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

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

?E-mail:ybzhao@xidian.edu.cn

Received Jan.4，2021;Revision accepted Mar.31，2021;Crosschecked Jan.12，2022;Published online Apr.29，2022

Abstract: Low-angle estimation for very high frequency (VHF) radar is a difficult problem due to the multipath effect in the radar field，especially in complex scenarios where the reflection condition is unknown.To deal with this problem，we propose an algorithm of target height and multipath attenuation joint estimation. The amplitude of the surface reflection coefficient is estimated by the characteristic of the data itself， and it is assumed that there is no reflected signal when the amplitude is very small. The phase of the surface reflection coefficient and the phase difference between the direct and reflected signals are searched as the same part，and this represents the multipath phase attenuation.The Cramer-Rao bound of the proposed algorithm is also derived. Finally， computer simulations and real data processing results show that the proposed algorithm has good estimation performance under complex scenarios and works well with only one snapshot.

Key words:Low-angle estimation;Very high frequency(VHF)radar;Complex scenarios;Multipath effect;Height estimation

1 Introduction

In recent decades， the problem of low-angle estimation for very high frequency (VHF) radar has attracted a lot of attention (Barton， 1974; Bosse et al.，1991; Xu et al.， 2013， 2014) because of the multipath effect， especially in complex scenarios (Wang et al.，2014) where the reflection condition is unknown. The direct and reflected signals passing along the Earth’s surface enter the radar main beam together. It is assumed that only the specular reflection signal is included in the reflected path signals， while the diffuse reflection signal can be ignored (Bosse et al.， 1991).The target height is calculated from the elevation angle and other information about the target， which is known for the height estimation，so the height estimation is equivalent to the elevation estimation. The direct and reflected signals cannot be distinguished in the time， Doppler， and space domains， resulting in degradation of target height estimation performance.In particular， the performance for complex scenarios is greatly reduced.

Subspace and maximum likelihood (ML) algorithms are currently the main methods to overcome this problem. Subspace algorithms such as music(Schmidt， 1986) and esprit (Roy and Kailath， 1989)usually require more snapshots， and cannot directly deal with coherent signals. Spatial smoothing (Shan et al.， 1985; Pillai and Kwon， 1989) can help reduce the coherence between the signals by losing a certain aperture before angle estimation.ML algorithms(Ziskind and Wax， 1988) can directly process coherent signals and can work even with only one snapshot.The refined maximum likelihood (RML) algorithm(Lo and Litva， 1991) makes full use of the prior knowledge of geometric information and the surface reflection coefficient， and uses a composite guide vector instead of a conventional guide vector. The RML algorithm performs well in simple scenarios where the reflection condition is known precisely， but not well in complex scenarios. The reflected signal can also be eliminated (Ahn et al.， 2010; Park et al.，2014)， but the information about the space domain in the reflected signal is lost. In Takahashi et al. (2010)and Wang et al. (2016)， the height of the reflector is treated as an unknown parameter， and is searched together with the target height. Thus， the sensitivity to the fluctuation of the reflector is reduced. However，the error of the surface reflection coefficient is not taken into account. In complex scenarios， the composite guide vector cannot be accurately calculated， resulting in performance degradation and even failure of the above algorithms. To overcome these problems， we propose an algorithm of target height and multipath attenuation joint estimation. This study considers the case of low antenna height and a far field target. The reflected signal is thought of as symmetric with respect to the horizontal plane of the direct signal. The amplitude of the surface reflection coefficient is estimated by the characteristic of the data itself， and it is considered that there is no reflected signal when the amplitude is very small. The phase of the surface reflection coefficient and the phase difference between the direct and reflected signals are searched as the same part， and this is the multipath phase attenuation.An alternating projection (AP) is used to reduce computation by iterative processing (Ziskind and Wax，1988; Heylen et al.， 2016). The proposed algorithm fully exploits the characteristics of data in complex scenarios and improves the performance of target height estimation.

2 Multipath signal model based on 4/3 Earth

In this section，we give the multipath signal model based on 4/3 Earth and briefly review the surface reflection coefficient.

