# Multiple Moving Targets Detection and Parameters Estimation in Strong Reverberation Environments

Authors: Ge Yu; Shengchun Piao
Journal: Shock and Vibration
 (2016)

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## Abstract

This paper considers the problem of multiple moving targets detection and parameters estimation (direction of arrival and range) in strong reverberation environments. As reverberation has a strong correlation with target echo, the performance of target detection and parameters estimation is significantly degraded in practical underwater environments. In this paper, we utilize two uniform circular arrays to receive plane wave of the linear frequency modulation signal reflected from far-field targets. On the basis of received signal, we build a variance matrix of multiple beams by using modal decomposition, conventional beamforming, and fractional Fourier transform (FrFT). We then propose a novel detection method and an estimation method of parameters based on the constructed image. A significant feature of the proposed methods is that our design does not involve any a priori knowledge about targets number and parameters of marine environments. Finally, we demonstrate via numerical simulation examples that the detection probability and the accuracy of estimated parameters of the proposed method are higher than the existing methods in both low signal-to-reverberation ratio and signal-to-noise ratio environment.


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## Body

## 1. Introduction

Targets detection and associated parameter estimation such as range, velocity, and direction of arrival (DOA) have various practical use applications in both military and civil. However, in shallow environments, reverberation as the main interference is generated by multiple reflection and diffusions of transmitted signal from rough boundaries interfaces (surface and bottom reverberation) [1]. The producing mechanism of reverberation means that it is highly correlated with target echo. As a result, it is very difficult to find a desired signal in the reverberation environment, especially when the signal-to-reverberation ratio (SRR) is relatively low. Therefore, we can draw a conclusion that reverberation results in high probability of false alarm and low detection probability in severe reverberation environment.In the case of reverberation, some methods are used to enhance SRR. Cheepurupalli and Konduri combined empirical mode decomposition (EMD) with adaptive boosting (AdaBoost), which is a learning algorithm, to process the echo signal in reverberation environment [2]. And as a result of AdaBoost, the accuracy and robustness of targets detection method are improved significantly. Similarly, the principle component inverse (PCI) algorithm has also been used to detect targets in sonar system. It is utilized to separate targets from the received signals and to eliminate the reverberation. With PCI algorithm, Ginolhac and Jourdain presented a method, which combined with beamforming, to resolve the rank problem and to get a better separation [1]. To make use of spatial information, matched-field processing (MFP) is developed to localize targets in range, depth, and azimuth [3]. It is a spatial-filtering technique and can be used in both passive and active sonar systems. Unfortunately, a stratified waveguide model of the acoustic environment is a necessary prior knowledge in this method [4, 5]. Considering utilizing Doppler shift features of moving target, fractional Fourier transform (FrFT) is widely applied to radial velocity and range estimation [6–8] and it performs well in low signal-to-noise ratio (SNR) condition, whereas the detection probability and accuracy of parameters estimation decrease along with the reduction of SRR.The methods which we have already listed above are applied by using traditional acoustic pressure sensor system. As acoustic vector array can be used to measure both pressure and particle velocity at a point in space [9, 10], in recent years, it has come to receiving particular attention. In particular, the vector sensor array has some characteristics which can bring many benefits such as high resolution and 4-dimensional information (pressure and particle velocities along the three orthogonal directions). Specifically, using a single vector sensor can estimate the azimuth elevation direction of the target [11].Spurred by the above pioneering works, we aim to develop an effective targets’ detection method and a parameters’ estimation method with active sonar system in severe reverberation environment. Two uniform circular sensor arrays are used in the proposed method. In the meanwhile, a linear frequency modulated (LFM) signal is used as the transmitted signal due to its high resolution and sensitivity of Doppler shift. The block diagram of the proposed method is shown in Figure1. For incorporating temporal information, spatial bearing distribution of received signal, and Doppler shift information of multiple moving targets into a new observed matrix, we can construct this new 3-dimensional (3D) matrix by utilizing modal preprocessing, fractional Fourier transform (FrFT), and conventional beamforming to divide the region of interest into N subregions in the signal preprocessing section. Then, in order to utilize the nonstationarity of reverberation, we calculate the changing trends of magnitudes in different subregions in the fractional Fourier domain to detect targets and to get the optimal values and the target subregion. With these optimal values, we can estimate the range and velocity of multiple moving targets. In addition, the target subregion can be exploited to suppress reverberation and get a high SRR value. Hence, the multiple signal classification method (MUSIC) can be applied to estimate the DOA of multiple targets in a high SRR environment.Figure 1
The block diagram of the proposed method for detection and parameters (range and direction of arrival) estimation of multiple moving targets.The main contributions of this paper are summarized as follows:(i)
Compared with the method in [6], the multiple moving targets can be detected in severe environment without any prior knowledge.(ii)
The proposed parameters estimation method can obtain significant performance in extremely low SNR and SRR environments.(iii)
With two uniform circular arrays, the proposed DOA estimation method can overcome the port and starboard ambiguity problem and solve the zero crossings at various different argument values in Bessel functions.(iv)
The detection probability and the precision of parameters estimation are not affected by the number of targets.The rest of the paper is organized as follows. The models of the received signals and active sonar system are given in Section2. In Section 3, signal preprocessing is introduced including modal preprocessing, Dolph-Chebyshev beamforming, and fractional Fourier transform. How to detect the moving target is presented in Section 4. Detailed discussions of parameters estimation are provided in Section 5. Simulation results and discussion are shown in Section 6. Conclusions and future work are drawn in the last section.We use the following notations in this paper:· denotes the vector norm. ·# represents the Moore-Penrose pseudoinverse. ·T denotes vector or matrix transpose.


