Transmit Beam Control in Low-Altitude Slow-Moving Small Targets Detection Based on Peak to Average Power Ratio Constraint

Abstract

When the radar system detects low-altitude, small, slow-moving (LSS) targets, the strong clutter interference from the ground will cause false alarms and affect the detection performance. In this paper, a phased array radar transmit beam steering algorithm is proposed to minimize strong interference from ground radiation. By minimizing the weighted vector norm and choosing variable sidelobe levels, the beam pattern can achieve deep notches in the ground-related area and maintain good main lobe detection performance. Furthermore, the designed beam should be insensitive to array mismatch and be robust. In addition, a peak-to-average power ratio (PAPR) constraint is introduced to fully utilize the transmitted energy. This optimization problem can be transformed into a second-order cone programming (SOCP) problem and solved using an off-the-shelf solver. The simulation results verify that the transmit beam synthesized by this method can meet the requirements of minimizing the main lobe loss and low side lobes on the ground side.

1. Introduction

At present, low-altitude, small, and slow-moving (LSS) targets are defined as flying at an altitude less than 1000 m, with a flying speed less than 55 m/s, with a radar cross-section (RCS) less than 2 m² [1]. The ground clutter obeys different probability distributions, so that the echo received by the radar will be overwhelmed by the clutter. At the same time, ground moving targets strongly interfere with the conventional radar system when detecting LSS targets, which makes the effective detection and tracking of LSS targets difficult. Therefore, in the military field, the enemy can carry out destructive missions under the cover of strong ground clutter and a ground-moving-target environment through LSS targets, such as an unmanned aerial vehicle (UAV), which is a threat and challenge to the conventional radar detection system. In the conventional radar detection system, monostatic radar is the most common. Therefore, monostatic radar should detect LSS targets effectively.

To improve the detection performance of LSS targets in an environment with strong echoes from ground-moving targets, many methods have been proposed. Many technologies are currently employed in detecting LSS targets, such as optical image-based detection [2] and wireless network-based detection [3,4,5]. Radar can be used all the time under all weather conditions and has become the main system for LSS detection [6]. Array radars have been widely used in the detection of LSS targets because of the good angular resolution and high sensitivity to slow-moving targets [7,8,9,10]. Compared with single antenna radars, phased array radars can enhance useful target echoes, while suppressing irrelevant interference and noise spatially through beamforming [11,12]. Based on phased array radar, most researchers are committed to designing weighting coefficients for receive beamforming. The method of the angle-Doppler-defocusing steering vector constraint proposed in [13] effectively detected fast-moving targets. Christopher D. Curtis proposed a linearly constrained minimum power algorithm with an additional quadratic constraint [14], which can be used to mitigate ground-clutter contamination on a weather surveillance radar. However, this algorithm requires covariance matrix of the echo signals, which cannot effectively suppress the clutter of the ground-moving target when there are not enough training data. For static transmit beamforming, the windowing method was used to suppress the echo from the ground moving targets. Although windowing can suppress the sidelobes effectively, it can also broaden the main lobe. Thus, it is important to propose a method that can maintain the main lobe capacity and deal with the clutter weather on the ground or in the air.

As a classical array signal processing method, beamforming has been widely studied in the dual functional radar-Communication (DFRC) system design in recent years. Joint radar and communications on a single platform can reduce the cost of the platform, share the spectrum, and enhance performance via the cooperation of radar and communication [15]. By encoding a communication message into radar waveform, radar can function as an information-embedding system [16]. The spatial degrees of freedom are exploited to synthesize multiple beams based on transmit beamforming towards several communication users and radar targets [17]. In [18,19], the transmit beamforming technology was applied to support communication and guarantee radar performance. An alternating projections two-stage iterative method was proposed to design multifunction waveforms. This method can maintain the main lobe performance. However, it has a high sidelobe of the radar transmit beam pattern. F. Liu [16] proposed a method to suppress the interference power at the communication receivers. This algorithm caused performance degradation at low SINR. J. Euziere [20] proposed a method for information embedding using a time-modulated array. The phases of the transmit weight vectors were adjusted from pulse to pulse in order to introduce variations in the sidelobe levels (SLLs) toward the intended communication receiver. However, it could not design multiple transmit power distribution patterns with the same main lobe for different time modulated arrays. In [21], a new technique with two weight vectors for the DFRC system was proposed. Sidelobe control of the transmit beamforming in tandem with waveform diversity enabled communication links using the same pulse radar spectrum. This method ensured the desired ASV was not mismatched.

