# Low-Grazing Angle Detection in Compound-Gaussian Clutter with Hybrid MIMO Radar

Authors: Jincan Ding; Haowen Chen; Hongqiang Wang; Xiang Li; Zhaowen Zhuang
Journal: International Journal of Antennas and Propagation
 (2013)

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## Abstract

This paper focuses on the target detection in low-grazing angle using a hybrid multiple-input multiple-output (MIMO) radar systems in compound-Gaussian clutter, where the multipath effects are very abundant. The performance of detection can be improved via utilizing the multipath echoes. First, the reflection coefficient considering the curved earth effect is derived. Then, the general signal model for MIMO radar is introduced in low-grazing angle; also, the generalized likelihood test (GLRT) and generalized likelihood ratio test-linear quadratic (GLRT-LQ) are derived with known covariance matrix. Via the numerical examples, it is shown that the derived GLRT-LQ detector outperforms the GLRT detector in low-grazing angle, and both performances can be enhanced markedly when the multipath effects are considered.


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## Body

## 1. Introduction

MIMO radar has gotten considerable attention in a novel class of radar system, where the term MIMO refers to the use of multiple-transmit as well as multiple-receive antennas. MIMO radar is categorized into two classes: the statistical MIMO radar and the colocated MIMO radar, depending on their antenna placement [1, 2]. The advantages of MIMO radar with colocated antennas have been studied extensively, which include improved detection performance, higher resolution [3], higher sensitivity to or detection of moving targets [4], and increased degrees of freedom for transmission beamforming [5]. MIMO radar with widely separated antennas can capture the spatial diversity of the target’s radar cross section (RCS) [6]. This spatial diversity provides the radar systems with the ability to support the improvement of the target parameter estimation [7, 8], high resolution target localization [9], and tracking performance [10]. The hybrid MIMO radars can obtain superiority both from colocated and separated MIMO radar. Thus, we focus on the hybrid MIMO radar system in this paper.Much published literature has concerned the issue of MIMO radar detection. Guan and Huang [11] investigated the detection problem of the MIMO radar system with distributed apertures in Gaussian colored noise and partially correlated observation channels. Tang et al. [12] introduced relative entropy as a measure to radar detection theory and analyzed the detection performance of MIMO radar and phased array radar. The authors in [13] investigated detection performance of MIMO radar for Rician target. In [14], the optimal detector in the Neyman-Pearson sense was derived for the statistical MIMO radar using orthogonal waveforms. The authors in [15] applied the Swerling models to target detection and derived the optimal test statistics for a statistical MIMO radar using nonorthogonal signal. For low-grazing angle detection of MIMO radar, the authors in [16] utilized the time reversal technique in a multipath environment to achieve high target detectability.Low-grazing angle targets are difficult to detect, which is one of the great threats propelling radar development. Otherwise, detection of low-altitude targets is of great significance to counter low-altitude air defense penetration. However, up to now, this problem has not been effectively resolved. Multipath effect plays an important role in the low-altitude target detection, by which the target echo signal is seriously polluted, even counteracted [17]. Two aspects can be considered for multipath: suppressing multipath and utilizing it. However, in a statistical sense, detection may be enhanced by the presence of multipath [18].In this paper, we consider low-grazing angle target detection in compound-Gaussian clutter for MIMO radar. The compound-Gaussian clutter represents the heavy-tailed clutter statistics that are distinctive of several scenarios, for example, high-resolution or low-grazing angle radars in the presence of sea or foliage clutter [19, 20]. To the end, the generalized likelihood ratio test (GLRT) and generalized likelihood ratio test-linear quadratic (GLRT-LQ) are derived.