## 2. Active Sonar System Model

In this paper, we have some assumptions to build the active sonar system model. First, we assume that the targets are located in the far field of the sensor array. Consequently, plane wave is used to represent target echo; second, the Gaussian noise at each sensor is stationary and temporally; third, the reverberation is locally stationary.With these assumptions, an active sonar system is established as shown in Figure2. The received array is made up of two uniform circular arrays. Particularly, each array has Q sensors with radius R1 and R2(R1>R2), respectively, and the qth sensor of ith array located at hqi≡hqi,φqi, where q=1,…,Q and i=1,2 in the polar coordinates system. Also, there are P targets located at up, with up=Dp denoting the distance between the pth target and origin of the array and ϕp is the incident angle of the pth target; u^p=up/up is a unit vector along the direction of the pth target.Figure 2
Geometry of two uniform circular arrays and targets.In reverberation environment, the received signalxqi(t) of the qth sensor in the ith array at time t can be described as(1)xqit=∑p=1Pspt+τpqi+nqit+rqit,where τpqi=hqi·up/c denotes the propagation delay of the echo from the pth target between the q sensor and the origin in the ith array and sp(t) is the pth target echo at the origin of the circular arrays. The expression of sp(t) is(2)spt=γpsσpt-τp,where σp is the time compression/stretching parameter; τp is the time delay; and γp is the range-dependent attenuation. nqi is additive white Gaussian additive noise at the (q,i) sensor which is assumed to be a zero-mean complex random process. rqi is reverberation which is considered as a sum of undesirable echoes issued from the transmitted signal received at the (q,i) sensor.As reverberation signal is dependent on the transmitted signal, it is closely bound up with target echo signal. Our scheme in this paper is to overcome the effects of reverberation and address the problem of estimating the range and DOA of multiple moving targets.


## 3. Signal Preprocessing

In the proposed method, LFM signal is used as transmitted signal due to the detection sensitivity and the high range resolution. However, some methods such as classical delay-and-sum beamforming and MUSIC are not applicable resulting from the wideband characteristics. In this section, we use three steps of signal preprocessing: Step1: modal preprocessing [6] is used to convert the echo signals into different spatial orthogonal modes; Step 2: we divide the spatial region of interest into several subregions to obtain the spatiotemporal information by virtue of Dolph-Chebyshev beamforming; Step 3: fractional Fourier transform (FrFT) is used to transform the received signal from frequency domain to fractional Fourier domain. With the help of these three signal processing methods, we finally reconstruct a 3-dimensional matrix, which contains spatial information, temporal information, and Doppler shift information of different moving targets.


### 3.1. Modal Preprocessing

In this subsection, we firstly use Fourier transform to convert the received signals from time domain to frequency domain. Then, (1) can be represented as(3)Xqiω=∑p=1PSpωejωτpqi+Nqiω+Rqiω,where Xqi(ω), Sp(ω), Nqi(ω), and Rqi(ω) are the Fourier domain expressions of xqi(t), sp(t), nqi(t), and rqi(t), respectively. With Jacobi-Anger expansion of a plane wave [12], ejωτpqi can be rewritten as(4)ejωτpqi=ejω/chqi·up=∑m=-MMJmωcRqiejmφqi+π/2-ϕp,where M=ω/cR1 is the truncation of the infinite series [13]. Thus, the signals received at the sensor arrays can be expressed as(5)Xω=X1ω+X2ω=JωASω+Nω+Rω,where Xi=[X1i(ω),X2i(ω),…,XQi(ω)]T(i=1,2), N=[N11+N12,…,NQ1+NQ2]T, and R=[R11+R12,…,RQ1+RQ2]T are all Q×1 matrices; other related matrices are described in the following.A is a (2M+1)×P frequency-independent steering matrix that can be written as(6)A=e-jMϕ1⋯e-jMϕP⋮⋱⋮ejMϕ1⋯ejMϕP.J is a Q×(2M+1) matrix shown as(7)J=KL1ω+L2ω,where K of size Q×(2M+1) satisfies(8)e-jMφ1+π/2⋯ejMφ1+π/2⋮⋮e-jMφQ+π/2⋯ejMφQ+π/2,Li is a diagonal matrix of size (2M+1)×(2M+1) defined as (9)J-MωRicJ-M+1ωRic⋱JMωRic,i=1,2.S is a P×1 vector which is composed of P target echoes that can be expressed as(10)S=γ1S1ω⋮γPSPω.As the location of each sensor is known in advance, the modal preprocessing [6] can be used to process the received signals at the sensor arrays by evaluating(11)Zω=J#ωX,where J#=(JTJ)-1JT is the pseudoinverse of J and Z is a (2M+1)×1 matrix.With the above modal preprocessing, we can transform wideband received signals at two circular arrays into signals at different modes. This is beneficial for the following conventional beamforming method.