In the complex environment, arrays often face various mismatches. Robust adaptive beamforming (RAB) technology was developed to mitigate the effect of uncertainty on an adaptive beamformer. Among them, the worst-case optimization is a widely studied method. Vorobyov [22] studied the RAB problem in the case of guidance vector mismatch. However, the mismatches in this paper could not deal with some extreme cases. Khabbazibasmenj [23] studied the RAB problem when the prior information of the steering vector was small enough to deal with different mismatches. However, the RAB algorithms needed cubic or greater computational cost in calculating the beamforming parameters. Therefore, researchers have developed a dimension reduction method to reduce the complexity [24,25] and improve the convergence speed. In the transmit beam control, the guidance vector mismatch must be controlled within a certain range. Considering the computational complexity, norm constraint is a good constraint [26]. In [27], Zhang proposed an RAB method based on second-order cone programming (SOCP). The RAB method with sidelobe level constraint was considered. Using worst-case optimization and norm constraints, this algorithm improved the robustness of the beam under steering vector mismatch.

In modern radar systems, various constraints should be considered to design adaptive beamforming. In [28], Cheng proposed a waveform design with constant modulus constraint. This method introduced an auxiliary variable to solve the non-convex quadratic equality constraint. In [29], to deal with intrapulse REM communication, an approach that can suitably trade off the reliability and the covertness features of the established communication was proposed. This method also considers imposing energy constraints to maximize transmit power. In [30], Cheng proposed a robust joint design of the transmit waveform and filtering structure for polarimetric radar. This method added energy constraints to maximize the transmitted signal energy. All the methods proposed in the above literature are applied to different types of radars, but in order to maximize the transmit power, the energy constraints are all considered invariably. Therefore, in radar, the energy constraint is a constraint that cannot be ignored. In [1], Xu proposed a constant modulus to deal with the peak to average power ratio (PAPR). This method achieved PAPR of 1 by directly normalizing the optimal weighting vector after solving a SOCP problem. However, this method is non-convex, and the two-step strategy method makes it impossible to guarantee that the first step still holds after the second step. Therefore, in this paper, combined with the previous RAB constraints, considering the actual situation of the limited power of the radar transmitter, a transmit energy constraint is imposed in the transmit beamforming problem. This can maximize the transmitter power and avoid the waste of the transmitter energy.

This paper proposes a transmit beam control algorithm for phased array radar to detect LSS targets. The proposed approach considers several practical requirements simultaneously, including robustness against the array mismatch, main lobe loss minimization, sidelobe suppression on the ground side, as well as a notch for specified directions. To avoid the energy wastage of beamforming and fully utilize the transmit power, the PAPR constraint is introduced to reduce transmission power loss and improve the LSS target detection performance of the radar in certain regions. The simulation experiment shows that the echo from the ground-moving-target region was suppressed effectively by the optimized beamformer, which also possessed satisfactory performance on the main lobe and sidelobes on the ground side. Generally, the major contributions of this paper include:

• The PAPR constraint is introduced for the radar transmitter power to improve the LSS target detection performance. This constraint can reduce the power loss of the radar transmitter, which is equivalent to increasing the transmit energy to a certain extent.
• To reduce the emission energy in the ground area, a sidelobe level constraint is devised. This constraint is imposed on the ground area to suppress the amplitude of radar transmitted beam pattern. It can reduce the energy of radar transmitting to the ground, thus fundamentally reducing the energy of ground clutter;
• To maintain the main lobe performance of transmit beamforming, amplitude invariant constraint is proposed for the direction of the desired signal, so that the transmit beam is directional and can have the basic function of radar detection of targets;

The structure of this paper is organized as follows. In Section 2, the signal model of uniform liner array (ULA) is given. Considering the PAPR constraint in the transmit beam control algorithm, we propose a new robust beamformer. Simulation results and performance analyses are provided in Section 3. In Section 4, we discuss the proposed method and analyze the limitations. Finally, we summarize the study in Section 5.

