I. Introduction
A radar is a sensor that detects the threedimensional position and radial velocity of a target, determining the target’s position by measuring its range, azimuth, and elevation in a spherical coordinate system. In particular, multibeam radar typically estimates the azimuth and elevation from the ratio of the amplitudes of the received signal between the sum beam and the difference beam—a process also known as monopulse processing [1].
However, the monopulse method has limitations, especially in the presence of multipath interference, which occurs when a target’s elevation is low. In such cases, the received signal is usually a combination of two signals reflected directly and indirectly from the target, resulting in constructive or destructive interference, depending on the phase relationship. This interference can distort the received signal’s amplitude, leading to errors in monopulse elevation estimation. This phenomenon is referred to as the multipath phenomenon [2–4].
To address this issue, researchers applied the maximum likelihood algorithm to simultaneously estimate the elevation of the received signals from both the direct and indirect paths in a multipath environment. This algorithm, which is based on the antenna reception model, aims to solve the problem caused by multipath interference. Two previous studies have been conducted in this condition [5, 6]. Notably, these methods are based on zeroforcing precoding [7].
In addition, research aimed at estimating the twodimensional (2D) angle in array antennas has been performed. For instance, one study proposed a method to simultaneously estimate the direction of arrival (DOA) and polarization information between nearfield sources and multiinput multioutput (MIMO) sensors in the nearfield area [8]. Another study moved a coprime linear array antenna in the vertical direction to operate it as a virtual coprime planar antenna [9], as a result of which, unlike the coprime liner antenna, 2D angle estimation could be conducted. Furthermore, 2D departure and arrival angle estimations using an Lshaped sparse MIMO radar have also been proposed [10].
Additionally, investigations into the DOA of coherent signals have been conducted. For example, a recent study presented a model that reliably simulated DOA using channel information [11]. This model, as mentioned in its corresponding study, offers versatility by not imposing any limitations on the number of radar channels, array arrangements, or the types of beamformers employed.
Along the same lines, another study focused on tackling the issue of DOA estimation in two dimensions for coherent sources when employing an electromagnetic vector sensor array configured with uniform rectangular array geometry [12]. To address this challenge, the researchers introduced three innovative parallel factor methods to effectively eliminate rank deficiency in the source matrix by reorganizing the data, ultimately leading to enhanced accuracy in DOA estimation.
Researchers have also explored the use of artificial intelligencebased techniques, such as convolutional neural networks (CNN) [13] and deep learning [14], to mitigate the effects of multipath interference on elevation estimation. Notably, CNNs have been employed to determine the disregarding receive beam.
Moreover, a minimax optimization approach was employed to improve the robustness of the maximum likelihood estimation [15], the results of which indicated that the optimization process aims to minimize the maximum error observed in the steering vector.
However, the system considered in this paper is a multifunctional phasedarray radar that needs to perform multiple tasks simultaneously in real time. In this context, to meet the constraints of limited memory and computing power, an algorithm that ensures realtime performance is crucial.
A potential algorithm for estimating multipath elevation has been discussed in [6]. However, it is characterized by an issue in which the elevation estimation error increases rapidly as the target elevation approaches zero—a phenomenon also known as “divergence.” This issue was addressed by Kim et al. [16], who proposed a realtime algorithm that utilized the previously measured nearfield pattern of an antenna to estimate the elevation in the divergence region. However, this algorithm was designed for mechanically rotated radars and did not account for active electronicallysteered array (AESA) radars, whose antenna beam pattern is distorted by electronical steering for azimuth or elevation direction.
In response, Kwon et al. [17] introduced a novel algorithm to accurately estimate multipath elevation in the divergence region without relying on the antenna’s nearfield pattern. However, this algorithm is not applicable to realtime systems.
In this paper, the authors propose a novel algorithm to estimate multipath elevation for lowaltitude targets in realtime systems without encountering divergence or relying on premeasured antenna nearfield patterns.
The structure of this paper is outlined as follows: Section II discusses related multipath elevation estimation algorithms, Section III presents the proposed algorithm in detail, Section IV conducts simulations to compare the performance of the proposed method with that of the other approaches, and Section V provides the conclusions drawn regarding the proposed algorithm.