### 3.2. Conventional Beamforming

Noting that the two concentric uniform circular arrays are omnidirectional and the modal components of the signals can be combined to form frequency invariant beamformers [14], we construct L beamformers to divide the observing region into L subregions in this subsection. Dolph-Chebyshev patterns are widely adopted when designing beamformer due to the narrow main-lobe and constant side-lobe level.We setw as the weight vector; the expression is(12)wθl=e-MjθlI-M,e-M-1jθlI-M-1,…,e-jθlI-1,2I0,ejθlI1,…,eM-1jθlIM-1,eMjθlIM,where θl is the look direction of the lth subregion and w is a 1×(2M+1) vector. Im is the weight of Dolph-Chebyshev pattern given in [15]. It follows from Dolph-Chebyshev beamforming that the output of the lth subregion can be expressed as(13)b~θl,ω=diag⁡wθlZω,where b~(θl,ω) is a (2M+1)×1 vector. The summation of the magnitude of the signals at 2M+1 modes is(14)bθl,ω=∑n=-MMb~n,forl=1,2,…,L.With beamformed output equation (14), we construct a new observation vector as(15)bω=bθ1,ω,…,bθL,ω.By beamforming, we can acquire the spatial distribution information of received signals. In the following, we employ fractional Fourier transform to obtain the temporal information and Doppler shift information of different targets.


### 3.3. Fractional Fourier Transform

Note that the fractional Fourier transform (FrFT), which is a generalization of the traditional Fourier transform, is a representation of signals using an orthonormal basis formed by LFM signals. Therefore, the certain angle FrFT of the LFM signals is a delta function. In general, the fractional Fourier transform of signalx(t) with transform angle α is defined as(16)Xαμ=∫-∞∞xtKαt,μdt,where Kα(t,μ) is the transformation kernel and satisfies(17)Kαt,μ=Γαej2πt2+μ2/2cot⁡α-μtcsc⁡αα≠nπδt-μα=2nπδt+μα=2n±1π,where Γα=1-jcot⁡α. In practical scenarios, discrete fractional Fourier transform (DFrFT) is used to deal with discrete received signals [16, 17]. With DFrFT, (16) can be naturally represented as(18)Xαu=Γαejπu2cot⁡α∫-∞∞ej2πcsc⁡αutejπt2cot⁡αxtdt.By substituting (18) into (15), the observing matrix can be given as(19)bαμ=bαθ1,μ,…,bαθl,μ,…,bαθL,μ.


## 3.1. Modal Preprocessing

In this subsection, we firstly use Fourier transform to convert the received signals from time domain to frequency domain. Then, (1) can be represented as(3)Xqiω=∑p=1PSpωejωτpqi+Nqiω+Rqiω,where Xqi(ω), Sp(ω), Nqi(ω), and Rqi(ω) are the Fourier domain expressions of xqi(t), sp(t), nqi(t), and rqi(t), respectively. With Jacobi-Anger expansion of a plane wave [12], ejωτpqi can be rewritten as(4)ejωτpqi=ejω/chqi·up=∑m=-MMJmωcRqiejmφqi+π/2-ϕp,where M=ω/cR1 is the truncation of the infinite series [13]. Thus, the signals received at the sensor arrays can be expressed as(5)Xω=X1ω+X2ω=JωASω+Nω+Rω,where Xi=[X1i(ω),X2i(ω),…,XQi(ω)]T(i=1,2), N=[N11+N12,…,NQ1+NQ2]T, and R=[R11+R12,…,RQ1+RQ2]T are all Q×1 matrices; other related matrices are described in the following.A is a (2M+1)×P frequency-independent steering matrix that can be written as(6)A=e-jMϕ1⋯e-jMϕP⋮⋱⋮ejMϕ1⋯ejMϕP.J is a Q×(2M+1) matrix shown as(7)J=KL1ω+L2ω,where K of size Q×(2M+1) satisfies(8)e-jMφ1+π/2⋯ejMφ1+π/2⋮⋮e-jMφQ+π/2⋯ejMφQ+π/2,Li is a diagonal matrix of size (2M+1)×(2M+1) defined as (9)J-MωRicJ-M+1ωRic⋱JMωRic,i=1,2.S is a P×1 vector which is composed of P target echoes that can be expressed as(10)S=γ1S1ω⋮γPSPω.As the location of each sensor is known in advance, the modal preprocessing [6] can be used to process the received signals at the sensor arrays by evaluating(11)Zω=J#ωX,where J#=(JTJ)-1JT is the pseudoinverse of J and Z is a (2M+1)×1 matrix.With the above modal preprocessing, we can transform wideband received signals at two circular arrays into signals at different modes. This is beneficial for the following conventional beamforming method.