## 2. Multipath Geometry Model

A point source at a distance ofRd from the receiver is considered. If the source is assumed to be a narrowband signal, it can be represented by
(1)x(t)=aej(ωt+φ),
where a is the amplitude, ω is the angular frequency, and φ is the initial phase. In the presence of multipath, the received by the receiver consists of two components, namely, the direct and indirect signal. For a simple multipath model of a flat earth, the direct signal is given by
(2)xd(t)=x(t)e-jκRd,
while indirect signal is
(3)xi(t)=x(t)ρejϕe-jκRi,
where ρejϕ is the complex reflection coefficient, κ=2π/λ is the wave number, λ is wavelength, target range Rd can be obtained from the time delay, and Ri is the total length of the indirect path. Thus, the total received signal is given by
(4)xr(t)=xd(t)+xi(t).To model the received signals more accurately, the curvature of the signal path due to refraction in the troposphere, in addition to the curvature of the earth itself, must be taken into account. The multipath geometry for a curved earth is given in Figure1.Figure 1
Multipath geometry for a curved earth.In (3), the term ρejϕ is the complex reflection coefficient. It generally consists of the Fresnel reflection coefficient divided into the vertical polarization Γv and horizontal polarization Γh, the divergence factor D due to a curved surface, and the surface roughness factor; that is, ρejϕ=Γ(v,h)Dρs. The vertical polarization and horizontal polarization Fresnel reflection coefficients are, respectively, as presented in [17]. Consider the following:
(5)Γv≃ψεc-1ψεc+1,(6)Γh≃ψ-εcψ+εc.
For horizontal polarization, ψ is the grazing angle and εc is the complex dielectric constant which is given by
(7)εc=εε0-j60λσ,
where ε/ε0 is the relative dielectric constant of the reflecting medium and σ is its conductivity. Thus, the Fresnel reflection coefficient is determined by the grazing angle under a deterministic condition.When an electromagnetic wave is incident on a round earth surface, the reflected wave diverges because of the earth’s curvature. Due to divergence, the reflected energy is defocused and radar power density is reduced. The divergence factor can be derived solely from geometrical considerations. A widely accepted approximation for the divergence factorD is given by
(8)D≃(1+2r1r2Rerψ)-1/2.The surface roughness factorρs is given by
(9)ρs=e-μ,μ={2[2πη]2,η≤0.1rad,0.16η2+7.42η+0.0468,otherwise,
and η is the surface roughness factor given by
(10)η=σHψλ
and σH is the root-mean-square (RMS) surface height irregularity. For simplicity, the diffuse component is treated as the incoherent white Gaussian noise.


## 3. MIMO Radar Multipath Signal

Consider a narrowband MIMO radar system withM~ and N~ subarrays for transmitting and receiving, respectively. The mth transmit and nth receive subarrays have, respectively, Mm and Nn closely spaced antennas. m=1,…,M~, and n=1,…,N~, M=M1+M2+⋯+MM~ and N=N1+N2+⋯+NN~ are the total numbers of transmit and receive antennas, respectively. We assume that the subarrays are sufficiently separated, and, hence, for each target, its RCSs for different transmit and receive subarray pairs are statistically independent of each other. The receive signal of MIMO radar can be expressed as [21]
(11)Y=A(ϕ)BϕS(ϕ)+Z,
where A(ϕ)=diag{a1(ϕ),a2(ϕ),…,aN~(ϕ)} is the steering matrix of receive subarrays and component {an(ϕ)}n=1N~ is the steering vector of nth receive subarray at direction ϕ. Bϕ denotes the RCSs for different transmit and receive subarray pairs with component {βmn,ϕ}m,n=1,1M~,N~. S(ϕ)=VT(ϕ)Φ denotes the transmit signal matrix, where Φ=[Φ1T,Φ2T,…,ΦM~T]T is the transmit waveform matrix, for each transmit subarray, the component ΦmM~×L=[{sm1(t)}T,{sm2(t)}T,…,{smMm(t)}T]T, and {sml(t)}ℓ=1Mm,  t=1,2,…,L is the probing waveform of m subarray; V(ϕ)=diag{v1(ϕ),v2(ϕ),…,vM~(ϕ)} is the steering matrix of transmit subarray and the component {vm(ϕ)}m=1M~ is the steering vector of mth transmit subarray at direction ϕ. Y=[Y1T,Y2T,…,YN~T] is the received data matrix and {Yn}n=1N~ denotes received signal of the nth subarray. Z=[z1,z2,…,zL] is the clutter matrix, each column {zℓ}ℓ=1L of which is modeled as spherically invariant random vectors (SIRV), and L is the number of data samples of the transmitted waveforms. We assume clutter distributing as the compound-Gaussian model, which represents the heavy-tailed clutter statistics that are distinctive of several scenarios, for example, high-resolution or low-grazing angle radars in the presence of sea or foliage clutter [19, 20]. The compound-Gaussian clutter z=uχ, where u and χ are the texture and speckle components of the compound model, respectively. The fast-changing χ is a realization of a stationary zero mean complex Gaussian process, and the slow-changing u is modeled as a nonnegative real random process [22].We rewrite the received signal (11) in vector form, given by
(12)ydd=Tβ+n,
where yNL×1=Vec(Y), TNL×M~N~=(ST⊗A), βM~N~×1=Vec(Bϕ), nNL×1=Vec(Z), Vec(•) is the vector operator, symbol ⊗ denotes the Kronecker product. Then and n is the compound Gaussian random vector with covariance matrix
(13)CnNL×NL=E[nn†]=P⊗C,P=diag{s12,s22,…,sL2}=diag{λ1,λ2,…,λL},
where P is considered deterministic matrix with unknown parameters {λℓ}ℓ=1L and † denotes conjugate transpose.


