I. Introduction
The problem of low-angle target tracking in surface-reflected multipath environments has been extensively investigated in recent years [1–9]. The radar received signal from a low-angle target is mainly composed of a direct path signal and a surface-reflected path signal, which are coherent and fall in almost the same range bin. Furthermore, the angle-of-arrivals (AoAs) of these two signal components lie within the array antenna beamwidth, meaning that the angular separation between the two is small.
In [1–3], the maximum likelihood (ML) cost function was formulated as a function of two AoAs by exploiting the geometrical relation between the AoAs of the direct and specular paths. The AoAs were estimated using the one-dimension parameter searching method, also referred to as refined ML estimation. In [4], the received signal was transformed into a three-dimensional (3D) beamspace, after which an ML function was implemented in the 3D beam domain, referred to as 3D BDML. The solution maximizing the ML function in the 3D beam domain was obtained by finding two roots for the second-order polynomial. As a result, the estimates for the AoAs were obtained in a closed form, thus not requiring a grid search. However, to estimate the AoA of the target using 3D BDML, the target needs to be located within the mainlobe region of three adjacent beams.
Furthermore, the multiple signal classification (MUSIC) algorithm a high-resolution subspace-based algorithm and its variants, such as the estimation of signal parameters via the rotational invariance technique (ESPRIT) and root-MUSIC, can be combined with the spatial smoothing technique [10–12] to estimate the AoAs of two coherent signal components with small angular separation. However, subspace estimation requires high computational complexity, and spatial smoothing reduces the effective aperture of the array, resulting in the degradation of angular resolution. Furthermore, root-MUSIC involves a numerical root-finding process, and the degree of the polynomial in the root-finding stage is usually high when a large number of antenna elements are involved.
In [5], adaptive monopulse beamforming techniques were developed in which beamforming weights were designed to minimize the beamforming gain on the AoA of the specular path signal, resulting in the suppression of the specular multipath component. However, this approach may reduce the effective aperture of the array due to linear constraints on beamforming weights (for interference nulling). Furthermore, to estimate the AoA of a target through the monopulse beamforming technique, the target must be located in the mainlobe region (e.g., the 3-dB beamwidth) of the sum beam. To the best of the authors’ knowledge, grid search-free and boresight angle-independent angle estimation for low-angle estimation in a surface-reflected multipath environment has not yet been addressed in the literature.
In this paper, the low-angle target tracking problem is tackled through successive interference cancellation (SIC) and single-tone frequency (STF) estimation. Notably, STF estimation in noise is a classical but important signal processing problem [13, 14]. Responding to this concern, fast and accurate frequency estimators for a single complex sinusoid in white Gaussian noise and its variants were developed. In [15], STF was exploited to estimate the AoA of a single source impinging into a linear antenna array. In contrast to [15], we exploit both SIC and STF to estimate AoAs in a coherent multipath environment. Specifically, instead of estimating two AoAs (of the direct and specular paths) simultaneously, the coherent specular multipath component is directly canceled in the received radar signal through SIC, after which the AoA of the direct path is accurately estimated by applying STF estimation to the interference-suppressed signal. Table 1 summarizes and compares relevant papers pertaining to this topic for the convenience of readers. The contributions of this paper with regard to addressing the low-angle target tracking problem are listed below:
• We propose an angle estimation algorithm that incorporates both SIC and STF (AE-SICSTF) to first cancel the reflected path component in the received radar signal by estimating the nuisance parameter and then accurately estimate the AoA of the direct path using STF.
• To further improve estimation accuracy, we develop an iterative process in which SIC and STF are sequentially repeated. Drawing on the AoA of the direct path estimated via STF, a more accurate AoA of the specular path is obtained by exploiting the geometrical relation between the AoAs of the direct and specular paths. This AoA is then employed to suppress the specular multipath in the received radar signal more precisely during the next SIC process. For STF, we use the generalized weighted linear predictor (GWLP) [14], which has an iterative structure. Furthermore, since a naive combination of SIC and STF results in a double-loop iterative algorithm, we construct AE-SICSTF as a single-loop iterative structure to mitigate computational complexity.
• Through computer simulations, it was verified that the proposed AE-SICSTF outperforms conventional angle estimation methods (e.g., 3D BDML, refined ML, root-MUSIC with spatial smoothing, and adaptive monopulse beamforming). Furthermore, computational complexity analysis confirmed that the complexity of each iteration in AE-SICSTF is comparable to that in 3D BDML. However, it took less than 10 iterations to converge the root mean square error (RMSE) performance using AE-SICSTF, resulting in substantially less computing time than the refined ML and root-MUSIC algorithms.