## 3.2. Conventional Beamforming

Noting that the two concentric uniform circular arrays are omnidirectional and the modal components of the signals can be combined to form frequency invariant beamformers [14], we construct L beamformers to divide the observing region into L subregions in this subsection. Dolph-Chebyshev patterns are widely adopted when designing beamformer due to the narrow main-lobe and constant side-lobe level.We setw as the weight vector; the expression is(12)wθl=e-MjθlI-M,e-M-1jθlI-M-1,…,e-jθlI-1,2I0,ejθlI1,…,eM-1jθlIM-1,eMjθlIM,where θl is the look direction of the lth subregion and w is a 1×(2M+1) vector. Im is the weight of Dolph-Chebyshev pattern given in [15]. It follows from Dolph-Chebyshev beamforming that the output of the lth subregion can be expressed as(13)b~θl,ω=diag⁡wθlZω,where b~(θl,ω) is a (2M+1)×1 vector. The summation of the magnitude of the signals at 2M+1 modes is(14)bθl,ω=∑n=-MMb~n,forl=1,2,…,L.With beamformed output equation (14), we construct a new observation vector as(15)bω=bθ1,ω,…,bθL,ω.By beamforming, we can acquire the spatial distribution information of received signals. In the following, we employ fractional Fourier transform to obtain the temporal information and Doppler shift information of different targets.


## 3.3. Fractional Fourier Transform

Note that the fractional Fourier transform (FrFT), which is a generalization of the traditional Fourier transform, is a representation of signals using an orthonormal basis formed by LFM signals. Therefore, the certain angle FrFT of the LFM signals is a delta function. In general, the fractional Fourier transform of signalx(t) with transform angle α is defined as(16)Xαμ=∫-∞∞xtKαt,μdt,where Kα(t,μ) is the transformation kernel and satisfies(17)Kαt,μ=Γαej2πt2+μ2/2cot⁡α-μtcsc⁡αα≠nπδt-μα=2nπδt+μα=2n±1π,where Γα=1-jcot⁡α. In practical scenarios, discrete fractional Fourier transform (DFrFT) is used to deal with discrete received signals [16, 17]. With DFrFT, (16) can be naturally represented as(18)Xαu=Γαejπu2cot⁡α∫-∞∞ej2πcsc⁡αutejπt2cot⁡αxtdt.By substituting (18) into (15), the observing matrix can be given as(19)bαμ=bαθ1,μ,…,bαθl,μ,…,bαθL,μ.


## 4. The Proposed Detection Method

In the previous section, we transform the received signals at different sensors into a new vector which contains spatial-temporal and Doppler shift information of moving targets in the fractional Fourier domain. In this section, we utilize the Doppler shift characteristic of moving targets to propose a novel detection method in a severe reverberation environment.In active sonar system, LFM signal as the transmitted signal can be written as(20)yt=ej2πχt2+f0t+ξ,where χ is the rate, f0 is the initial frequency, and ξ is the initial phase. In discrete fractional Fourier transform, the optimal transform angle α~ is dependent on rate χ of LFM signal. The interrelationship between rate and optimal angle is(21)α~=-arctan⁡Fs2/Ω2χ,where Ω is the total number of time samples and Fs is the sampling rate. In practice, when we consider the moving targets, χ is varying due to Doppler shift effect. Hence, it is impossible to know the optimal transform angle in advance. Fortunately, an advantage of Doppler shift effect is that different target echoes have different χ values as well as optimal transform angles. Further, the energy of a signal is centralized on the certain LFM basis and its maximum amplitude will be at a particular μ. Therefore, with the above two characteristics, the multiple target echoes can be conveniently separated in a nonreverberation environment [6]; consequently, it is easy to get the optimal α~ and u~ values in fractional Fourier domain [8].However, the strong reverberation interference signal results in a low SRR as illustrated in Figure3. In this case, using Doppler shift is not enough to detect targets. Therefore, we consider temporal and spatial information to detect multiple moving targets with Doppler shift information which makes the optimal transform angle α and location μ change in fractional Fourier domain [6]. When αp and μp are the optimal transform angle and location of the pth target in the lp subregion, we have(22)bαpθlp,μp≫bαpθl,μp,∀lp≠l.This is illustrated using an example in Figure 4 with the plots of different bαp(μj)(j=1,…,6), where μp=μ4 is the optimal value and θlp=θ30 is the optimal direction. It is easy to observe that the fluctuation of curve μp=μ4 is much more considerable than any other curve. In other words, the variance of vector bαp(μp) is most significant of all. Consequently, the variances of b can be used to measure the fluctuation degree by evaluating(23)Vα,u=var⁡bαu,u∈-FsT2,FsT2,α∈-0.2,-0.1.Figure 5 shows the simulation result of the variance calculation of all subregions. It can be clearly found that the target could be detected at the condition of low SRR and SNR. The number of targets is equal to the number of spikes. It means that the optimal (αp,μp) values of different targets can be acquired according to the locations of spikes.Figure 3
The fractional Fourier transform of received signal in SNR = −15 dB and SRR = −15 dB environment.Figure 4
The magnitudes of 6 curves ofbαp(μp) in SNR = −15 dB and SRR = −15 dB.Figure 5
The variance values of different subregions in fractional Fourier domain. The environment is SNR = −15 dB and SRR = −15 dB. There are two moving targets: one is located at[1450,1550] m with velocity being [4,5] m/s; the other is located at [1600,1500] m with velocity being [5,4.5] m/s.