## 4. Multipath Signal Model of MIMO Radar

In the presence of multipath, consider atmosphere refraction and the curved earth effect; the reflected signals from a point target of MIMO radar include four parts: directly-directly path, directly-reflected path, and reflected-directly path, reflected-reflected path. Assume the point target is located atX0=(x0,y0) and reflected point in ground is located at Xi=(xi,yi),  i=1,2. Figure 2 illustrates a four-way MIMO radar propagation model with multipath.Multipath MIMO radar.
(a)
Directly-directly path
(b)
Directly-reflected path
(c)
Reflected-directly path
(d)
Reflected-reflected pathThe directly-directly path echo signal is given by (12). The directly-reflected path echo signal is
(14)ydr=T(kdr⊙β)+n,
where Kdr is M~N~×1 amplitude of reflect coefficient and symbol ⊙ represents the Hadamard product.The reflected-directly path echo signal is(15)yrd=T(krd⊙β)+n.The reflected-reflected path echo signal is(16)yrr=T(krr⊙β)+n.Thus, the received signal of MIMO radar with multipath is(17)ymp=ydd+ydr+yrd+yrr=T((1+kdr+krd+krr)⊙β)+n=T(k⊙β)+n,
where k=1+kdr+krd+krr and 1M~N~ is an M~N~×1 one vector.


## 5. MIMO Radar Detector in Compound-Gaussian Clutter

### 5.1. GLRT Detector Design

The problem of detecting with MIMO radar can be formulated in terms of the following binary hypotheses test:(18)H0:ymp=n,H1:ymp=T(k⊙β)+n.Standard GLRT is the following decision rule:(19)maxβ,λ1,…,λLf(Y∣H1,λ1,…,λL,β)maxλ1,…,λLf(Y∣H0,λ1,…,λL)>H1<H0γG,
where f(Y∣H1,λ1,…,λL,β) and f(Y∣H1,λ1,…,λL) denote the probability density functions (pdfs) of the data under H1 and H0, respectively. And the pdfs can be written, respectively, as
(20)f(Y∣H1,λ1,…,λL,β)=1πNLdet(Cn)exp{-tr(ℑ)},
where ℑ=Cn-1(y-T(k⊙β))×(y-T(K⊙β))†, and
(21)f(Y∣H0,λ1,…,λL)=1πNLdet(Cn)exp{-tr(Cn-1yy†)},
where det(•) and tr denote the determinant and the trace of a matrix, respectively.The log-likelihood function of (21) is
(22)lnf(Y∣H0,λ1,…,λL)=-NLlnπ-Llndet(C)-N∑ℓ=1L‍lnλℓ-∑ℓ=1L‍y†(Eℓℓ⊗C-1)yλℓ,
where Eℓℓ denotes the elementary matrix with component e(ℓ,ℓ)=1 and zero for others. Then, it is easy to obtain the Maximum Likelihood (ML) estimator of λ under H0; that is
(23)λ^ℓ=y†(Eℓℓ⊗C-1)yN.According to [21], we rewrite the log-likelihood function of (20) as
(24)lnf(Y∣H1,λ1,…,λL,β)=-NLlnπ-Llndet(C)-N∑ℓ=1L‍lnλℓ-∑ℓ=1L‍(y-T(k⊙β))†(Eℓℓ⊗C-1)(y-T(k⊙β))λℓ.Thus, the ML estimator of{λℓ}ℓ=1L is
(25)λ^ℓ=(y-T(k⊙β))†(Eℓℓ⊗C-1)(y-T(k⊙β))N.The estimatorβis [23]
(26)β^=((Tk)†Cn-1(Tk))(Tk)†Cn-1y.Substituting the estimator{λ^ℓ}ℓ=1L, β^under H1 and H0 into (19)–(21), the final GLRT becomes
(27)∏ℓ=1L‍y†(Eℓℓ⊗C-1)y(y-T(k⊙β^))†(Eℓℓ⊗C-1)(y-T(k⊙β^))>H1<H0γG.