2. Methodology

2.1. Notation & Acronyms

We use boldfaced lowercase letters (e.g., $\mathit{a}$) to represent vectors and boldfaced uppercase letters (e.g., $A$) to represent matrices. The superscripts ‘T’ and ‘H’ represent the transpose and Hermitian conjugate transpose. $\Vert \Vert$ and $\left|\right|$ denote the vector Euclidean-norm and absolute value. $E\left\{\right\}$ represents the statistical expectation.

For the convenience of readers, all acronyms used in this manuscript is concluded in

Table 1. List of acronyms.

Acronym	Full Name	Acronym	Full Name
LSS	low altitude, small, and slow-moving	SLLs	sidelobe levels
RCS	radar cross section	RAB	robust adaptive beamforming
UAV	unmanned aerial vehicle	PAPR	peak to average power ratio
DFRC	dual functional radar-communication	ULA	uniform liner array
SOCP	second-order cone programming

according to the order they appeared.

2.2. Signal Model of LSS Detection

The following array models in this paper all use the classic ULA model. Suppose there is a ULA composed of $M$ array elements spaced with each other by $d$, as shown in Figure 1. LSS targets can reflect the radar transmit signals. The radar receives echoes from the LSS targets.

If the transmitted signal by each array element reaches a certain point in the direction, the wave path difference is

$$\mathit{\lambda }\left(\theta \right)={\left[0,d\sin (\theta ),2d\sin (\theta ),3d\sin (\theta ),\dots ,\left(M-1\right)d\sin (\theta )\right]}^T$$

(1)

Suppose the propagation velocity of electromagnetic wave is C, and the antenna carrier frequency is f; then, the spatial phase difference vector is

$$\mathit{\phi }\left(\theta \right)={\left[0,2\pi f_{0}d\sin \left(\theta \right)/c,2\pi f_{0}2d\sin \left(\theta \right)/c,2\pi f_{0}3d\sin \left(\theta \right)/c,\dots ,2\pi f_0\left(M-1\right)d\sin \left(\theta \right)/c\right]}^T$$

(2)

We define the weighting vector of each element of the array as

$$\mathit{w}={\left[w_1,w_2,w_3,\dots ,w_M\right]}^T$$

(3)

Suppose the signal transmitted by the array is s(t); at the same time, all array elements transmit the same signal. The received signal at a point in direction a in the far field is expressed as

$$y\left(t\right)=w_{1}s\left(t\right)+w_{2}s\left(t\right)e^{j2\pi f_{0}d\sin \left(\theta \right)/c}+w_{3}s\left(t\right)e^{j2\pi f_{0}2d\sin \left(\theta \right)/c}+\dots +w_{m}s\left(t\right)e^{j2\pi f_0(M-1)d\sin \left(\theta \right)/c}$$

(4)

We define the array direction vector as

$$\mathit{a}\left(\theta \right)={\left[1,e^{-j2\pi f_{0}d\sin \left(\theta \right)/c},e^{-j2\pi f_{0}2d\sin \left(\theta \right)/c},e^{-j2\pi f_{0}3d\sin \left(\theta \right)/c},\dots ,e^{-j2\pi f_0\left(M-1\right)d\sin \left(\theta \right)/c}\right]}^T$$

(5)

Equation (4) is simplified to

$$y\left(t\right)={\mathit{w}}^H\ast \mathit{a}\left(\theta \right)\ast s\left(t\right)$$

(6)

We define the array pattern as

$$F\left(\theta \right)={\mathit{\omega }}^H\ast \mathit{a}\left(\theta \right)$$

(7)

2.3. Proposed Algorithm

In the practical application of LSS target detection, the actual environment is complex. The space between the array elements is not constant, due to errors in the array element positions. There may also be amplitude and phase errors in array element channels, distorting the beam formation. Errors reduce the spatial filtering capability of the beamformer, resulting in the reduction in beam robustness and the deterioration of the LSS target detection performance. Thus, when designing the beamformer, the array mismatch must be considered. Firstly, the beam response in the real circumstances can be expressed as

$$\widehat{F}\left(\theta \right)={\widehat{\mathit{\omega }}}^{\mathrm{H}}\widehat{\mathit{a}}\left(\theta \right)={\displaystyle \sum _{n=1}^M{\widehat{g}}_n}{e}^{j\left({\widehat{\phi }}_n-2\pi {\widehat{p}}_n\sin \left(\theta \right)/\lambda \right)}$$