## 5. The Proposed Parameters Estimation Method

In this section, we intend to develop a method for the estimation of velocity, range, and direction of arrival (DOA) with the knowledge of the number of targets and the optimal values(αp,up) of each target.


### 5.1. Range Estimation

The transmitted signal with a rectangular window of durationΥ can be written as(24)st=ytrect⁡t-Υ/22.From (24), the received echo of the pth target can be obtained as(25)xpt=sσpt-τp,where τp is the delay parameter and σp is the compression/stretching parameter of the pth target. By substituting (18) into (25), the received signal in fractional Fourier domain is expressed as(26)Xpu=Γαej2πu2/2cot⁡α+χτp2-f0τp+ξ∫t2t1ej2πχσp2+1/2cot⁡αt2ej2πf0σp-2χσpτ-ucsc⁡αtdt,where t1=(Υ+τ)/σ and t2=τ/σ, respectively. With (21) and optimal transform angle αp, the rate χp of the pth target echo can be easily calculated by(27)χp=Fs22tan⁡-αpΩ.It then follows that(28)σp=χpχ.According to [12], we can decompose ∫ej(px2+2qx)dx as(29)∫ejpx2+2qxdx=π2pe-jq2/pCpx+qp+jSpx+qp,where(30)Cx=∫0xcos⁡πt22dt,Sx=∫0xsin⁡πt22dt.Under (29) and (30), (26) can be rewritten as(31)Xpu=Γαej2πu2/2cot⁡α+χτ2-f0τ+ξπ2he-jk2/hCht+kh+jSht+kht2t1,where h=2πχσp2Ω/Fs2+0.5cot⁡α, k=πf0κσp-2χκσpτp-ucsc⁡α, and g(u)=ejπu22χσp2cot⁡α-1+2ucsc⁡αf0σp-2χσpτp-f0σ-2χσpτp2/2χσp2+cot⁡α and κ=T/Fs is the scaling parameter.Note that, in (31), the center values μ~ of main lobe in the real part are determined by the Fresnel cosine integral C(x), which crosses zeros only at x=0 as shown in Figure 6. Therefore, we can get the zeros crossing, respectively, at(32)μ1=ht1+πκσpf0-2χτpπcsc⁡αp,(33)μ2=ht2+πκσpf0-2χτpπcsc⁡αp.As a consequence, we can get the center value μ~ as(34)μ~=μp=ht1+t2+2πκσpf0-2χτpπcsc⁡αp.By substituting the estimates μp, αp, and σp into (34), the delay parameter of the pth target can be obtained as(35)τp=2πf0κσp-πμpcsc⁡αp+ht1+t24πκχσp.Furthermore, the pth target distance Dp is given as(36)Dp=τpc2.Figure 6
The main lobe formed by the subtraction of the two Fresnel cosine integrals in the FrFT of a received signal.


### 5.2. Direction of Arrival (DOA) Estimation

In this subsection, we propose a DOA estimation method using the optimal subregionlp and optimal values (αp,up). By substituting (18) with the optimal values αp and θlp into (13), the subregion lp in the fractional Fourier domain can be written as b~αp(μ,θlp). Then, a bandpass filter with center frequency μp is applied to reduce the interference among the targets and effectively increase the SRR and SNR values. As a result, we can get a new vector b~αp,up(μ,θlp). Finally, utilizing the property of the FrFT, we can transform signal from b~αp,up(μ,θlp) to b~αp,up(ω,θlp).As the transmitted signal is wideband, the truncation value varies along with frequency. We assume that the minimal truncation isM1=2π(f0)/cR1 and the maximal truncation is M2=2π(f0+B)/cR1, where B is the bandwidth of the transmitted signal. That is to say, we have M2-M1+1 subbands. Each subband vector can be expressed as b~αp,upm(ω,θlp), for m=M1,…,M2, and the dimension of each subband vector is (2m+1)×1. Then, according to [18], we can calculate the pth target direction ϕpm with the mth frequency subband b~αp,upm(ω,θlp) of the pth target. With the above development, the DOA of the pth target can be obtained as(37)ϕp=1M2-M1+1∑m=M1M2ϕpm.


