Mainlobe maintenance for FDA-MIMO radar Open Access

Adaptive digital beamforming (ADBF) is widely used in communications, sonar, radar, and other fields. When the target signal is mixed with interference and noise signals, the target signal and the interference are likely to overlap in the time and frequency domains, leading to difficulties in signal reconstruction.^1–4 Adaptive beamforming techniques can achieve spatial filtering by adjusting the weight vector of the emitted signal, so that the desired target signal can be collected and the effect of the interfering signal can be minimized. However, with the continuous development of electronics and the increasing complexity of the electromagnetic environment, interfering signals can enter the mainlobe of the radar at the same angle as the target signal, which leads to the inability of conventional multiple input mulitple output (MIMO) system radars to use angular dimensional filtering to counteract interference.⁵ At the same time, conventional beamforming algorithms produce mainlobe distortion and high sidelobe distortion in the face of interference in the distance dimension of the mainlobe. This problem is exacerbated when there are multiple disturbances in the mainlobe. In effect, the elimination of mainlobe interference comes at the cost of weakened target detection in multiple regions within the mainlobe. At this point, the mainlobe conformal capability of the ADBF becomes particularly important, as the radar’s ability to resist mainlobe interference is greatly enhanced if the interference can be suppressed without creating mainlobe distortion.

Many attempts have been made to suppress interference while preserving the shape of the mainlobe. A block matrix preprocessing (BMP) method is proposed in Ref. 6, which blocks the mainlobe interference so that the interference signal is removed before the adaptive processing, thus achieving some conformality a priori condition. Reference 7 performs a weighted compensation algorithm based on the BMP, which can avoid the mainlobe peak offset, but the algorithm complexity is high. Both Refs. 8 and 9 use Eigen-projection Matrix Processing (EMP) to solve the waveform distortion problem, but the former has a mainlobe beam shift; the latter uses a combination of EMP and covariance matrix reconstruction, which can improve the main beam shift of the EMP algorithm to some extent, and has some robustness, but both are only applicable to the case of a single mainlobe interference. Not only that, when the mainlobe interference and the sidelobe interference power are similar, the zero value of the adaptive array map of the EMP algorithm will disappear, leading to a serious performance degradation. In summary, the current difficulty in beam conformality lies in maintaining accurate primary beam alignment and outputting a radar beam with a robust waveform in the face of multiple intra-mainlobe interferences and only knowing the position of the target signal.

The deceptive target targeted in this paper is a special interference that cannot be identified in the distance domain within the mainlobe, which is difficult to be suppressed in the angular dimension by conventional phased array MIMO radar.¹⁰ Frequency diverse array (FDA) radar introduces a small frequency offset between each array element, which gives it the freedom in the distance dimension and thus can suppress the deceptive interference in the distance domain.¹¹ In order to more effectively solve the multiple distance dimension spoofing interference within the radar mainlobe, this paper uses the FDA-MIMO radar platform for anti-interference experiments, and the proposed algorithm is experimentally verified to be able to output a complete beam with a high signal-to-interference-noise ratio (SINR) while outputting an aligned mainlobe and a low sidelobe.

Consider a uniform linear array composed of M transmitting array element and N receiving array element, the carrier frequency of each transmitting array element signal is different, that is, f_m = f₀ + (m − 1)Δf, m = 1, 2, …, M, where f₀ is the reference frequency and Δf is the frequency increment between adjacent elements. Under the far-field condition, the transmitted signal of the m array element is

The signal transmitted from the array element m to the target can be expressed as

where r_n = r₀ − nd sin θ, r₀ is the distance from the first transmitting array element to the target, r_m is the element of m, θ is the incident angle, d is the distance between the array elements, and c is the speed of light. With a target at $(θ0r̂0)$⁠, the transmitted signal reaching the target can be expressed as

where s(t, θ, r₀) is the steering vector of the signal; $αθ0,r̂0$ is the weighted vector at the emission to better focus the radar beam at the target location and can be expressed as

and s(t, θ, r₀) can be expressed as

After the transmitted signal reaches the target, it is reflected and incident to the receiving array. The FDA-MIMO radar regime has a high degree of freedom and is able to distinguish between signal and interference in the distance dimension. In this paper, the system is used for signal processing, and its signal processing framework is shown in Fig. 1.