(8)

where $\widehat{\mathit{\omega }}$ is the real weight vector. $\widehat{\mathit{a}}\left(\theta \right)$ is the real array steer vector. ${\widehat{g}}_n$ and ${\widehat{\phi }}_n$ are the real amplitude and phase weight coefficient, respectively. ${\widehat{p}}_n$ is the real position of the nth array element. When the error is small, the expected amplitude square of the actual beam response can be expressed as

$$\begin{array}{r}\mathrm{E}\left\{{\left|\widehat{F}\left(\theta \right)\right|}^2\right\}={\left|\widehat{F}\left(\theta \right)\right|}^2{e}^{-\left({\sigma }_{\mathrm{p}}^2+{\sigma }_{\mathsf{\phi }}^2\right)}+{\displaystyle \sum _{n=1}^M{g}_n^2}\left({\sigma }_{\mathrm{p}}^2+{\sigma }_{\mathsf{\phi }}^2+{\sigma }_{\mathrm{g}}^2\right)\\ ={\left|\widehat{F}\left(\theta \right)\right|}^2{e}^{-\left({\sigma }_{\mathrm{p}}^2+{\sigma }_{\mathsf{\phi }}^2\right)}+{\Vert \mathit{\omega }\Vert }^2\left({\sigma }_{\mathrm{p}}^2+{\sigma }_{\mathsf{\phi }}^2+{\sigma }_{\mathrm{g}}^2\right)\end{array}$$

(9)

where ${\sigma }_{\mathrm{p}}^2$ is the variance of the array element position, ${\sigma }_{\mathrm{g}}^2$ and ${\sigma }_{\mathsf{\phi }}^2$ are the variance of the amplitude weight coefficient and phase weight coefficient, respectively. The first part in (9) is the ideal beam formation multiplied by an attenuation factor, which affects the overall size of the beam formation in each azimuth and does not affect the beam gain. The second part is the product of the Euclidean norm ${\Vert \mathit{\omega }\Vert }^2$ and the sum of ${\sigma }_{\mathrm{p}}^2$, ${\sigma }_{\mathrm{g}}^2$, and ${\sigma }_{\mathsf{\phi }}^2$, which mainly affects the sidelobes of the beam pattern. In the case of a certain error, the smaller the weighted vector norm ${\Vert \mathit{\omega }\Vert }^2$, the higher the robustness of the beam.

To ensure the detection of the desired target, we set the array gain in the target direction to be constant.

$${\mathit{\omega }}^{\mathrm{H}}\mathit{a}\left({\theta }_0\right)=1$$

(10)

Secondly, to achieve interference suppression effectively, the notch level of the beam pattern is an important means.

$$\left|{\mathit{\omega }}^{\mathrm{H}}\mathit{a}\left({\theta }_{\mathrm{q}}\right)\right|\le \eta ,\hspace{1em}\forall {\theta }_{\mathrm{q}}\in {\mathsf{\Theta }}_{\mathrm{q}}$$

(11)

where ${\mathsf{\Theta }}_{\mathrm{q}}$ is the region corresponding to the ground moving targets, with ${\theta }_{\mathrm{q}}$ as the discrete direction in the notch region. $\eta$ is the upper level, which can be set according to the actual requirements.

According to (10), we can obtain the following formula

$$1={\left|{\mathit{\omega }}^{\mathrm{H}}a\left({\theta }_0\right)\right|}^2\le {\Vert \mathit{\omega }\Vert }^2{\Vert a\left({\theta }_0\right)\Vert }^2=M{\Vert \mathit{\omega }\Vert }^2$$

(12)

In engineering practice, in order to improve the transmission efficiency as much as possible, the radar transmitter amplifier usually works in the saturated state; so, it is impossible to modulate the waveform amplitude. Therefore, in order to make full use of the amplified power of the transmitter amplifier, the transmitted waveform is often required to have PARP or constant mode characteristics.