The rest of this paper is organized as follows. In Section II, the signal model for a low-angle tracking radar system is introduced. In Section III, we briefly review two conventional low-angle estimation algorithms—3D BDML and refined ML. In Section IV, AE-SICSTF is proposed, in which the SIC process for specular path suppression and the STF process for target AoA estimation are iteratively processed. In Section V, we provide the simulation results and analyze the computational complexity of the proposed AE-SICSTF. Finally, Section VI concludes the paper.
II. System Model
Fig. 1 presents a low-angle tracking radar system with a uniform linear array of M antennas, in which M antenna elements are vertically spaced by half wavelength to estimate the elevation angle of a target flying at a low altitude. Here, hr and ht denote the altitude of the center of the antenna array and that of a flying target above the sea surface, respectively. The distance of the direct path from the target to the center of the antenna array is denoted as θ1, while the AoA of the specular path reflected onto the surface is referred to as θ2. In general, it is considered that the target is in the far field and the direct and specular path signals are in the same range bin. Accordingly, the nth snapshot of the received signal data samples,
x ( n ) ∈ C M × 1
where aM (ui) is an array response vector, expressed as follows:
Here, ui can be given as ui = sin(θi), where i = 1, 2; ci (n), with i = 1, 2, is the complex envelope associated with the direct/specular paths at the nth snapshot; and v(n) is a zero-mean spatially and temporarily white additive Gaussian noise vector with variance
σ n 2
Since the target is considered to be in the far field, the direct/specular paths have a similar path loss. Therefore, the relationship between c1 (n) and c2 (n) can be formulated as follows [2, 3]:
where ε is the relative fading coefficient, expressed as:
where Δψ is the phase difference between the direct and specular path signal components and ρ is the surface reflection coefficient, which is generally nearly equal to −1 for a small incident angle θ [1, 2]. Effectively, the received signal can be rewritten as follows:
From Fig. 1, the geometrical relation between θ1 and θ2 can be derived as follows:
Therefore, u2 can be expressed as a function of u1:
and (5) can be effectively exploited to suppress the specular multipath in the received radar signal.
III. Previous Works Relevant to Low-Angle Tracking Radar System
In this section, we briefly review two conventional low-angle estimation algorithms 3D BDML and refined ML [1–3].
1. 3D BDML
To reduce computational complexity, the received signal is transformed into a 3D beamspace using this algorithm. Specifically, the nth beamspace snapshot vector,
x B ( n ) ∈ C 3 × 1
where SM is the beamforming matrix, expressed as:
and
b ( u i ) = S M H a M ( u i ) u c = - h r R t
The ML estimation problem in beamspace can be formulated as follows:
where B(u1, u2) ≜ [b(u1), b(u2)]. Therefore, the least square solution for c(n) can be given by cLS(n) = (BT(u1, u2)B(u1, u2)−1BT(u1, u2)xB(n). Substituting cLS(n) into Eq. (8), the ML estimation problem can be rewritten as follows:
where
2. Refined ML
Drawing on Eq. (3), the log-likelihood function can be expressed as follows:
where w(u1, u2) = [aM(u1) + εaM (u2)]. By eliminating the nuisance parameters c1 (n) and
σ n 2
Drawing on Eq. (3), the log-likelihood function can be expressed as follows:
where
P w ( u 1 , u 2 ) = w ( u 1 , u 2 ) w H ( u 1 , u 2 ) w H ( u 1 , u 2 ) w ( u 1 , u 2 )
which requires a one-dimensional grid search over u1.
IV. Angle Estimation using SIC and STF
The 3D BDML is an attractive algorithm for addressing computational complexity because its dimension reduction capabilities through beamspace transformation offer a closed form solution that does not require a grid search process. However, to estimate the AoA of a target using 3D BDML, the target should be located within the mainlobe region of three adjacent beams. Meanwhile, refined ML involves performing a grid-searching process, which necessitates high computational complexity. To overcome these weaknesses of previous works, we propose the AE-SICSTF, which features two main processes—SIC and STF. In AE-SICSTF, the coherent specular multipath component in the received radar signal is canceled through SIC, after which the AoA of the direct path is accurately estimated by applying STF estimation to the interference-suppressed signal.
