CHEN Sheng, ZHAO Yongbo, PANG Xiaojiao, HU Yili, and CAO Chenghu

National Laboratory of Radar Signal Processing, Xidian University, Xi’an 710071, China

Abstract: Low elevation estimation, which has attracted wide attention due to the presence of specular multipath, is essential for tracking radars. Frequency agility not only has the advantage of strong anti-interference ability, but also can enhance the performance of tracking radars. A frequency-agile refined maximum likelihood (RML) algorithm based on optimal fusion is proposed. The algorithm constructs an optimization problem, which minimizes the mean square error (MSE) of angle estimation.Thereby, the optimal weight at different frequency points is obtained for fusing the angle estimation. Through theoretical analysis and simulation, the frequency-agile RML algorithm based on optimal fusion can improve the accuracy of angle estimation effectively.

Keywords: frequency-agile, maximum likelihood, multipath signal, low-angle estimation.

1. Introduction

In recent decades, the problem of tracking low-angle targets has attracted wide attention [1-17] due to the presence of specular multipath. Radar receives the direct path signal and reflects the path signal passing through the Earth surface. It is assumed that the reflected path signal only contains the specular reflection signal [1-3]. The direct path signal and the specular reflection signal cannot be distinguished in the time domain, Doppler domain and space domain, resulting in a significant decrement in the performance of the target angle estimation.

To overcome this problem, the subspace algorithms and the maximum likelihood (ML) algorithm are good candidates. The subspace algorithms such as multiple signal classification (MUSIC) [18] and estimating signal parameter via rotational invariance techniques (ESPRIT)[19] require a higher signal-to-noise ratio (SNR) and snapshot numbers, and cannot directly dealing with coherent signals. Spatial smoothing [20,21] is usually used to solve coherent signals before angle estimation. Unfortunately, the accuracy of angle estimation is reduced via spatial smoothing due to the effective aperture loss.

The ML algorithm [22] can directly process coherent signals and can work even in the case of single snapshot.On the basis of the ML algorithm, the refined maximum likelihood (RML) algorithm makes full use of the prior knowledge of geometric information and surface reflection coefficient of multipath signal models [3]. The RML algorithm has attracted much attention due to its fascinating properties and potential improvement.

In [4], the RML algorithm with a quadrature correction term was proposed to diminish the impact of incoherent reflection signals. In [5-7], the height of the reflector was treated as an unknown parameter, and the height of the reflector and the target were searched, respectively.Thus the robustness to the fluctuation of the reflector was enhanced. The reflected signal was regarded as interference and was eliminated in [8,9], but the information about the space domain in the reflected signal was lost.

Frequency agility not only has the advantage of a strong anti-interference ability [23], but also can enhance the performance of tracking radars [10-14]. In [10,11],the intra-frame frequency-agile RML algorithm was studied to enhance the tracking performance. In [12,13], the intra-frame frequency-agile RML algorithm performed better in many aspects compared with the intra-frame frequency-agile eigenvector analysis algorithm using a multipath model (EAMM). In [14], an inter-frame frequencyagile RML algorithm, which minimized the mean square error (MSE) of angle estimation via adaptively adjusting the working frequency in the process of target tracking,was proposed. The theoretical MSE of angle estimation of the RML algorithm was analyzed, and the result showed that the MSE was affected by the target angle, the signalto-noise ratio (SNR) and the working frequency.

In [24], Swerling Ⅱ and Ⅳ targets were considered for the frequency-agile radar. The echoes of different frames have fluctuation, so the SNR of different frames is different. The echoes of different frames have different angle estimation accuracies due to different frequency points and SNRs. Fusing the estimation at different frequency points with the optimal weight is considered. To improve the accuracy of angle estimation for the frequency-agile radar, an intra-frame frequency-agile RML algorithm based on optimal fusion is proposed. An optimization problem,which minimizes the MSE of the angle estimation, is constructed. Then, the optimal weight, which is obtained by solving the optimization problem, is used for fusing different estimations to improve the accuracy of angle estimation.

This paper is structured as follows. Section 2 gives a multipath model of the low-angle target. Section 3 reviews the conventional intra-frame frequency-agile RML algorithm, and an intra-frame frequency-agile RML algorithm based on optimal fusion is proposed. In Section 4,the performance of the proposed algorithm and the conventional frequency-agile algorithms by computer simulation are presented. Finally, Section 5 concludes this paper.