PAPR is defined as the ratio of the peak power to the average power

$${\zeta }_{\mathrm{PAPR}}=\frac{\underset{i=1,2,\dots ,M}{\max }{\left|{\omega }_i\right|}^2}{\overline{{\Vert \mathit{\omega }\Vert }^2}}=\frac{\underset{i=1,2,\dots ,M}{\max }{\left|{\omega }_i\right|}^2}{{\Vert \mathit{\omega }\Vert }^2}M$$

(13)

When (12) is equal, we obtain

$${\Vert \mathit{\omega }\Vert }^2\ge 1/N$$

(14)

Therefore, the PAPR constraint has an upper limit.

$${\zeta }_{\mathrm{PAPR}}=M\frac{\underset{i=1,2,\dots ,M}{\max }{\left|{\omega }_i\right|}^2}{{\Vert \mathit{\omega }\Vert }^2}\le M\frac{\underset{i=1,2,\dots ,M}{\max }{\left|{\omega }_i\right|}^2}{1/M}=M^2\underset{i=1,2,\dots ,M}{\max }{\left|{\omega }_i\right|}^2$$

(15)

In real transmit beam control, the amplitude coefficient must not be infinity. Without losing generality, we can assume the upper limit is $\epsilon$.

$$\underset{i=1,2,\dots ,M}{\max }{\left|{\omega }_i\right|}^2\le \epsilon$$

(16)

We can obtain

$$M^2\underset{i=1,2,\dots ,M}{\max }{\left|{\omega }_i\right|}^2\le M^2\epsilon$$

(17)

$${\zeta }_{\mathrm{PAPR}}=M\frac{\underset{i=1,2,\dots ,M}{\max }{\left|{\omega }_i\right|}^2}{{\Vert \mathit{\omega }\Vert }^2}\le M^2\epsilon$$

(18)

If we synthesize (10), (11), and (16), they can be integrated and transformed into the following convex optimization formula.

$$\begin{array}{l}\min {\Vert \mathit{\omega }\Vert }^2\\ \mathrm{s}.\mathrm{t}.{\mathit{\omega }}^H\mathit{a}\left({\theta }_0\right)=1\\ \left|{\mathit{\omega }}^H\mathit{a}\left({\theta }_{\mathrm{q}}\right)\right|\le \eta ,\forall {\theta }_{\mathrm{q}}\in {\mathsf{\Theta }}_{\mathrm{q}}\\ \underset{i=1,2,\dots ,M}{\max }\left|{\omega }_i\right|\le \sqrt{\epsilon }\end{array}$$

(19)

Figure 2 gives a bird’s eye view of the whole system.

3. Simulation Analysis of the Proposed Method

Three experiments were carried out to demonstrate the effectiveness of the LSS target detection. In all simulations, we assumed that the number of array elements $N$ = 32, with interelement spacing $d=\lambda /2$. The desired target was at ${\theta }_0$ = 5°. The region ${\mathsf{\Theta }}_{\mathrm{q}}$ of the ground-moving targets was $\left[-{0.2}^{°},{0.2}^{°}\right]$. The scanning range was $\left[-{90}^{°},{90}^{°}\right]$. For comparison, the static beamformer given by ${\mathit{\omega }}_{\mathrm{cbf}}={\mathit{a}}^{\mathrm{H}}\left({\theta }_0\right)$ was also tested due to its good directivity. Static beamformers have great main lobe performance and low sidelobe power compared with the adaptive transmit beamforming, such as MVDR beamformers.

To show the transmit beamforming performance in LSS target detection, the indicator to measure the property of sidelobe suppression was the first sidelobe amplified power. The first sidelobe amplified power is calculated as follows

$$P_{sidelobe}=\mathit{\omega }\ast \mathit{a}(\theta ),\theta ={\theta }_{sidelobe}$$

(20)

${\theta }_{sidelobe}$ is the first sidelobe pointing angle in the ground area. The lower the sidelobe-amplified power transmitted, the less emitting energy irradiates the ground clutter. Thus, radar can improve the LSS target detection performance in the ground regions instead of being submerged by strong ground clutter.

In this study, firstly, the beampattern of the proposed method was compared with static beamformers in order to show the beampattern promotion in experiment 1. Then, to more intuitively understand the influence of sidelobe level and PAPR constraints on the beamformer, the parameter perturbation experiment was designed, and the simulation experiment structure under different parameter settings was tested in experiments 2 and 3.