### 5.2. GLRT-LQ Detector Design

We rewrite the detection problem as(28)H0:ym,n=zm,n,H1:ym,n=ρMmNnβ(m,n)Tm,n+zm,n,
where ρMmNn=(1+ρMmNn(dr)+ρMmNn(rd)+ρMmNn(rr)); ρMmNn(dr), ρMmNn(rd), ρMmNn(rr) are the amplitudes of reflect coefficient, because the grazing angles are different; the reflect coefficient ρMmNn(dr)≠ρMmNn(rd)≠ρMmNn(rr).As the transmit-receive subarrays are widely separated, the clutter returns can be considered to be independent; hence, the low-grazing angle likelihood ratio test (LRT) detector for MIMO radar in the compound-Gaussian clutter is given by(29)logp(y∣H0)p(y∣H1)>H1<H0γG-LQ.If we assume that covariance matrixCc is known and according to [24], p(y∣H0) and p(y∣H1) are replaced by their Bayesian estimates, and, asymptotically, the generalized likelihood ratio test-linear quadratic (GLRT-LQ), extended to the MIMO case, is given by
(30)Λ(Y)=p(Y∣H1)p(Y∣H0)=∏m,n‍p(ym,n∣H1)∏m,n‍p(ym,n∣H0)=∏m,n‍[1×(1-|Tm,n†Cm,n-1ym,n|2(Tm,n†Cm,n-1Tm,n)(ym,n†Mm,n-1ym,n))-1]>H1<H0γG-LQ,
where Cm,n is the covariance matrix for the m-n transmit-receive pair.According to [24], the probability of false alarm Pfa is given by
(31)Pfa=P(∏i=1I‍Λi(yi)>γG-LQ∣H0)=γG-LQ-L+1∑i=0I-1‍(W-I)ii!(lnγG-LQ)i,
where I=M~N~ and W=MmNn.The probability of detectionPd is given by
(32)Pd=P(∏i=1I‍Λi(yi)>γG-LQ∣H0).For a given signal-to-clutter ratio (SCR), denoted bySCRm,n, the amplitude of β(m,n) is given by
(33)|β(m,n)|=SCRm,nσ2,
where σ2 is the clutter power. In this paper, we consider that |β(m,n)| is the same for all m and n.


