when the strong jamming signal enters the main lobe range, the reception-gain of all schemes for small signals is greatly reduced, and the small signal information loss is large. However, CC algorithm can still improve the effect of small-signal information loss to a certain extent, which proves the feasibility of the proposed algorithm. incoming direction. We confirmed that the jamming signal enters the main lobe range, and that the other simulation conditions were the same as Section 5.1. At this time, the SINR of the received signal is shown in Figure 8. It can be seen that when the strong jamming signal enters the main lobe range, the reception-gain of all schemes for small signals is greatly reduced, and the small signal information loss is large. However, CC algorithm can still improve the effect of small-signal information loss to a certain extent, which proves the feasibility of the proposed algorithm.

**Figure 8.** Comparison of small signal anti-jamming performance under strong main lobe jamming. **Figure 8.** Comparison of small signal anti-jamming performance under strong main lobe jamming.

#### **6. Discussion**

**7. Conclusions**

**6. Discussion** In summary, the scheme proposed in this paper has good small-signal anti-jamming performance. Compared with the traversal search algorithm and the traditional adaptive beamforming MVDR algorithm, the CC algorithm we adopt not only improves the receiv-In summary, the scheme proposed in this paper has good small-signal anti-jamming performance. Compared with the traversal search algorithm and the traditional adaptive beamforming MVDR algorithm, the CC algorithm we adopt not only improves the receiving gain of small signals under strong jamming, but also can be used even when strong jamming

ing gain of small signals under strong jamming, but also can be used even when strong jamming enters the range of the main lobe. To a certain extent, the information loss of

In addition, we also conducted an in-depth analysis of the parameter characteristics of single DMA. This solution can effectively improve the receiving SINR of small signals when dealing with strong far-field electromagnetic jamming scenarios, as shown in Figure 5. This method of adjusting the number of DMA bits or the metamaterial element spacing can improve the anti-jamming effect of small signals to a certain extent, and the specific

This paper studies the problem of anti-jamming matching-reception of small-signal anti-jamming based on single DMA under strong electromagnetic jamming. In view of the limited dynamic range of signal processing of traditional antenna receivers, and the fact that the small signal cannot be received normally in the scene of strong electromagnetic jamming on the battlefield, this paper proposes a small-signal anti-jamming receptionenhancement model based on a single DMA to ensure the maximum SINR as the optimization goal, transforming the non-convex optimization model into a quasi-convex SDR problem. We use the CC transform algorithm to find the optimal DMA array-element codeword-state matrix that maximizes the received SINR. The simulation results show that DMA has more advantages than traditional array antennas. With better spatial beamforming capability, the scheme can resist strong jamming. Even if the jamming intensity is high, the scheme can form attenuation in the jamming direction, and form a main lobe

parameters should be reasonably configured according to the actual scene.

enters the range of the main lobe. To a certain extent, the information loss of small signals is reduced, which proves the feasibility and superiority of this scheme.

In addition, we also conducted an in-depth analysis of the parameter characteristics of single DMA. This solution can effectively improve the receiving SINR of small signals when dealing with strong far-field electromagnetic jamming scenarios, as shown in Figure 5. This method of adjusting the number of DMA bits or the metamaterial element spacing can improve the anti-jamming effect of small signals to a certain extent, and the specific parameters should be reasonably configured according to the actual scene.

#### **7. Conclusions**

This paper studies the problem of anti-jamming matching-reception of small-signal anti-jamming based on single DMA under strong electromagnetic jamming. In view of the limited dynamic range of signal processing of traditional antenna receivers, and the fact that the small signal cannot be received normally in the scene of strong electromagnetic jamming on the battlefield, this paper proposes a small-signal anti-jamming reception-enhancement model based on a single DMA to ensure the maximum SINR as the optimization goal, transforming the non-convex optimization model into a quasi-convex SDR problem. We use the CC transform algorithm to find the optimal DMA array-element codeword-state matrix that maximizes the received SINR. The simulation results show that DMA has more advantages than traditional array antennas. With better spatial beamforming capability, the scheme can resist strong jamming. Even if the jamming intensity is high, the scheme can form attenuation in the jamming direction, and form a main lobe in the direction of the desired small signal, and the small signal can obtain a large receiving gain.

Additionally, this paper provides a feasible solution for the direction of small-signal anti-jamming in the field of strong electromagnetic countermeasures. In the future, we will be able to study the anti-jamming performance of multiple DMA line-arrays, or two-dimensional arrays for wireless communication, from both spatial and temporal dimensions in combination with the digital coding and control characteristics of DMA. In addition, DMA is coded and regulated based on FPGA, so it can be matched to and compatible with existing wireless communication software and hardware architectures. Compared with traditional antenna arrays, the communication overhead is further reduced. These advantages of DMA make it possible to innovate and upgrade existing wireless communication systems.

**Author Contributions:** Y.W. and L.J. proposed the methodology; Y.W. conducted the theoretical analysis as well as the simulation verification; Y.L. and L.J. managed and coordinated responsibility for the research activity planning and execution; Y.W. wrote the original draft, which was reviewed and edited by Y.L. and Y.H. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work is supported by the National Natural Science Foundation of China (No. U22A2001), the National Key Research and Development Program of China (2022YFB2902202), and the Song Shan Laboratory Program (Included in the management of Major Science and Technology Program of Henan Province): 221100211300-03.

**Data Availability Statement:** Not applicable.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


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