1. SIC Process for Specular Path Suppression
To suppress the specular multipath component, the complex envelope of c(n), n = 1, … , N as well as the AoA of the specular multipath need to be estimated. For this purpose, an initial estimation of the AoA of the specular multipath, û2, has to be obtained. Although this can be achieved by exploiting the low-complexity 3D BDML algorithm [4], it requires the target to be located within the mainlobe region of three adjacent beams. Therefore, we initialized the AoA of the specular path using the following method:
Drawing on Eq. (4), we arrive at the following equations for small θ1 and θ2:
Next, the bisector angle (
θ B = θ 1 + θ 2 2
We initialized û1 as any value satisfying û1 ≥ uB(= sin θB ≈ θB) and, for convenience, û1 and û2 are considered as follow:
Subsequently, we define the aggregated array response vector using the following equation:
where ∊̂ denotes an estimate of the aggregate reflection coefficient in Eq. (2), expressed as:
Therefore, the received signal in Eq. (3) can be modeled as x(n) = ŵc(n) + v(n), and the minimum mean square error (MMSE) estimate for c(n) can be given as:
Notably, with no statistical information on noise and the complex envelope c(n) (i.e.,
σ n 2 σ c 2
2. STF Process for Target AoA Estimation
The xc(n) in Eq. (22) can be modeled as follows:
where v′(n) is a composite of noise and residual interference. In other words, v′(n) = v(n) + vr(n), with vr(n)= εaM(u2)c(n) − ∊̂aM(û2)ĉ(n). By adopting a similar approach as in [15], the spatially-correlated signal y(k) can be formulated as follows:
(21)
where
ω 1 = 2 π d λ u 1
It is evident that Algorithm 1 has an iterative structure, while the initial frequency estimate of ω̂1 can be obtained from a conventional non-iterative STF estimation algorithm [13]. Since the STF is determined from the AoA of the direct path in the proposed system model, the initial frequency is obtained from the initially estimated û2 and Eq. (5). Furthermore, motivated by Algorithm 1, we propose an iterative algorithm in which the SIC and STF processes are repeated to improve estimation accuracy. Specifically, drawing on the AoA of the direct path estimated via STF, a more accurate AoA of the specular path is obtained by exploiting the geometrical relation in Eq. (5) between the AoAs of the direct and specular paths, which is then used to suppress the specular multipath in the received radar signal more precisely during the SIC in an iterative manner. Specifically, from Eq. (5), û2 can be obtained in terms of û1 as follows:
Based on the above discussion, the AE-SICSTF can be summarized as in Algorithm 2.
Fig. 2 illustrates the flowchart for AE-SICSTF (Algorithm 2).
Remark 1
Since beamforming weight is not utilized in AE-SICSTF the AoA is estimated by formulating a spatially correlated signal from the received signal using Eq. (21) it is a boresight angle-independent estimation method. If the 3D BDML algorithm is applicable (i.e., the target is located within the mainlobe region of three adjacent beams), we can coarsely estimate the initial AoA using 3D BDML and proceed with Algorithm 2 to obtain a more accurate estimate of the target AoA. See Fig. 4 for the convergence behavior with regard to the two initialization methods.
V. Simulation Results
Extensive computer simulations were conducted to validate the performance of the proposed AE-SICSTF algorithm in a low-angle sea environment. When conducting the simulations, we assumed that the target is moving toward the radar with constant velocity and height, with its range varying from 20,000 m to 2,000 m. The detailed simulation parameters are summarized in Table 2.
Notably, in Figs. 3–6, the Swerling I model [17, 18] is considered for the radar cross section (RCS) (σt), which is assumed to be constant during a single scan but fluctuates independently across scans according to the exponential distribution, expressed as follows:
where σ̄t is the mean of the RCS. Fig. 3 exhibits the average of the estimated angles as the range of the target varies from 20,000 m to 2,000 m, with SNR = 30 dB. It is observed that the proposed method with Niter = 1 tracks the target’s trajectory well, although there are occasional jumps in the estimated angles. As the number of iterations increases, the estimated angles converge to reach the true target angle. In Fig. 4, the convergence behaviors of the angles estimated using the AE-SICSTF algorithm, considering two different initialization methods (initialization using Eq. 14 and 3D BDML), are shown for Rt = {6000, 12000} m. It is noted that it takes less than 10 iterations to reach convergence, regardless of the target range and initialization method.
Furthermore, Fig. 5 presents the results obtained on evaluating the RMSE of the proposed AE-SICSTF at Niter = {1, 4, 8, 20, 30}, with the range of the target varying from 20,000 m to 2,000 m and SNR = 30 dB. For comparison, the RMSEs of the conventional 3D BDML [4] and root-MUSIC with smoothing [12] algorithms are also presented. It is evident that as the number of iterations (Niter) increase, AE-SICSTF exhibits better RMSE performance, achieving lower values than the conventional angle estimation methods.
Fig. 6 illustrates the RMSE of the proposed AE-SICSTF with Niter = {1, 8, 20, 30} for various SNRs. For comparison, the RMSEs of the conventional 3D BDML [4], the refined ML [2], the root-MUSIC with smoothing [12], and adaptive monopulse beamforming [5] algorithms are shown as well. Fig. 6 depicts that as the number of iterations (Niter) increase, the proposed AE-SICSTF exhibits better RMSE performance, attaining lower values than the conventional angle estimation methods. Specifically, when Niter = 1, the RMSE of the AE-SICSTF is comparable to that of the root-MUSIC with smoothing. Moreover, at Niter > 1, it outperforms the conventional schemes. In other words, it is apparent that by canceling the reflected multipath signal effectively the AoA of the direct path can be estimated more accurately. In addition, it took 20 iterations to reach convergence in RMSE performance.