## 5.1. GLRT Detector Design

The problem of detecting with MIMO radar can be formulated in terms of the following binary hypotheses test:(18)H0:ymp=n,H1:ymp=T(k⊙β)+n.Standard GLRT is the following decision rule:(19)maxβ,λ1,…,λLf(Y∣H1,λ1,…,λL,β)maxλ1,…,λLf(Y∣H0,λ1,…,λL)>H1<H0γG,
where f(Y∣H1,λ1,…,λL,β) and f(Y∣H1,λ1,…,λL) denote the probability density functions (pdfs) of the data under H1 and H0, respectively. And the pdfs can be written, respectively, as
(20)f(Y∣H1,λ1,…,λL,β)=1πNLdet(Cn)exp{-tr(ℑ)},
where ℑ=Cn-1(y-T(k⊙β))×(y-T(K⊙β))†, and
(21)f(Y∣H0,λ1,…,λL)=1πNLdet(Cn)exp{-tr(Cn-1yy†)},
where det(•) and tr denote the determinant and the trace of a matrix, respectively.The log-likelihood function of (21) is
(22)lnf(Y∣H0,λ1,…,λL)=-NLlnπ-Llndet(C)-N∑ℓ=1L‍lnλℓ-∑ℓ=1L‍y†(Eℓℓ⊗C-1)yλℓ,
where Eℓℓ denotes the elementary matrix with component e(ℓ,ℓ)=1 and zero for others. Then, it is easy to obtain the Maximum Likelihood (ML) estimator of λ under H0; that is
(23)λ^ℓ=y†(Eℓℓ⊗C-1)yN.According to [21], we rewrite the log-likelihood function of (20) as
(24)lnf(Y∣H1,λ1,…,λL,β)=-NLlnπ-Llndet(C)-N∑ℓ=1L‍lnλℓ-∑ℓ=1L‍(y-T(k⊙β))†(Eℓℓ⊗C-1)(y-T(k⊙β))λℓ.Thus, the ML estimator of{λℓ}ℓ=1L is
(25)λ^ℓ=(y-T(k⊙β))†(Eℓℓ⊗C-1)(y-T(k⊙β))N.The estimatorβis [23]
(26)β^=((Tk)†Cn-1(Tk))(Tk)†Cn-1y.Substituting the estimator{λ^ℓ}ℓ=1L, β^under H1 and H0 into (19)–(21), the final GLRT becomes
(27)∏ℓ=1L‍y†(Eℓℓ⊗C-1)y(y-T(k⊙β^))†(Eℓℓ⊗C-1)(y-T(k⊙β^))>H1<H0γG.


## 5.2. GLRT-LQ Detector Design

We rewrite the detection problem as(28)H0:ym,n=zm,n,H1:ym,n=ρMmNnβ(m,n)Tm,n+zm,n,
where ρMmNn=(1+ρMmNn(dr)+ρMmNn(rd)+ρMmNn(rr)); ρMmNn(dr), ρMmNn(rd), ρMmNn(rr) are the amplitudes of reflect coefficient, because the grazing angles are different; the reflect coefficient ρMmNn(dr)≠ρMmNn(rd)≠ρMmNn(rr).As the transmit-receive subarrays are widely separated, the clutter returns can be considered to be independent; hence, the low-grazing angle likelihood ratio test (LRT) detector for MIMO radar in the compound-Gaussian clutter is given by(29)logp(y∣H0)p(y∣H1)>H1<H0γG-LQ.If we assume that covariance matrixCc is known and according to [24], p(y∣H0) and p(y∣H1) are replaced by their Bayesian estimates, and, asymptotically, the generalized likelihood ratio test-linear quadratic (GLRT-LQ), extended to the MIMO case, is given by
(30)Λ(Y)=p(Y∣H1)p(Y∣H0)=∏m,n‍p(ym,n∣H1)∏m,n‍p(ym,n∣H0)=∏m,n‍[1×(1-|Tm,n†Cm,n-1ym,n|2(Tm,n†Cm,n-1Tm,n)(ym,n†Mm,n-1ym,n))-1]>H1<H0γG-LQ,
where Cm,n is the covariance matrix for the m-n transmit-receive pair.According to [24], the probability of false alarm Pfa is given by
(31)Pfa=P(∏i=1I‍Λi(yi)>γG-LQ∣H0)=γG-LQ-L+1∑i=0I-1‍(W-I)ii!(lnγG-LQ)i,
where I=M~N~ and W=MmNn.The probability of detectionPd is given by
(32)Pd=P(∏i=1I‍Λi(yi)>γG-LQ∣H0).For a given signal-to-clutter ratio (SCR), denoted bySCRm,n, the amplitude of β(m,n) is given by
(33)|β(m,n)|=SCRm,nσ2,
where σ2 is the clutter power. In this paper, we consider that |β(m,n)| is the same for all m and n.