## 5.1. Range Estimation

The transmitted signal with a rectangular window of durationΥ can be written as(24)st=ytrect⁡t-Υ/22.From (24), the received echo of the pth target can be obtained as(25)xpt=sσpt-τp,where τp is the delay parameter and σp is the compression/stretching parameter of the pth target. By substituting (18) into (25), the received signal in fractional Fourier domain is expressed as(26)Xpu=Γαej2πu2/2cot⁡α+χτp2-f0τp+ξ∫t2t1ej2πχσp2+1/2cot⁡αt2ej2πf0σp-2χσpτ-ucsc⁡αtdt,where t1=(Υ+τ)/σ and t2=τ/σ, respectively. With (21) and optimal transform angle αp, the rate χp of the pth target echo can be easily calculated by(27)χp=Fs22tan⁡-αpΩ.It then follows that(28)σp=χpχ.According to [12], we can decompose ∫ej(px2+2qx)dx as(29)∫ejpx2+2qxdx=π2pe-jq2/pCpx+qp+jSpx+qp,where(30)Cx=∫0xcos⁡πt22dt,Sx=∫0xsin⁡πt22dt.Under (29) and (30), (26) can be rewritten as(31)Xpu=Γαej2πu2/2cot⁡α+χτ2-f0τ+ξπ2he-jk2/hCht+kh+jSht+kht2t1,where h=2πχσp2Ω/Fs2+0.5cot⁡α, k=πf0κσp-2χκσpτp-ucsc⁡α, and g(u)=ejπu22χσp2cot⁡α-1+2ucsc⁡αf0σp-2χσpτp-f0σ-2χσpτp2/2χσp2+cot⁡α and κ=T/Fs is the scaling parameter.Note that, in (31), the center values μ~ of main lobe in the real part are determined by the Fresnel cosine integral C(x), which crosses zeros only at x=0 as shown in Figure 6. Therefore, we can get the zeros crossing, respectively, at(32)μ1=ht1+πκσpf0-2χτpπcsc⁡αp,(33)μ2=ht2+πκσpf0-2χτpπcsc⁡αp.As a consequence, we can get the center value μ~ as(34)μ~=μp=ht1+t2+2πκσpf0-2χτpπcsc⁡αp.By substituting the estimates μp, αp, and σp into (34), the delay parameter of the pth target can be obtained as(35)τp=2πf0κσp-πμpcsc⁡αp+ht1+t24πκχσp.Furthermore, the pth target distance Dp is given as(36)Dp=τpc2.Figure 6
The main lobe formed by the subtraction of the two Fresnel cosine integrals in the FrFT of a received signal.


## 5.2. Direction of Arrival (DOA) Estimation

In this subsection, we propose a DOA estimation method using the optimal subregionlp and optimal values (αp,up). By substituting (18) with the optimal values αp and θlp into (13), the subregion lp in the fractional Fourier domain can be written as b~αp(μ,θlp). Then, a bandpass filter with center frequency μp is applied to reduce the interference among the targets and effectively increase the SRR and SNR values. As a result, we can get a new vector b~αp,up(μ,θlp). Finally, utilizing the property of the FrFT, we can transform signal from b~αp,up(μ,θlp) to b~αp,up(ω,θlp).As the transmitted signal is wideband, the truncation value varies along with frequency. We assume that the minimal truncation isM1=2π(f0)/cR1 and the maximal truncation is M2=2π(f0+B)/cR1, where B is the bandwidth of the transmitted signal. That is to say, we have M2-M1+1 subbands. Each subband vector can be expressed as b~αp,upm(ω,θlp), for m=M1,…,M2, and the dimension of each subband vector is (2m+1)×1. Then, according to [18], we can calculate the pth target direction ϕpm with the mth frequency subband b~αp,upm(ω,θlp) of the pth target. With the above development, the DOA of the pth target can be obtained as(37)ϕp=1M2-M1+1∑m=M1M2ϕpm.


## 6. Simulation Results

The numerical example considered here is an LFM signal as a transmitted signal, which swept from 2.1 kHz to 3.05 kHz over pulse duration of 0.1 s. The initial phase is zero. Signal propagation speed is 1500 m/s. The sampling frequencyFs equals 10 kHz. The total number of time samples is 80001. The bottom reverberation model is based on ray acoustics theory and follows the large number theorem [19]. The reverberation is assumed to be nonstationary and colored. The additive white Gaussian noise is zero-mean complex random process at each sensor.In order to test the performances of multiple targets detection and parameters estimation, we consider two scenarios. Scenario I: we have one target which is located at[1700,1500] m in the Cartesian coordinates and moves with a radial velocity of [4,5] m/s. Scenario II: there are two adjacent targets. One is located at [1500,1500] m; the other is located at [1550,1500] m. The radial velocities of the two targets are [5,5] m/s and [4,4.5] m/s in different directions.