1. Notations
In this paper, the italicized lowercase letters (such as s), the italicized capital letters (such as M), and the bold lowercase letters (such as v), denote scalars, matrices, and vectors, respectively. Furthermore, v ^ denotes the estimation of vector v. Additionally, (M)^{H} and (M)^{−1} denote the conjugate transpose and inverse of matrix M, respectively. The s×s identity matrix is denoted by I_{s}, while v and v denote the L^{2} norm and absolute values, respectively.
II. Related Works
1. Double Null Algorithm
In a multipath environment, an antenna receives the signals reflected by a target from a direct path and an indirect path. In this context, 3D beam domain maximum likelihood (BDML) [6] estimation uses reception models in a multipath environment for elevation estimation. Considering θ_{d} as the elevation of the target in the direct path and θ_{i} as the elevation of a target in the indirect path, the double null algorithm serves to minimize the estimation error of both θ_{d} and θ_{i}. The reception model is shown in Fig. 1.
The model uses a linear antenna array with l=1,2,3,…,N_{l}, N_{l} being the number of antennas. Furthermore, the antenna is uniformly spaced by d, assuming 5 receive beams.
For signal reception modeling, a beamforming matrix W[N_{l}×5] and a steering matrix A[N_{l}×2] are used, where W is a function of N_{l}, d, and the spacing of the receive beams. Notably, 5 columns of W are considered due to the 5 receive beams. Meanwhile, A is a function θ_{d} and θ_{i}, with 2 columns of A referring to the two reception paths: θ_{d} and θ_{i}. Considering c as the complex envelope of the received signal corresponding to θ_{d} and θ_{i}, if the modeled received signal is x ^ , it can be expressed as follows:
Additionally, considering that the radar measurement is x, the optimization equation for minimizing the received signal estimation error is as follows:
Furthermore, the optimal solution for c, as mentioned above, can be obtained through zero forcing as follows:
Eq. (4) represents the energy function of the double null algorithm, where I_{5} is the identity matrix of size 5, corresponding to the 5 receive beams. In summary, the double null algorithm seeks to find the solution to θ_{d} and θ_{i} in the optimization problem.
In general, to use optimization algorithms, the entire energy function is generated, while the optimal value is found within the domain and constraints. However, in the case of the double null algorithm, 3D BDML must be performed to generate a single point value of the energy function. This means that the energy function can be obtained only when this process is performed for each point in the domain of the energy function.
The optimization algorithm should ideally be applied to the energy function generated by the abovementioned process. However, this is impossible due to the realtime limitations of the system considered in this study. Therefore, whenever the energy function value of each point is calculated, the minimum value is obtained by comparing it with the value of the previously calculated point. This is referred to as the double null algorithm.
The double null algorithm provides accurate elevation estimations in a multipath environment. It constructs the energy function using Eq. (4) and then finds the minimum value. This suggests that a unique minimum value must exist in the search area to make the algorithm converge to one value. Fig. 2 illustrates a scenario which has a unique minimum value. The blue area in Fig. 2 signifies the area with low energy, with the minimum value indicated using a white square.
The diagonal line in Fig. 2 appears when θ_{d} =θ_{i}. It is evident that the energy values in the diagonal line, originating from calculating the inverse matrix of the energy function, diverge.
Fig. 3 compares the elevation estimation performance of the monopulse method and the double null algorithm. At target elevations less than 3.15°, the multipath phenomenon occurs, and the received signal level is distorted. Notably, Fig. 3 confirms that the elevation estimation error of the monopulse method increases as elevation decreases.
Compared to the monopulse method, the elevation estimation of the double null algorithm is significantly more accurate. The double null algorithm can be described in Algorithm 1.
2. Divergence of the Double Null Algorithm
In general, when a single energy function has a unique global minimum, the double null algorithm converges to the global minimum.
However, if the target elevation approaches 0, a region with low energy appears in the form of a cross in the energy function map, exhibiting more than one local minimum. This leads to divergence in the double null algorithm, as shown in Fig. 4. Although the true target elevation is near 0°, the double null algorithm diverges and estimates the elevation to be way over 1°. As a result, the elevation estimation error becomes larger than the monopulse results in the region, as shown in Fig. 5.