3.1. Experiment 1: Proposed Beampattern versus Different Beamformers

Comparing the algorithm proposed in this paper with static beamforming, constant modulus and sidelobe constraint in [27], the simulation results are shown in Figure 3. The beam pointed in the desired direction are consistent with static beamforming, which indicates that the main peak detection performance is preserved. Compared with the other two methods, the sidelobe level of the ground target of the proposed method is much lower than the other two methods in

Table 2. Region amplified power improvement and PAPR.

Method	Actual PAPR	Ground Region Amplified Power (dB)
Static	1	−13.24
Proposed method	1.1475	−30.00
Constant modulus	1	−21.29
Sidelobe constraint	1.4760	−30.41

. This can greatly reduce the energy of the transmitter irradiating the ground target, thereby significantly suppressing the ground moving target area. The sidelobe constraint method can achieve the sidelobe level of the proposed method, but the PAPR is much higher, which will cause a great waste of transmission energy. Therefore, the algorithm has a good effect on beam pointing and clutter suppression.

3.2. Experiment 2: Beampattern under Different Sidelobe Thresholds

We gave different values of clutter suppression in Formula (19) to carry out the simulation experiment. The results are shown in Figure 4. When the values of the constraint were −20 dB, −25 dB, −30 dB, −35 dB, −40 dB, and −50 dB, respectively, the direction of the beam in the desired direction remained the same, but there were different null depths in the clutter suppression area in

Table 3. Region amplified power improvement and PAPR under different sidelobe thresholds.

Sidelobe Thresholds (dB)	Actual PAPR	Ground Region Amplified Power (dB)
−20	1.1819	−20
−25	1.1625	−25
−30	1.1475	−30
−35	1.1375	−35
−40	1.1314	−40
−50	1.1257	−50

. With the increase in the absolute value of the constraint, the depth of the clutter suppression region gradually deepened, but the sidelobe was also raised slightly. As the sidelobe level decreases, the PAPR also decreases, but the area that can be confined also decreases. This shows that the method in this paper can set optimization parameters according to specific needs.

3.3. Experiment 3: Beampattern under Different PAPR Tolerances

We took the values 1.5, 1.4, 1.3, and 1.2 with different PAPR constraints in Formula (19) for this simulation. The results are shown in Figure 5. The influence of the three on the null depth in the clutter suppression area was not obvious, but the sidelobe was significantly too high with the decrease in the value. As the PAPR decreases, the side lobes to the right of the desired signal begin to gradually increase, which can lead to an increased amplitude of the clutter in this region, causing additional problems. Therefore, combining the results of experiment 2 and experiment 3, it was necessary to set the PAPR beam reduction value and clutter region suppression value, respectively, according to the actual situation, so as to achieve the required results.

4. Discussion

Conventional RAB algorithms can be robust against steering vector deviations. However, the traditional RAB does not consider the actual working condition of the radar transmitter, which will lead to the waste of transmitter power. In order to make full use of the transmitter efficiency, the PAPR of the weighting coefficient should be set to 1 in theory, but this makes the optimization problem non-convex. Therefore, the algorithm in this paper proposes an RAB method under PAPR constraint. This method not only has distinct practical physical meaning, but also is a convex optimization problem, that is, an analytical solution can be obtained.

Although this paper proposes an RAB algorithm under low PAPR, the method in this paper does not achieve a PAPR of 1, which is the next direction for future optimization.

5. Conclusions

To deal with LSS detecting under the background of strong interference from ground-moving targets, this paper proposed an SOCP-based transmit beam control algorithm to suppress the ground clutter, especially generated by the ground-moving targets. It can be observed from the transmitted beam pattern that the transmitted energy of the ground clutter can be controlled at a level far below the main lobe by using the algorithm proposed in this paper, so that the energy can irradiate a large area of the ground as little as possible. Thus, the LSS target will not be submerged by strong clutter when receiving echoes. With the consideration of the full use of emission power, the PARP was further implemented. The proposed algorithm set the amplitude coefficient upper bound to keep the PARP within limits. Simulation results show that it can achieve almost no energy loss in the desired direction. In the sidelobe direction, it can set the sidelobe level of the beam pattern as required. In terms of emission efficiency, it can achieve PAPR nearly equal to 1. Therefore, the algorithm can synthesize beams that meet the requirements of deep notch and low sidelobe in specific directions.

Although the PAPR constraint in this paper can improve the transmit power of the radar transmitter, it still does not achieve 100% utilization. Follow-up research can consider increasing the efficiency to 100%.