### 6.1. Detection Probability

Computer simulations are first carried out for the detection probability. In scenario I, the detection probability curves of the proposed method are illustrated in Figure7, with the range of SRR being from −20 dB to 3 dB and different SNR values. It can be seen that the proposed detection method performs well although in very low SRR scenario. In addition, the effect caused by additive Gaussian noise is less than that of reverberation.Figure 7
Detection probabilities with two detection methods over different SNR and SRR values.Furthermore, we also compare the detection probability of the proposed method with the existing method in [6]. It can be observed from Figure 7 that both methods can get good probability detection in a high SRR environment. However, when the SRR value is less than −10 dB, the performance of the compared method declines sharply. Particularly, when the SRR is equivalent to −11 dB, it is hard to detect the target using the compared method. However, the proposed method still keeps a fine performance in an extremely low SRR environment. This is because the proposed method makes use of the differences of the Doppler shift between target echo and reverberation to detect target.In scenario II, as shown in Figure5, two adjacent targets can be easily distinguished. The simulation results which are presented in Figure 8 show that there is no mutual influence between the two targets by the proposed method and compared method. Specifically, the performance of the proposed detection method is better than the compared method in low SRR environment.Figure 8
Detection probabilities of two targets with proposed detection methods over different SRR values and SNR is equal to −15 dB.


### 6.2. Parameters Estimation

In this subsection, parameters’ (range and DOA) estimation of multiple targets simulation results are shown. Root-mean-square (RMS) error is used to measure the performance of the proposed method. The expression of RMS error is defined as(38)RMSep=1N∑n=1Ne^pn-ep2,where e^pn is the nth estimation value of the pth target and ep is the true value of the pth target. N is the total number of Monte Carlo runs. In the following simulations, N is equal to 100.


#### 6.2.1. Range Estimation

In this subsection, we first consider scenario I to test the parameters estimation at low SRR values. Figure9 shows the RMS range errors of single target with the SRR values varying from −10 dB to 2 dB. We can see that the range RMS errors decrease with some fluctuations along with the increase of SRR. This result from the main interference-reverberation has a great influence on the center of peaks, which determines the accuracy of parameters estimation. In consideration of the reverberation distribution being uniform [20], the effect caused by reverberation is random. Besides, we consider different environments with SNR chosen as −10 dB, −15 dB, and −20 dB, respectively. Note that the variation of SNR has a negligible effect on the accuracy of range estimation.Figure 9
The RMS range errors of single target over different SNR and SRR values.Figure10 provides the simulation results in scenario II. Allowing for the accuracy of parameters is only determined by the widths of peaks. The proposed parameters’ estimation method is not restricted and influenced by the number of targets.Figure 10
The RMS range errors of two targets over different SRR values and SNR is equal to −15 dB.


#### 6.2.2. DOA Estimation

Figure11 displays the DOA errors in scenario I with different SNR and SRR values and compares the proposed DOA estimation method with the method given in [6]. The SRR varies from −14 dB to −2 dB and SNR is equivalent to −10 dB and −15 dB. The proposed method maintains good accuracy even in a low SRR environment. Also, the SNR has little impact on the accuracy. Both methods perform well in a high SRR environment. By using the beamforming to suppress reverberation, we can obtain high SRR values. This indicates that a higher accuracy can be guaranteed by the proposed method as compared to the method in [6], when considering a low SRR condition. In different SNR scenarios, the compared method has been more greatly affected than the proposed method. Furthermore, there are no interactions between these two targets due to the use of a bandpass filter to separate the received signals of multiple targets into P subband in fractional Fourier domain.Figure 11
The RMS DOA errors with two methods over different SRR values and SNR is equal to −15 dB.


## 6.1. Detection Probability

Computer simulations are first carried out for the detection probability. In scenario I, the detection probability curves of the proposed method are illustrated in Figure7, with the range of SRR being from −20 dB to 3 dB and different SNR values. It can be seen that the proposed detection method performs well although in very low SRR scenario. In addition, the effect caused by additive Gaussian noise is less than that of reverberation.Figure 7
Detection probabilities with two detection methods over different SNR and SRR values.Furthermore, we also compare the detection probability of the proposed method with the existing method in [6]. It can be observed from Figure 7 that both methods can get good probability detection in a high SRR environment. However, when the SRR value is less than −10 dB, the performance of the compared method declines sharply. Particularly, when the SRR is equivalent to −11 dB, it is hard to detect the target using the compared method. However, the proposed method still keeps a fine performance in an extremely low SRR environment. This is because the proposed method makes use of the differences of the Doppler shift between target echo and reverberation to detect target.In scenario II, as shown in Figure5, two adjacent targets can be easily distinguished. The simulation results which are presented in Figure 8 show that there is no mutual influence between the two targets by the proposed method and compared method. Specifically, the performance of the proposed detection method is better than the compared method in low SRR environment.Figure 8
Detection probabilities of two targets with proposed detection methods over different SRR values and SNR is equal to −15 dB.