2.1 Multipath signal model

Electromagnetic waves do not travel along a straight line on the Earth.This is because of the effects of the Earth’s atmosphere and its curved surface(Ayasli，1986;Teti，2000;Griesser and Balanis，2003).However， it can be equivalent to traveling along a straight line with appropriate approximation. Using the 4/3 Earth model (Beckmann and Spizzichino， 1987;Mahafza，2013)，the effective Earth radius is

whereR0is real Earth radius. Based on the above approximation， the geometry for a 4/3 Earth model with multipath propagation is shown in Fig. 1. The linear array with theMelements receives the two paths of signal echoes from the different directions of arrival.One path returns directly from the target to the radar antenna， whereas the other path returns from the reflecting surface， whereθ1andθ2are the incident angles of the direct signal and the reflected signal，respectively.The height of the array radar center ishr.The distance between two adjacent array elements isd.The height of the target isht.The distance from the target to the array radar center isRd.

Fig.1 Geometry for a 4/3 Earth model with multipath propagation

After pulse compression and Doppler filtering，the signal received by array radar can be written as

whereαis the complex amplitude related to target characteristics，wis the composite array steering vector including direct and reflected signals，anda(θ)∈CM，called the array steering vector，is defined as

whereλis the working wavelength，andεis the attenuation coefficient，satisfying

whereρis the surface reflection coefficient. The details ofρwill be discussed later.φis the phase difference between the direct and reflected signals，and ΔRis the wave path difference between the direct and reflected signals.nis the Gaussian white noise vector with zero mean.This is not correlated with the target signals. The variance matrix of the noise vector isσ2IM.σ2represents the noise power of a single array element， andIMrepresents an identity matrix of sizeM×M. In addition， from the multipath geometry in Fig. 1， we give the calculation ofθ2and ΔR. The height of the target is

The angleΦis

where

The anglesΦ1andΦ2are

Then，R1andR2can be calculated by △OABand△OBC.We have

Similarly，the relationship betweenθ1andθ2can be calculated by △ABC.We have

From the above formula， ΔRandθ2can be easily obtained:

2.2 Surface reflection coefficient analysis

According to the electromagnetic scattering theory，the surface reflection coefficientρdepends on the working frequency， type of surface， polarization of the signal，surface roughness，etc.It can be expressed as(Lo and Litva，1991)

whereρo，ρs， andDare the smooth surface reflection coefficient， rough surface factor， and divergent factor，respectively. The smooth surface reflection coefficientρois generally divided into horizontal and vertical polarizations (Beckmann and Spizzichino， 1987;Mahafza，2013)，as

whereγis the glancing angle，as

andεcis the complex dielectric constant related to the reflection surface. The rough surface factorρsis given by(Lo and Litva，1991)

where

andσhis the root-mean-squared value of the reflection surface. The divergent factorDis given by (Lo and Litva，1991)

3 Target height and multipath attenuation joint estimation

For complex scenarios where the reflection condition is unknown，θ2，ρ， andφcannot be accurately calculated. This will result in performance degradation and even failure of conventional height estimation algorithms. Next， we analyze these three parameters in detail.

3.1 Incident angle of the reflected signal θ2

For a far-field target， the direct wave ray is approximately parallel to the reflected ray in the case of low antenna height. The reflection area is very close to the radar. Thus， the reflection on the radar side can be approximated by plane reflection. Then，we obtain the approximation of

Fig. 2 shows the difference between |θ2| andθ1against the target elevation angle. Consider a digital array radar equipped with a uniform linear array. Sethr=5 m andRd=100 km.It can be seen that the difference between|θ2|andθ1can be negligible.The above approximation is acceptable. Therefore， in the following discussion，we believe that Eq.(26)holds.

Fig.2 The angle difference against the target elevation angle

3.2 Surface reflection coefficient ρ

The surface reflection coefficientρcan be divided into amplitude and phase，as

whereρAdenotes the amplitude of the surface reflection coefficient， andψdenotes its phase. The preciseρAandψare difficult to obtain. For vertical polarization，ψvaries with the glancing angle， but for horizontal polarization，ψis close to 180°. In complex scenarios，an error inψresults in reduction of the target height estimation performance， but the approximate value of the phase for horizontal polarization can play a role in estimatingρA.Fortunately，horizontal polarization is often used for low-angle tracking. Thus， we discuss mainly the case of horizontal polarization.