In order to eliminate the influence of the FDA time parameter¹² and activate the distance characteristics, the distance–angle correlation of the FDA is generally activated by adding multiple matched filters to each receiver array. The signal processing is designed in such a way that after the received signal is counted in each receiver array element, the received transmit signal of all channels is separated by N narrowband filters in each channel, and then, the separated echo data are rearranged, so that they become an NM × 1 vector, and thus, the rearranged data can be beam scanned with

The received data part is also weighted, and its weight vector w_r can be expressed as

Therefore, the weight vector of the signal of the n transmitting array element and received by the m receiving array element can be expressed as

After weighting, the total received signal output is

Referring to the signal processing model for FDA-MIMO radar presented in Sec. I, when a spoofed target is present, the received data after matched filtering can be represented in the vector form as

In (10), it is considered that the target signal, the deception interference signal, and the noise are all statistically independent, where y_s is the target signal and n is the noise vector, which is considered to be Gaussian white noise with variance. y_i is the received deception signal. The k received signal y(k) at the snapshot can be written as

The output of the beamformer Y(k) can be written as

where w^H is the beamforming weight vector and $w=[w1,w2,…,wMN]T$⁠.

The ideal Capon beamformer minimizes the output power of the array and maximizes the output (SINR) by receiving the desired signal without distortion. Its mathematical principle can be expressed as

where R_i+n is the interference-noise covariance matrix. The obtained weight vector can be expressed as

However, the interference-noise covariance matrix cannot be obtained directly under realistic conditions, and it is necessary to estimate the training data, that is, construct a sampling covariance matrix R_s, which can be expressed as

where K is the number of training samples.

This solution of replacing the interference noise covariance matrix with a sample covariance matrix is called the sample matrix inverse (SMI) based on the minimum variance distortionless response. However, the SMI algorithm still cannot guarantee beam integrity in the face of mainlobe interference, and its performance is severely degraded by the presence of interference, so a reconstruction of the covariance matrix is required. In Sec. II B, the proposed interference-noise covariance matrix reconstruction method is presented.

At high signal-to-noise ratios (SNRs), the interference-noise covariance matrix contains the desired signal, which is then treated as an interference signal and leads to serious degradation of the beamformer’s performance. To solve this problem, the interference-noise covariance matrix needs to be pre-processed to exclude any regions that may contain the desired signal. In this paper, a two-dimensional angle–distance plane is constructed and discretized into many position points with angle–distance coordinates. One of the regions containing only the target signal is taken as a known region, and then, integration is performed over the remaining complementary regions that do not contain the desired signal. Finally, the interference-noise covariance matrix that does not contain the target signal is obtained.

Assuming that the desired signal lies in the transmit–receive frequency domain Θ, which does not contain the spoofed target, the region Θ is considered to be known, which can be determined by a low-resolution angular scan method, and the width of the region can be represented by Θ_θ and Θ_R, respectively. The distribution of the desired, dummy, and actual target locations is shown in Fig. 2, where the interference is considered to be distance-dimensional spoofing interference.

The conventional Capon power spectrum is a high resolution spectrum without sidelobes and reflects the power distribution in the transmit–receive region. The power at different frequency points over the whole planar region can be represented by the Capon power spectrum, which can be combined with the power spectrum and the guide vector when reconstructing the interference noise covariance matrix. First, the two-dimensional power spectrum in the transmit–receive region can be expressed as

In the formula, $Pcaponθ0,r̂0$ represents the power intensity of the received data at the point of $θ0,r̂0$⁠. The complementary area of Θ is expressed as $Θ̄$⁠, that is, the part outside the area where the target point is located in Fig. 2 only contains spoofing targets in the transmit–receive area. Therefore, the Capon spectrum is integrated over $Θ̄$⁠, and the reconstructed interference covariance matrix R_i+n can be expressed as