3. Selective Double Null Algorithm
To overcome the divergence phenomenon arising from the double null algorithm, Kim et al. [16] proposed a selective double null algorithm. Considering D and x as the signal magnitude for each receive beam obtained by measuring the nearfield pattern and the target in a multipath environment, respectively, and β as the elevation for each measured nearfield point, the following discriminant equation can be formulated to determine the occurrence of the multipath phenomenon, as noted in [16]:
If the multipath effect does not occur, Q_{d} will appear to be close to 0. Conversely, as the multipath effect increases, Q_{d} will gradually approach 1. Notably, in the selective double null algorithm, the double null algorithm is only used when Q_{d} is greater than a certain value; the elevation is estimated using the monopulse method otherwise. The improved elevation estimation results obtained using the selective double null algorithm are shown in Fig. 6.
Fig. 6 confirms that the selective double null algorithm (Algorithm 2) shows better performance than the double null algorithm. However, it still suffers from the limitation of requiring the measurement of the antenna’s nearfield pattern, which makes it difficult for application in AESA radars.
III. Proposed Algorithm
1. LowAltitude Double Null Algorithm
In this paper, an algorithm capable of estimating the elevation in a divergence region without the need for previously measured antenna nearfield patterns is proposed, termed the lowelevation double null algorithm. The elevation range for the use of this method, with θ_{min} and θ_{max} being the minimum and maximum elevations, respectively, can be defined as follows:
Here, θ_{d} is a vector of θ_{d} ranging from θ_{min} to θ_{max}, with an interval of Δθ_{d}, while θ_{i} is symmetric to θ_{d}.
An example of the energy function that results in divergence is shown in Fig. 7.
The lowenergy region is distributed in a rotated Lshape, represented by the bluecolored region in Fig. 7. Even though the true target elevation is nearzero degrees, the double null’s elevation estimation results show 2.65° (indicated by the white square mark).
To solve this case of divergence, the following assumption is made: the index of the lowenergy region’s center is considered the elevation to be estimated. The proposed algorithm was then designed based on this assumption (Algorithm 3).
Algorithm 3
1. 
Find

2.  Save the minimum index i_{m}(θ_{i}) of E(θ_{d},θ_{i}) in the θ_{d} direction while performing Algorithm 1 
3.  Condition (C): The values of i_{m}(θ_{i}) differ by 1 or less for l times consecutively. 
4.  Calculate θ_{l} = θ_{d}(i_{m}(θ_{i})) 
5. 

Notably, the proposed algorithm is a modification of the double null algorithm. Experimentally, l is determined to be 40%–50% of the number of θ_{d} grids.
In this study, Algorithms 1 and 3 were both designed to be executed in the same main loop. After acquiring both θ_{D} and θ_{L}, the elevation was selected based on the flow described in Fig. 8. Notably, since the proposed algorithm has been developed to attain more accurate estimations of elevation when the target elevation approaches near zero, it is needed to use this algorithm conditionally.
2. Finding θ_{Thr} for the Proposed Algorithm
θ_{Thr} signifies the boundary value above which multipath effects become negligible. More specifically, θ_{Thr} is the starting point at which both the monopulse and multipath elevations begin to accurately estimate the target elevation. To calculate θ_{Thr}, the elevations of the monopulse method and the double null algorithm were compared to arrive at θ_{Thr} =3.15°, based on Fig. 3.
3. Selective Operation of the Elevation Estimation Algorithms
Fig. 8 presents a flowchart describing the elevation estimation process using the lowaltitude double null algorithm. The operational concept was developed to increase elevation estimation accuracy by selectively using elevation estimation values according to specific conditions. The first condition, shown at the top right corner of Fig. 8, involves a comparison of θ_{M} and the decision parameter θ_{Thr}.
If θ_{M} is larger than θ_{Thr}, θ_{M} is selected. This means that the target elevation is high enough to neglect the multipath effect. The value of θ_{Thr} is dealt with in the previous section.
The second condition, noted in the middle section of Fig. 8, pertains to checking for the divergence of θ_{D}. If Condition R is satisfied, it is determined that θ_{D} diverges. Thus, θ_{M} or θ_{L} is selected. On the other hand, if Condition R is not satisfied, θ_{D} is considered reliable. As a result, θ_{M} or θ_{D} is selected.