Consider a digital array radar equipped with a uniform linear array withMelements where the distance between two adjacent array elements isd.Ignoring the effect of noise and assuming that there is no reflected signal， the signal received by array radar can be written as

which shows that there is no relationship between the signal and the reflection coefficient. Next， ignoring the effect of noise and assuming the existence of a reflected signal， the signal received by array radar can be written as

Then，we obtain the amplitude ofx:

whereX∈CM，Xidenotes theithelement ofX. Next，we takeyas a time series and obtain the spectrum ofyby the Fourier transform，as

whereyidenotes theithelement ofy，yi=0，i=M，M+1，…，N，Nis the number of samples of the Fourier transform， andF(k) denotes thekthelement ofF.Then，we obtain the maximum element ofFas

We callQthe multipath characteristic value(MCV)， which depends on the reflection coefficient，target elevation angle， and antenna height. Although the target elevation angle cannot accurately be known，the rough estimate of the elevation angle can be used as a priori knowledge. The details of rough estimation of the elevation angle will be discussed in Section 3.4. The error ofψand the antenna height are generally acceptable for the estimation ofρA.In addition，a small error inρAhas little effect on the performance of height estimation (Wang et al.， 2016). Then MCV depends only onρA， so we can establish the relationship between MCV andρAto estimateρA. More concretely，the estimate ofρAcan be written as

Consider a digital array radar equipped with a uniform linear array with 16 elements where the distance between two adjacent array elements is 0.5 m.Sethr=5 m，Rd=100 km，ht=3500 m，λ=1 m， andα=1.The amplitudes of the signals without noise against the number of array elements are shown in Fig. 3.From Fig. 3， it can be observed that the amplitude of the multipath signal varies with the number of elements. Next， setN=128.The spectrum amplitude ofyand the case where there is no reflected signal are shown in Fig. 4. There is a distinct difference in the maximum values of the spectrum amplitude between the two cases.The steps for estimation ofρAare summarized in Algorithm 1.

Fig.3 The amplitude of signal against the number of array elements

Fig.4 The spectrum amplitude against the number of array elements

3.3 Phase difference between direct and reflected signals

The phase difference between the direct and reflected signalsφis a very important parameter for height estimation.The antenna height can be searched as an unknown parameter to obtain the phase difference (Wang et al.， 2016). However， there is also an error in the phase of the surface reflection coefficientψ. From Eq. (5)， it can be seen that the effects ofφandψo(hù)n the signal are equivalent. Thus， we treat them as the same part， called multipath phase attenuation. Then， the composite array steering vector can be written as

whereφ=ψ-φis the multipath phase attenuation.

3.4 Target height and multipath attenuation joint estimation by alternating projection

Previously，we analyzed the key factors that affect the performance of height estimation under complex reflecting surface conditions. The main idea of the proposed algorithm is to estimate the amplitude of the surface reflection coefficient first， and then to conduct a joint search for target height and multipath phase attenuation by AP.

The log-likelihood function can be written as(Zhu et al.，2017)

where [·]Hdenotes the conjugate transpose operation，and

The ML estimates of the target elevation angle and multipath phase attenuation are the values corresponding to the largest peak in the amplitude of the function:

?

It requires a lot of computation to solve the above problem directly. The AP technique reduces computation by iterative processing. The technique solves the one-dimensional maximization problem by fixing all the other parameters at every iteration.By Ziskind and Wax (1988)， the algorithm must converge to a local maximum. The initial estimation method in Ziskind and Wax(1988)has good performance，which ensures that the local maximum is the global maximum.

First， for the initial estimation ofθ1， which is the rough estimation mentioned earlier， we use the method in Ziskind and Wax (1988) and take Eq. (26)into account. Then， we obtain the initial estimate ofθ1as

where

Second， we use the method in Algorithm 1 to estimate the amplitude of the reflection coefficient.Then， we start the iteration to estimate the elevation angle. The estimate ofφat the (k+1)thiteration is obtained by

The estimate ofθ1at the (k+1)thiteration is obtained by

Next， from the idea of AP， we repeat the above iteration process until the results of the two adjacent iterations are the same.

Note that whenρˉAis less than 0.2，ρ^Awill be set to zero. That is， there is no reflected signal in the target echo. Thus， the estimate ofθ1can be obtained directly by

where ||·||2denotes the 2-norm. Finally， we calculate the target heightby Eq. (6). The proposed algorithm is summarized in Algorithm 2.

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4 Cramer-Rao bound

Assume that there areLdiscrete samples of the target echo. Based on the above analysis， the signal received by array radar can be written as

The log-likelihood function can be written as

The Fisher information matrix is given by

where

real(·) denotes the real part， and * denotes the Hadamard product.Define

We obtain the Cramer-Rao bound (CRB) ofθandφas follows:

In what follows， we show the estimated rootmean-squared error (RMSE) of the target elevation angle of the proposed algorithm and the CRB against the signal-to-noise ratio(SNR).We define SNR as

Setρto 0.9ejπ. The other simulation conditions are the same as in Fig.4.In the case of a single snapshot，the estimated RMSE of the target elevation angle of the proposed algorithm and the CRB against SNR are shown in Fig. 5. The estimated RMSE of the target elevation angle is smaller as SNR increases. The higher the SNR， the closer the estimated RMSE is to the CRB.