2. Multipath signal model

The geometry for a smooth earth model with the multipath propagation is shown in Fig.1. The linear array withMelements receives two paths of signal echoes from different directions of arrival. One path returns directly from the target to the radar antenna, whereas the other pathreturns from thereflecting surface,whereθ1andθ2aretheincident anglesof the direct signal and the reflectedsignal, respectively. The height ofthearray radar centre ishr. The distance between two adjacent arrayelements isd. The height of thetarget isht. The distance from thetargetto thearray radarcentre isRd.

For the array radar system withKworking frequency points, the receiving signal at thekth frequency point can be written as

where λkis the working wavelength. εkis the fading coefficient satisfying

where ΔRis the wave path-difference between the direct signal and the specular reflection signal, and ρ is the surface reflection coefficient. The surface reflection coefficient ρ is generally expressed as ρ=ΓDρs, where Γ denotes the smooth surface reflection coefficient,Dis the divergence factor and ρsis the surface roughness factor[3], but it can be regarded as a constant for the scenario in this paper [15]. nk∈CM×1is the Gaussian white noise vector with zero mean, which is not correlated with the target signals. The variance matrix of the noise vector isand IMrepresents an identity matrix of sizeM×M. In addition, according to the multipath geometry in Fig. 1, θ2can be easily obtained by

3. Intra-frame frequency-agile RML algorithm based on optimal fusion

In this section, the conventional intra-frame frequencyagile RML algorithm is briefly reviewed. Then, its performance is analyzed. Based on the MSE of the angle estimation in [14], an intra-frame frequency-agile RML algorithm based on optimal fusion is proposed for lowangle estimation.

3.1 Conventional intra-frame frequency-agile RML algorithm

After the target is detected, the range of the target can be used for the angle estimation. The surface reflection coefficient, the height of the array radar centre and the range of the target are considered as the prior knowledge to improve the accuracy of angle estimation in the RML algorithm [14]. For the array radar system withKworking frequency points, the log-likelihood (LL) function of the signal model can be written as

where

The LL function can be simplified by removing the parameterswith the method in [10]. The frequency-agile RML estimation of the target angle is the one which corresponds to the largest peak in the magnitude of the function as follows:

3.2 Intra-frame frequency-agile RML algorithm based on optimal fusion

Considering the case of single frequency, the RML estimation of the angle θ1at thekth frequency point is as follows:

In the case of high SNR, the MSE of the angle estimation (8) using the RML algorithm [14] is given by

where

withdw,k(θ1) indicating the derivative vector of the array steering vectorwk(θ) with respect to the angle θ at θ1.The vectorswk(θ) anddw,k(θ1) are relative to the target angle and working frequency. Therefore, the MSE of the RML algorithm is affected by the target angle and the working frequency. Meanwhile, there exists the fluctuation in the echoes of different frames, so the SNR is variable. Namely, the echoes of different frames have different angle estimation accuracies.

To improve the accuracy of angle estimation, fusing the different estimations with the optimal weight is considered. The weight of the high accuracy should be larger,on the contrary, the weight of the low accuracy should be smaller. Thus, the new estimation can be obtained as

wherevkis the weight of thekth estimation. The MSE of the estimation is given that

The main idea of the frequency-agile RML algorithm based on optimal fusion is to minimize the MSE of (12).Therefore, the following optimization problem is constructed by

The optimal weight can be obtained by solving the above problem. The objective function can be derived that

We set the derivatives of (14) with respect toto zero, respectively, and we get

Then, we have

wherek,l=2,3,···,K. Substituting (16) in (15), we get

Then, the only stationary point can be obtained from(17), and we get

It is easy to see that (18) also satisfies the case ofk=1and the objective function value at the stationary point is less than the objective function value at boundary points.Thus, the objective function value at the stationary point is the minimum value. Then, the optimal weight is given that

Furthermore, supposing that the target angle can be obtained as a priori knowledge, the optimal fusion weight can be obtained to minimize the mean square error of the angle estimation. However, the target angle cannot accurately be known, otherwise it is not necessary to make this estimation. In fact, the angle estimation corresponding to each frequency can be used as a priori knowledge to calculate the optimal fusion weight.

Then, we can get the optimal estimation of the target elevation angle as follows:

The MSE of the optimal estimation is given that

Compared with the conventional intra-frame frequencyagile RML algorithm [10], the proposed algorithm fuses the estimation with the optimal weight to improve the accuracy of angle estimation. The proposed intra-frame frequency-agile RML algorithm based on optimal fusion is summarized below.

Step 1According to (8), use the single frequency RML algorithm to calculate the LL function, and obtain the angle estimation corresponding to each frequency from the LL function.

Step 2According to (20), estimateby the received data.

Step 3According to (21), calculate the optimal fusion weight.