The third condition, presented at the bottom left of Fig. 8, is related to the reliability of θ_{L}. If Condition L is satisfied, θ_{L} is selected, and θ_{M} is selected otherwise.
The last condition at the bottom right corner of Fig. 8 pertains to the reliability of θ_{D}. If Condition D is satisfied, θ_{D} is selected, and θ_{M} is selected otherwise.
Each condition is summarized in Algorithm 4.
Algorithm 4
IV. Simulation
1. Simulation Settings
Simulations were performed considering a scenario where the lowaltitude target approached the radar, maintaining a constant altitude (Table 1).
The signaltonoise ratio (SNR) is based on the value at the farthest distance, which increased by the factor of R^{4} as the target got closer. A total of 25 scenarios were analyzed by performing 5 types of SNRs for the 5 types of altitudes. Finally, 1,667 scan results were acquired for each scenario, after which the root mean square errors (RMSEs) for the four algorithms were obtained to compare their performances.
The simulation parameters—surface dielectric constant, surface conductivity, surface roughness, and vegetation—were optimized to simulate a marine environment.
2. Results of Elevation Estimation
The performance results of SNR5 at Altitude1 are illustrated in Fig. 9, where the blue dotted line denotes the monopulse method, the red dotted line pertains to the double null algorithm, the light green solid line represents the selective double null algorithm, the purple solid line indicates the proposed algorithm, and the black solid line signifies the simulated target.
It is evident that under the most severe conditions, all θ_{D}, θ_{M} and θ_{L} values diverged by a distance of over 50 km. However, it can also be confirmed that the proposed algorithm achieved the most accurate results.
The comparison results for SNR5 at Altitude5 are shown in Fig. 10. In this scenario, the distorted result of θ_{M} resulting from the presence of the multipath effect is clearly revealed. In contrast, θ_{D} and θ_{L} show the most accurate results. In this case, the RMSE of the proposed algorithm is slightly higher than that of θ_{D}, which can be attributed to the fact that the proposed algorithm required 8 scans at the initial stage for calculating the average and standard deviation of the elevation values.
3. Performance Comparison of the Algorithms in Terms of RMSE
The RMSE results by altitude are shown in Figs. 11–15. Furthermore, a Cramer–Rao bound (CRB) was numerically calculated using the Monte Carlo simulation. Compared to the other algorithms, the proposed algorithm showed the results closest to the CRB.
In Figs. 11–15, the blue dotted line denotes the results of the monopulse methods, the red solid line with an “x” marker indicates the double null algorithm, the light green solid line with an “o” marker represents the selective double null algorithm, the purple solid line with a “hexagon” marker refers to the proposed algorithm, and the light blue solid line with a “square” marker is the CRB.
Except for one case, the proposed algorithm exhibited the best performance among all the compared algorithms.
The singular exception occurred in the SNR5 at Altitude5 scenario, illustrated in Fig. 10. To confirm the relative accuracy of the proposed algorithm, the relative RMSE value was defined in terms of the formula noted below:
In this formula, RMSE_{p} refers to the RMSE of the proposed algorithm, RMSE_{s} is the RMSE of the selective double null algorithm, RMSE_{m} indicates the RMSE of the monopulse method, and RMSE_{d} denotes the RMSE of the double null algorithm. Furthermore, RMSE_{p}_{,}_{s}, RMSE_{p}_{,}_{m}, RMSE_{p}_{,}_{d} are the relative RMSEs of the proposed algorithm in terms of the RMSEs of the selective double null algorithm, monopulse method, and double null algorithm, respectively. In each of the 25 scenarios, the average RMSE was calculated to compare the relative performance, the results of which are shown in Table 2. Compared to the selective double null algorithm, the proposed algorithm exhibited an improvement in RMSE by 25.6743% on average.
V. Conclusion
This paper proposed a lowaltitude double null algorithm and selective operation of elevation estimation algorithms to estimate the elevation of lowaltitude targets in a multipath environment. The performances of popular elevation estimation algorithms, including monopulse, double null, and selective double null, were compared to those of the proposed algorithm through simulation. In most cases, the proposed algorithm showed the best performance. However, an exceptional case was also observed, resulting from the initialization process that calculated the average and standard deviation from the elevation values using 8 initial scans. This limitation of the proposed algorithm should be addressed by conducting further research.