Fig.5 RMSE of the target elevation angle against SNR

Fig. 6 RMSE of the target elevation angle (a) and target height (b) estimation against SNR with a smooth reflecting surface for 64 snapshots

Fig. 7 RMSE of the target elevation angle (a) and target height (b) estimation against SNR with a smooth reflecting surface for a single snapshot

5 Simulation and real data processing results

5.1 Simulation results

In different scenarios， the proposed algorithm is compared with the RML algorithm (Lo and Litva，1991)， ML-AP algorithm (Ziskind and Wax， 1988)，and height and reflection surface joint estimation(HRJE) algorithm (Wang et al.， 2016). The processing methods of these four algorithms for key parameters are shown in Table 1.

Table 1 The processing methods of four algorithms for key parameters

Consider a digital array radar equipped with a uniform linear array with 16 elements where thedistance between two adjacent array elements isd=0.5 m. Setλ=1 m，hr=5 m，andRd=100 km.The number of Monte Carlo trials is set at 200 in all these simulations.All the following simulations follow the above conditions. These simulations are performed using MATLAB R2016b on a PC with Intel i7-7700，3.60 GHz core frequency，and 8 GB RAM.

First，the ideal scenario with a smooth reflecting surface is considered. Setht=3500 m，ρ=0.9ejπ， and the target elevation angle is 1.67°. The RMSE of the target elevation angle and height against SNR of these algorithms mentioned earlier are shown in Figs.6 and 7，for 64 snapshots and a single snapshot，respectively. For the case of 64 snapshots， the RML algorithm， ML-AP algorithm， and proposed algorithm perform Doppler filtering before height estimation.From Figs.6 and 7，it can be seen that the RML algorithm is optimal and that the proposed algorithm performs better than the ML-AP algorithm and the HRJE algorithm in the ideal scenario.This is because the RML algorithm uses the most accurate information and the proposed algorithm tries its best to obtain the most accurate information. The running time of different methods for one Monte Carlo trial is shown in Table 2. The search region for the target elevation angle is [0.1°， 10°]， and the search interval is 0.01°.The RML algorithm has the minimum running time because of the one-dimensional search of a single target. In the ML-AP algorithm， the direct and reflected signals are treated as two targets rather than two parts of the composite guide vector. The ML-AP algorithm has the maximum running time because of multiple one-dimensional searches of two targets. The HRJE algorithm and the proposed algorithm take less time than the ML-AP algorithm because of multiple one-dimensional searches of a single target.The proposed algorithm takes less time than the HRJE algorithm， because the proposed algorithm does not need to calculate the phase difference between the direct and reflected signalsφ.

Table 2 Running time of different methods

Second， the complex scenario with an unknown reflecting surface is considered. Setρ=0.7ejπ. Assume the error ofhris 1 m， the error ofρAis 0.1， and the error ofρψis 10°.The radar receives only one snapshot. The RMSE of the target elevation angle and height against SNR of these algorithms mentioned earlier are shown in Figs.8 and 9，forht=3500 m andht=7000 m， respectively. The target elevation angles for Figs. 8 and 9 are 1.67° and 3.10°， respectively.The performance of the RML algorithm is improved little with the increase of SNR because of the use of inaccurate information， and the three other algorithms are not so sensitive to the error. The proposed algorithm performs better than the other algorithms.

Then a more complex scenario with an unknown reflecting surface is considered. Setht=3500 m andρ=0.7ejπ.Assume the error ofhris 3 m，the error ofρAis 0.3， and the error ofρψis 30°. The radar receives only one snapshot. The RMSE of the target height against the SNR of these algorithms mentioned earlier is shown in Fig. 10. The performances of the RML algorithm and the HRJE algorithm are improved little with the increase of SNR. The performance of the HRJE algorithm is greatly reduced because of the large amplitude error of the reflection coefficient.Compared with other algorithms，the advantages of the proposed algorithm are more obvious in the more complex scenario.