It can be seen that the reconstructed interfering noise covariance matrix includes the information of the deceived target’s sending and receiving steering vectors, but excludes the steering vector information of the desired signal. For formula (17), it is difficult to directly integrate it. In order to facilitate the calculation, the entire region is discretized into U = U_θ × U_R points, the angle domain has U_θ points, and its angle interval is Δr, the distance domain has a point, and the distance interval is. At the same time, there are $UΘ̄$ points in the complementary area $Θ̄$⁠, each point corresponds to an angle and distance, and the position points contained in it are (⁠$θ0,1,r̂0,1$⁠), $(θ0,2,r̂0,2)$…, $(θ0,UΘ̄,r̂0,UΘ̄)$⁠. Then, replacing the process of integration with summation, Eq. (17) can be rewritten as

This method uses the reconstructed interference plus noise covariance matrix to exclude the effect of the desired signal components on the beamforming performance. Next, the idea of the worst-case performance optimization (WCPO) algorithm will be used to optimize the emission power vector.

The standard Capon beamformer requires distortion-free reception of the desired signal, but this is not absolutely guaranteed when the true guide vector is unknown and in an uncertain set. In order to set a distortion-free response constraint on the true guide vector, the WCPO algorithm ensures distortion-free reception of the desired signal by imposing a distortion-free response constraint on all possible guide vectors in the uncertain set. This can be expressed specifically as

where a_s is the desired steering vector of the true target, which contains the mismatch value e of the true target steering vector, and the norm of the mismatch value is bounded by a known constant ɛ, which can be expressed as $as=a+e,e≤ε$⁠; $Rε$ is the set to which the desired steering vector a_s belongs.

It can be seen that the set $Rε$ is a spherical indeterminate set, and for each $as∈Rε$⁠, $wHas≥1$ represents a non-convex quadratic constraint function, that is to say, the above problem is equivalent to a quadratic constraint problem with countless non-convex constraints, which is a nondeterministic polynominal (NP)-hard problem. To solve this problem, it is necessary to transform the constraints, convert it into a convex problem equivalently, and then use the interior point method to solve it. Within the set $Rε$⁠, for all $e≤ε$ and a_s, using the Cauchy–Schwarz inequality. There are

Therefore, the semi-infinite non-convex quadratic constraint problem in Eq. (19) can be rewritten as

It should be noted that the constraint condition w in Eq. (21) will not change the result when the phase rotation is performed at any angle. Therefore, the constraint condition can be added without loss of generality, and let w^Ha be a real number, that is,

Therefore, Eq. (21) can be re-expressed as

However, the reconstructed covariance matrix is not a positive semi-definite matrix and cannot be directly applied to the CVX toolbox in place of the sample covariance matrix. Therefore, it is necessary to introduce a quadratic penalty term to convert R_s into a positive definite matrix, which can be rewritten as

Then, through the Cholesky decomposition, the matrix R_s can be decomposed into the product of the lower triangular matrix and its conjugate transpose, which can be expressed as

Ultimately, the constraint problem can be written as

At this point, we have equivalently converted the general non-convex quadratically constrained quadratic programming problem into a second-order cone programming problem, which can be solved using a polynomial-time algorithm, the basic theory of the WCPO algorithm, and the optimization of the WCPO algorithm is completed by replacing the original disturbance noise covariance matrix with the reconstructed covariance matrix. This method is similar to other convex problems where the CVX toolbox can be used to find the optimal solution. In summary, the specific algorithmic steps of this paper are shown in Table I.

The complexity of the optimized WCPO algorithm mainly depends on the reconstruction process of the covariance matrix and the solution of the second-order cone programming problem. It should be noted that since the number of array elements is M, the dimension of the transmit–receive steering vector is MN × 1. In the covariance matrix reconstruction process, it includes two parts, the noise interference covariance matrix and the construction of the desired target covariance matrix. Obviously, the computational complexity is O((NM)³), which has the same order of complexity of the SMI algorithm, diagonal loading (DL) algorithm, and WCPO algorithm. Therefore, the computation time of this algorithm is almost the same as that of the traditional algorithm.