## 6.2. Parameters Estimation

In this subsection, parameters’ (range and DOA) estimation of multiple targets simulation results are shown. Root-mean-square (RMS) error is used to measure the performance of the proposed method. The expression of RMS error is defined as(38)RMSep=1N∑n=1Ne^pn-ep2,where e^pn is the nth estimation value of the pth target and ep is the true value of the pth target. N is the total number of Monte Carlo runs. In the following simulations, N is equal to 100.


### 6.2.1. Range Estimation

In this subsection, we first consider scenario I to test the parameters estimation at low SRR values. Figure9 shows the RMS range errors of single target with the SRR values varying from −10 dB to 2 dB. We can see that the range RMS errors decrease with some fluctuations along with the increase of SRR. This result from the main interference-reverberation has a great influence on the center of peaks, which determines the accuracy of parameters estimation. In consideration of the reverberation distribution being uniform [20], the effect caused by reverberation is random. Besides, we consider different environments with SNR chosen as −10 dB, −15 dB, and −20 dB, respectively. Note that the variation of SNR has a negligible effect on the accuracy of range estimation.Figure 9
The RMS range errors of single target over different SNR and SRR values.Figure10 provides the simulation results in scenario II. Allowing for the accuracy of parameters is only determined by the widths of peaks. The proposed parameters’ estimation method is not restricted and influenced by the number of targets.Figure 10
The RMS range errors of two targets over different SRR values and SNR is equal to −15 dB.


### 6.2.2. DOA Estimation

Figure11 displays the DOA errors in scenario I with different SNR and SRR values and compares the proposed DOA estimation method with the method given in [6]. The SRR varies from −14 dB to −2 dB and SNR is equivalent to −10 dB and −15 dB. The proposed method maintains good accuracy even in a low SRR environment. Also, the SNR has little impact on the accuracy. Both methods perform well in a high SRR environment. By using the beamforming to suppress reverberation, we can obtain high SRR values. This indicates that a higher accuracy can be guaranteed by the proposed method as compared to the method in [6], when considering a low SRR condition. In different SNR scenarios, the compared method has been more greatly affected than the proposed method. Furthermore, there are no interactions between these two targets due to the use of a bandpass filter to separate the received signals of multiple targets into P subband in fractional Fourier domain.Figure 11
The RMS DOA errors with two methods over different SRR values and SNR is equal to −15 dB.


## 6.2.1. Range Estimation

In this subsection, we first consider scenario I to test the parameters estimation at low SRR values. Figure9 shows the RMS range errors of single target with the SRR values varying from −10 dB to 2 dB. We can see that the range RMS errors decrease with some fluctuations along with the increase of SRR. This result from the main interference-reverberation has a great influence on the center of peaks, which determines the accuracy of parameters estimation. In consideration of the reverberation distribution being uniform [20], the effect caused by reverberation is random. Besides, we consider different environments with SNR chosen as −10 dB, −15 dB, and −20 dB, respectively. Note that the variation of SNR has a negligible effect on the accuracy of range estimation.Figure 9
The RMS range errors of single target over different SNR and SRR values.Figure10 provides the simulation results in scenario II. Allowing for the accuracy of parameters is only determined by the widths of peaks. The proposed parameters’ estimation method is not restricted and influenced by the number of targets.Figure 10
The RMS range errors of two targets over different SRR values and SNR is equal to −15 dB.


## 6.2.2. DOA Estimation

Figure11 displays the DOA errors in scenario I with different SNR and SRR values and compares the proposed DOA estimation method with the method given in [6]. The SRR varies from −14 dB to −2 dB and SNR is equivalent to −10 dB and −15 dB. The proposed method maintains good accuracy even in a low SRR environment. Also, the SNR has little impact on the accuracy. Both methods perform well in a high SRR environment. By using the beamforming to suppress reverberation, we can obtain high SRR values. This indicates that a higher accuracy can be guaranteed by the proposed method as compared to the method in [6], when considering a low SRR condition. In different SNR scenarios, the compared method has been more greatly affected than the proposed method. Furthermore, there are no interactions between these two targets due to the use of a bandpass filter to separate the received signals of multiple targets into P subband in fractional Fourier domain.Figure 11
The RMS DOA errors with two methods over different SRR values and SNR is equal to −15 dB.


## 7. Conclusion

In this paper, we proposed methods for detecting and estimating parameters (range and DOA) of multiple moving targets in a severe reverberation environment. Modal preprocessing, conventional beamforming, and fractional Fourier transform have been used to form a 3-dimensional matrix. With the properties of moving targets and reverberation, this matrix can be applied to detect multiple targets and estimate the parameters in strong reverberation environment. Specifically, the proposed scheme does not involve any a priori knowledge of environment parameters and number of targets. The simulation results demonstrate that the provided method can perform with high detection probabilities and maintain good accuracies of parameters estimation even in low SNR and SRR conditions. More importantly, the performances of the proposed method are not affected by the number of targets. The future work will focus on extending the method to detect and estimate parameters in a 3-dimensional reverberation environment.



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*Source: 5274371-2016-09-22.xml*