Fig. 8 RMSE of the target elevation angle (a) and target height (b) estimation against SNR with a complex reflecting surface for ht=3500 m

Fig. 9 RMSE of the target elevation angle (a) and target height (b) estimation against SNR with a complex reflecting surface for ht=7000 m

Fig. 10 RMSE of height estimation against SNR with a more complex reflecting surface for ht=3500 m

Finally， a special scenario with an unknown reflecting surface is considered. Suppose that there is an obstacle blocking the reflected signal from entering the antenna at some time. That is to say， it is uncertain whether the reflected signal is included in a certain echo. Setht=3500 m andρ=0.9ejπ. The radar receives only one snapshot. In the following simulation， whether the target signal contains the reflected signal is random in each Monte Carlo trial.The RMSE of the target height against the SNR of the algorithms is shown in Fig. 11.The conclusion is similar， and the performance of the proposed algorithm is also better than those of the others.

Fig. 11 RMSE of height estimation against SNR with an unknown multipath for ht=3500 m

For the proposed algorithm，the number of iterations to convergence decreases with the increase of SNR. For example， it usually takes four iterations to obtain the result for 0 dB， three iterations for 20 dB，and only two iterations for higher SNR.

5.2 Real data processing results

The real data under complex scenarios are applied to further verify the performance of the proposed algorithm. The array radar is equipped with a uniform linear array with 18 elements， where a distance between two adjacent array elements isd=0.55λ. The beam width is about 5.13°. The height of the reflecting surface and the surface reflection coefficient cannot be accurately obtained. The radar works in the meter wave band and receives only one snapshot. The real data are processed by pulse compression before height estimation.

The terrain scenario is a hilly area of great undulation. The target is an airplane that is first heading for the radar and then away from it. Figs. 12 and 13 show the target distance and azimuth change against frames， respectively. The distance between the airplane and radar first decreases from 235 km to 197 km and then increases to 210 km. Fig. 14 shows the processing results of the real data using RML， ML-AP，HRJE， and the proposed algorithm. The target elevation angle first increases from 1.3° to 2° and then decreases to 1.7°. The target height is about 9 km. It can be seen that the proposed algorithm has better performance in elevation angle and height estimations，and that the other algorithms have large estimation errors and less robustness. For the proposed algorithm， the number of iterations to convergence is two or three.In case of 2/3 of the frames，the elevation error of the proposed algorithm is less than 0.25°， that is， 1/20 of the beam width， and the height error is less than 1 km. In most frames， the elevation error of the proposed algorithm is less than 0.5°， that is， 1/10 of the beam width， and the height error is less than 2 km. Although the estimation error of the proposed algorithm is slightly larger at the 22ndand 61stframes，the overall advantage is still obvious.

Fig.12 Target distance against frames

Fig.13 Target azimuth against frames

Fig. 14 Estimation of the target elevation angle (a)， elevation angle error (b)， target height (c)， and height error (d)against frames

6 Conclusions

In this paper， the problem of low-angle estimation for VHF radar with complex scenarios is discussed.The reflected signal is thought of as symmetric with respect to the horizontal plane of the direct signal.The amplitude of the surface reflection coefficient is estimated by the characteristic of the data itself， and it is considered that there is no reflected signal when the amplitude is very small.The phase of the surface reflection coefficient and the phase difference between the direct and reflected signals are searched as the same part， which is the multipath phase attenuation.Alternating projection is used to reduce computation.

The proposed algorithm fully exploits the characteristics of data in complex scenarios and improves the performance of target height estimation.Computer simulations and real data processing results show the superiority of the proposed algorithm under complex scenarios， and the proposed algorithm works well with only one snapshot.

This paper discusses only the case of one target and one path reflection. In the case of multiple targets，the target echoes will affect each other.However，multiple targets can be separated in the range， Doppler，and angle dimensions by signal processing. If there are multiple targets in the same range unit， Doppler channel， and beam direction， the estimation performance will be degraded. In the case of multipaths，the signal model cannot match the actual data well.This will also lead to degradation of the estimation performance.

This paper discusses mainly the case of horizontal polarization. For vertical polarization， the amplitude of the surface reflection coefficient is still difficult to estimate.

This paper discusses the case where the absolute values of the incident angle and reflection angle are equal. In the case of high antenna height or a near field target， the absolute values of the incident angle and reflection angle are not equal. The reflection angle needs to be introduced as an unknown parameter，which greatly increases the complexity of the algorithm.

Contributors

Sheng CHEN designed the research. Sheng CHEN and Yongbo ZHAO processed the data. Sheng CHEN drafted the paper. Yili HU， Chenghu CAO， and Xiaojiao PANG helped organize the paper. Sheng CHEN and Yongbo ZHAO revised and finalized the paper.

Compliance with ethics guidelines

Sheng CHEN， Yongbo ZHAO， Yili HU， Chenghu CAO，and Xiaojiao PANG declare that they have no conflict of interest.