Simulation results are presented in this section to verify the performance of the proposed algorithm in the FDA-MIMO platform. According to the performance requirements and implementation conditions, the reference frequency of the radar is set to 3 GHz, and one target source and five interfering sources are set within the mainlobe. The sensor noise is assumed to be Gaussian white noise in both space and time, and the transmit and receive arrays are set to (12, 12), with the arrays spaced at half-wavelengths. Figure 3 compares the beam formation effects of the conventional WCPO algorithm, SMI algorithm, DL algorithm, and the proposed algorithm. The interference distribution is as follows: one angular dimension of deceptive interference in the mainlobe and one angular dimension of deceptive interference in the sidelobe. The locations of the signal, mainlobe interference, and sidelobe interference are (0°, 30 km), (2°, 30 km), and (20°, 30 km). The target signal is represented by a yellow circle, and the interference signal is represented by a red square. To better reflect the performance of algorithms, the power of each interference is the same in this paper. It can be seen that when there is a single interference in the mainlobe, each algorithm produces a mainlobe depression problem, and the mainlobe is mismatched. The proposed algorithm can keep the beam intact when the same power of interference is present in and outside the mainlobe at the same time.

Next, the performance of each algorithm in the presence of multiple mainlobe interferences is examined. As shown in Fig. 4, multiple disturbances were introduced in the distance and angle dimensions of the mainlobe, with the target and disturbance coordinates being (0°, 30 km), (2°, 30 km), (−2°, 30 km), (0°, 32 km), (0°, 34 km), and (0°, 36 km), respectively. It can be seen that after the introduction of multiple intra-mainlobe interferences, the performance of each algorithm receives a significant impact. With multiple interferences, the WCPO algorithm also improves the sidelobe, the mainlobe depression problem remains severe, and the mainlobe also produces a significant offset; the SMI algorithm, after the addition of multiple interferences, has a more severe mainlobe depression problem and the target angle even produces a zero trap, which will lead to the radar. The zero trap of the DL algorithm is not as severe as that of the SMI algorithm, but it also generates significant mainlobe offset and sidelobe elevation problems. The algorithm is highly robust to interference within the mainlobe.

In the next step, Fig. 5 compares the output SNRs of the above beamforming algorithms under different SNR conditions. The improved WCPO algorithm proposed in this paper is compared with the DL, WCPO, and SMI algorithms at an SINR of 0 dB and a snapshot number of 100. It can be seen that under the above conditions, as the SNR increases, the output performance of each algorithm decreases due to signal self-cancellation. The proposed algorithm increases steadily with the input SNR and does not affect the performance due to the increase in target signal energy. The output SINR is slightly better than the conventional algorithm at low input SNR, while the performance is far better than several other algorithms at the input SNR when more than −5 dB. Figure 6 compares the variation of the output SINR of several beamforming algorithms with the number of snapshots N. It can be seen that the performance of the algorithms proposed in this paper is relatively stable at different snapshot numbers, and there is only a small degradation in the performance of the algorithms at smaller snapshot numbers. Figure 7 depicts the comparison of the output SINR of the improved WCPO algorithm and the WCPO algorithm under different conditions with an SNR of 0 dB. It can be seen that the proposed algorithm has good robustness for the selection of ɛ. It is worth noting that for the WCPO algorithm, it is more difficult to choose the appropriate ɛ. Therefore, the algorithm in this paper has more obvious advantages in practical applications.

In this paper, an improved worst-case performance optimization algorithm is proposed based on the interference-noise covariance matrix reconstruction method. After theoretical study and simulation, the following conclusions are obtained:

1. The proposed algorithm is able to maintain the integrity of the waveform in the case of multiple interferences within the mainlobe, without the problems of elevated sidelobes and shifted mainlobes.

2. The covariance matrix reconstruction solves the drawback of traditional algorithms where the output SINR decreases as the input SNR increases. What is more, the proposed algorithm has a higher output SINR than the conventional algorithm for different input SNR and snapshot number conditions.

The authors declare that they have no known competing financial interests or personal relationships that could have influenced the work reported in this paper.

The data that support the findings of this study are available from the corresponding author upon reasonable request.