## 6. Numerical Simulations

This section is devoted to the performance assessment of the GLRT and GLRT-LQ detectors in low-grazing angle for MIMO radar, when the texture component of clutter distributed as gamma distribution, leading to the wellknownK clutter model. Since the closed-form expressions of the GLRT detector for the probability of the detection and of alarm are not available, we resort to standard Monte Carlo.In our first example, we, respectively, analyze the GlRT-LQ detectors considering multipath effect and without considering multipath effect. Assume MIMO radar is with three transmit antennas and two receive antennas, the heights of transmit arrays are fixed at 100 m, 200 m, and 300 m, the height of receive arrays are fixed at 100 m, and 200 m, and the target’s height is fixed at 200 m. The(p,q), the element of the covariance matrix of the speckle component, is chosen as
(34)Cp,q=σ2×0.9|p-q|×exp[j(π2)(p-q)].
Here, we select σ2=10.17, L=16. We define the signal-to-clutter pulse noise ratio (SCNR) by [25]
(35)SCNR=1L∑ℓ=1L[T(k⊙β)]†[T(k⊙β)]E{u}tr{Cn}.Figure3 depicts the detection performance using GLRT-LQ detectors, as a function of the SCNR. The probability of false alarm is fixed at Pfa=10-4. For the given SCNR, the detection performance with multipath outperforms the one without considering multipath effect.Figure 3
GLRT-LQ detector performance in low-grazing angle.Figure4 depicts the detection performance using GLRT detector, as a function of the SCNR. For the given SCNR, the detection performance with multipath outperforms the one without considering multipath effect.Figure 4
GLRT detector performance in low-grazing angle.Figure5 depicts the performance comparison between GLRT-LQ and GLRT detectors. From Figure 5, GLRT-LQ detector outperforms the GLRT detector in low-grazing angle for MIMO radar, respectively, with and without considering multipath effects.Figure 5
Comparison of GLRT-LQ detector and GLRT detector.Figures6 and 7 depict the detection performance of GLRT-LQ and GLRT detector with different antenna numbers, respectively, as a function of SCNR. The probability of false alarm is set at Pfa=10-4; the transmit antenna and receive antenna are set at 2, 4, and 6, respectively. Figures 6 and 7 show that GLRT-LQ or GLRT detector can obtain better detection performance when there are more number of transmit antennas and receive antennas.Figure 6
Detection performance of GLRT-LQ detector with different numbers.Figure 7
Detection performance of GLRT detector with different numbers.Figures8 and 9 depict the detection performance of GLRT-LQ and GLRT detector with target height, respectively; the heights of target are fixed at 200 m, 400 m, and 600 m and the probabilities of false alarm are fixed at Pfa=10-4. Figures 8 and 9 show that the detection performance varies with the height of target. We can see that the performance increases with the height of target under the low-grazing scene. However, the performance does not always increase with the height of target, just as Figure 10. When the height of target is 1600 m, the condition of low-grazing angle is not satisfied. If we still take it for low-grazing angle, the detection performance will decrease.Figure 8
GLRT-LQ detector detection performance varies with the height of target.Figure 9
GLRT detector detection performance varies with the height of target.Figure 10
GLRT-LQ detector detection performance varies with the height of target.


## 7. Conclusion

In this paper, we have introduced the concept of reflection coefficient under considering curved earth effect and introduced general signal model for MIMO radar in low-grazing angle, firstly. Then, we have derived the GLRT-LQ and GLRT detectors, respectively. Furthermore, we have compared the performance of GLRT-LQ and GLRT detector for MIMO radar between with multipath and without multipath effects. The simulation results have shown the importance of multipath effects for target detection in low-grazing angle and demonstrated that GLRT-LQ detector outperforms the GLRT detector in low-grazing angle.



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