Step 4According to (22), calculate the optimal estimation of the target elevation angle.

4. Simulation results

In this section, the accuracy of the proposed estimator is compared with that of the conventional RML algorithm[10] and the EAMM algorithm [13] based on frequencyagile. The effectiveness of the proposed algorithm is verified via computer simulation results. Consider a digital array radar equipped with uniform linear array with 16 elements where the distance between two adjacent array element(si)sd=0.5m.Sethr=5m,Rd=150km,ρ=0.9expjπ.Thenumber of Monte Carlotrialsissetas 2 000 in all these simulations. The echo models of Swerling Ⅱ and Ⅳ are considered for the simulations, because the fast fluctuating is accompanied by greater SNR changes relative to the slow fluctuating, advantages of the optimal fusion can be better extracted. Define the detection SNR of the direct signal at a single frequency point as

The EAMM algorithm requires a higher SNR and snapshot number than the RML algorithm. Thus we first give the simulation results of the proposed algorithm and the RML algorithm based on frequency-agile with single snapshot.

Assumethatθ1=3.5°, therearetwodiscretefreroot-mean-squared error (RMSE) of the target angle versus the SNR of the conventional RML algorithm and the proposed algorithm are shown in Fig. 2, for the echo models of Swerling Ⅱ and Ⅳ, respectively. From Fig. 2,it can be observed that the proposed algorithm outperforms the conventional RML algorithm. Furthermore, the theoretical results are consistent with the experimental results in a high SNR. However, the theoretical results are not consistent with the experimental results in a low SNR due to the approximate loss.

Fig. 2 RMSE of angle estimation against SNR

Assume the SNR is set as 10 dB. The RMSE of the target angle versus the target angle of the conventional RML algorithm and the proposed algorithm are shown in Fig. 3,for the echo models of Swerling Ⅱ and Ⅳ, respectively.

Fig. 3 RMSE of angle estimation against angle

From Fig. 3, it can be observed that the proposed algorithm outperforms the conventional RML algorithm. In addition, the RMSE is larger when the angle is small, because the phase of (3) gets closer to 180° as the angle decreases. This causes the direct signal and the specular reflection signal to cancel each other out, making the RMSE larger.

Assume the SNR is set as 10 dB and θ1=3.5°. There are 10 discrete frequency points, ranging from 288 MHz to 315 MHz with a frequency step of 3 MHz. The RMSE of the target angle versus the number of frequency points of the conventional RML algorithm and the proposed algorithm are shown in Fig. 4, for the echo models of Swerling Ⅱ and Ⅳ, respectively. From Fig. 4, it can be observed that the proposed algorithm outperforms the conventional RML algorithm. Furthermore, the theoretical results are consistent with the experimental results in case of more frequency points. For both algorithms, the performance improvement decreases when the number of frequency points is greater than 5.

Fig. 4 RMSE of angle estimation against frequency points number

Then, we give the simulation results of the proposed algorithm, the conventional RML algorithm and the EAMM algorithm based on frequency-agile with 16 snapshots. The proposed algorithm and the conventional RML algorithm perform pulse accumulation before the angle estimation. The other simulation conditions are the same as Fig. 2. The RMSE of the target angle versus the SNR of these three algorithms are shown in Fig. 5, for the echo models of Swerling Ⅱ and Ⅳ, respectively.

Fig. 5 RMSE of angle estimation against SNR

From Fig. 5, it can be observed that the proposed algorithm outperforms the other algorithms. The RMSE of the EAMM algorithm is larger than that of the conventional RML algorithm, especially in a low SNR.

For the case of multiple snapshots, it can be seen that the proposed algorithm mainly contains three computing procedures of pulse accumulation, LL function calculation, and optimal weight calculation. AssumeLis the number of snapshots andQis the number of spatial search grid. The computational complexity of the pulse accumulation isO(KLMlog(L)), the computational complexity of computing the LL function isO(QKM2) and the computational complexity of computing the optimal weight isO(K2M2). The computation complexity of these three algorithms are summarized in Table 1. Generally speaking, the computation complexity of the proposed algorithm is larger than the computation complexity of the conventional RML algorithm but smaller than the computation complexity of the EAMM algorithm.

Table 1 Computation complexity

5. Conclusions

In this paper, an intra-frame frequency-agile RML algorithm based on optimal fusion is proposed for lowangle estimation. The estimations for each frequency point are fused to minimize the MSE of angle estimation.Although the theoretical results are not consistent with the experimental results in a low SNR due to the approximate loss, compared with the conventional algorithm,the proposed algorithm still obtains better performance of angle estimation.