Fig. 7 traces the RMSE of the proposed AE-SICSTF with Niter = {1, 8, 20, 30} for various surface reflection coefficients ρ. Here, the RCS is assumed to have a constant envelope with a random phase, as in [7]. From Fig. 7, it is evident that as ρ increases, the RMSE also increases. This implies that a larger ρ induces more interference, which is attributable to the surface-reflected multipath. Furthermore, when SNR = 30 dB, it took less than 10 iterations to reach a convergence in RMSE performance.
Since the proposed AE-SICSTF exploits the geometrical relationship between the AoAs of the direct and specular paths, as in Eq. (4), the estimation error of the target range may degrade its performance. To investigate the effect of estimation error on target range estimation, we exploited R̂t = ( = Rt (1 + δ)) instead of Rt in Eq. (4), where δ is a random variable uniformly distributed over [
- Δ 2 , Δ 2
1. Comparison of Computational Complexity
In this section, the computational complexity of the proposed AE-SICSTF in Algorithm 2 is analyzed. In Step 1 of Algorithm 2, Eq. (14) is exploited to obtain the initial AoAs, which required two floating operations (flops). In addition, O(MN) flops were required in Step 2 of Algorithm 2 and O(M2) flops were required in Steps 3 and 4. Therefore, the computational complexity of the proposed AE-SICSTF in Algorithm 2 can be expressed as O(NiterM × (min(M, N))), where Niter is the number of iterations required for the convergence of AE-SICSTF. Meanwhile, the beamspace transformation in Eq. (6) required 3MN flops, while the computation of eigenvectors for the 3 × 3 matrix required O(33) flops. Hence, the computational complexity of 3D BDML is considered as O(MN) flops. Since N ≤ M, the complexity of AE-SICSTF is found to be Niter times higher than that of 3D BDML.
Fig. 9 presents the results obtained on evaluating the computing time for varying M at SNR = 30 dB using MATLAB R2022a with Intel Core i9-12900K CPU@3.2 GHz and 64 GB RAM. For comparison, the computing time of the conventional 3D BDML [4], the refined ML [2], the adaptive monopulse beamforming [5], and the root-MUSIC with smoothing [12] algorithms are also presented. In refined ML, the number of grids for a one-dimension parameter search is set to 800, while the number of iterations for adaptive monopulse beamforming is set to 20. Meanwhile, since the computing time of refined ML primarily depends on the number of searching grids, it is less sensitive to the number of antennas. In contrast, for root-MUSIC with spatial smoothing, the degree of the polynomial in the root-finding stage is proportional to the number of antenna elements. As a result, it requires more computing time.
Fig. 9 shows that the computing time for the proposed AE-SICSTF increases with an increase in the number of antennas, M. In addition, AE-SICSTF with Niter = 8 is faster than that with Niter = 20, indicating lower computing time than refined ML and root-MUSIC with spatial smoothing.
In Table 3, we list the specific computing time for various estimation schemes when M = {20, 30}. We note that the root-MUSIC with spatial smoothing gives a computing time of 2.53 seconds, while AE-SICSTF with Niter = 8 takes 0.112 seconds for M = 20. Therefore, considering the RMSE results in Fig. 6, the proposed method is more accurate and 95.6% (= (2.53 − 0.112)/2.53) faster than the root-MUSIC with spatial smoothing.
VI. Conclusion
This paper addresses the low-angle target tracking problem using SIC and STF estimation methods. SIC and STF are jointly exploited to estimate AoAs in a coherent multipath environment. Specifically, instead of estimating two AoAs (of direct/specular paths) simultaneously, the coherent specular multipath component is directly canceled in the received radar signal through SIC, after which the AoA of the direct path is accurately estimated by applying STF estimation to the interference-suppressed signal. We developed a simple iterative approach that draws on the AoA of the direct path, estimated via STF, to obtain a more accurate AoA of the specular path by exploiting the geometrical relation between the AoAs of the specular multipaths in the received radar signal more precisely using SIC in an iterative way. Computer simulations verified that the proposed AE-SICSTF outperforms conventional angle estimation methods (e.g., 3D BDML, refined ML, adaptive monopulse beamforming, and root-MUSIC with spatial smoothing). Furthermore, although computational complexity analysis showed that the complexity of each iteration in AE-SICSTF is comparable to that in 3D BDML, the former required less than 10 iterations to converge the RMSE performance, thus necessitating considerably less computing time than the refined ML and root-MUSIC algorithms.
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