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

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.

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 of 𝑅_𝑑 from the receiver is considered. If the source is assumed to be a narrowband signal, it can be represented by 𝑥(𝑡)=𝑎𝑒^{𝑗(𝜔𝑡+𝜑)}, (1) 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 𝑥_𝑑(𝑡)=𝑥(𝑡)𝑒^{−𝑗𝜅𝑅𝑑}, (2) while indirect signal is 𝑥_𝑖(𝑡)=𝑥(𝑡)𝜌𝑒^𝑗𝜙𝑒^{−𝑗𝜅𝑅𝑖}, (3) where 𝜌𝑒^𝑗𝜙 is the complex reflection coefficient, 𝜅=2𝜋/𝜆 is the wave number, 𝜆 is wavelength, target range 𝑅_𝑑 can be obtained from the time delay, and 𝑅_𝑖 is the total length of the indirect path. Thus, the total received signal is given by 𝑥_𝑟(𝑡)=𝑥_𝑑(𝑡)+𝑥_𝑖(𝑡). (4)

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 Figure 1.

Figure 1 

Multipath geometry for a curved earth.

In (3), the term 𝜌𝑒^𝑗𝜙 is the complex reflection coefficient. It generally consists of the Fresnel reflection coefficient divided into the vertical polarization Γ_𝑣 and horizontal polarization Γ_ℎ, the divergence factor 𝐷 due to a curved surface, and the surface roughness factor; that is, 𝜌𝑒^𝑗𝜙=Γ_(𝑣,ℎ)𝐷𝜌_𝑠. The vertical polarization and horizontal polarization Fresnel reflection coefficients are, respectively, as presented in [17]. Consider the following: Γ_𝑣≃𝜓√𝜀_𝑐−1𝜓√𝜀_𝑐,Γ+1_ℎ≃𝜓−√𝜀_𝑐𝜓+√𝜀_𝑐. (5)(6) For horizontal polarization, 𝜓 is the grazing angle and 𝜀_𝑐 is the complex dielectric constant which is given by 𝜀_𝑐=𝜀𝜀₀−𝑗60𝜆𝜎, (7) 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 factor 𝐷 is given by 𝐷≃1+2𝑟₁𝑟₂𝑅_𝑒𝑟𝜓^−1/2. (8)

The surface roughness factor 𝜌_𝑠 is given by 𝜌_𝑠=𝑒^−𝜇,2𝜇=2𝜋𝜂²,𝜂≤0.1rad,0.16𝜂²+7.42𝜂+0.0468,otherwise, (9) and 𝜂 is the surface roughness factor given by 𝜎𝜂=^𝐻𝜓𝜆 (10) and 𝜎_𝐻 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 with 𝑀 and 𝑁 subarrays for transmitting and receiving, respectively. The 𝑚th transmit and 𝑛th receive subarrays have, respectively, 𝑀_𝑚 and 𝑁_𝑛 closely spaced antennas. 𝑚=1,…,𝑀, and 𝑛=1,…,𝑁, 𝑀=𝑀₁+𝑀₂𝑀+⋯+𝑀 and 𝑁=𝑁₁+𝑁₂𝑁+⋯+𝑁 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] 𝐘=𝐀𝜙𝐁_𝜙𝐒𝜙+𝐙, (11) where 𝐀(𝜙)=diag{𝐚₁(𝜙),𝐚₂𝑁(𝜙),…,𝐚(𝜙)} is the steering matrix of receive subarrays and component {𝐚_𝑛(𝜙)}𝑁_𝑛=1 is the steering vector of 𝑛th receive subarray at direction 𝜙. 𝐁_𝜙 denotes the RCSs for different transmit and receive subarray pairs with component {𝛽_{𝑚𝑛,𝜙}}𝑀_,𝑁_{𝑚,𝑛=1,1}. 𝐒(𝜙)=𝐕^𝑇(𝜙)Φ denotes the transmit signal matrix, where Φ=[Φ^𝑇₁,Φ^𝑇₂,…,Φ^𝑇𝑀]^𝑇 is the transmit waveform matrix, for each transmit subarray, the component Φ𝑀_𝑚×𝐿=[{𝑠_𝑚1(𝑡)}^𝑇,{𝑠_𝑚2(𝑡)}^𝑇,…,{𝑠_𝑚𝑀𝑚(𝑡)}^𝑇]^𝑇, and {𝑠_𝑚𝑙(𝑡)}^𝑀𝑚_ℓ=1,  𝑡=1,2,…,𝐿 is the probing waveform of 𝑚 subarray; 𝐕(𝜙)=diag{𝐯₁(𝜙),𝐯₂𝑀(𝜙),…,𝐯(𝜙)} is the steering matrix of transmit subarray and the component {𝐯_𝑚(𝜙)}𝑀_𝑚=1 is the steering vector of 𝑚th transmit subarray at direction 𝜙. 𝐘=[𝑌^𝑇₁,𝑌^𝑇₂,…,𝑌^𝑇𝑁] is the received data matrix and {𝑌_𝑛}𝑁_𝑛=1 denotes received signal of the 𝑛th subarray. 𝐙=[𝐳₁,𝐳₂,…,𝐳_𝐿] is the clutter matrix, each column {𝐳_ℓ}^𝐿_ℓ=1 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 𝐲_dd=𝐓𝜷+𝐧, (12) where 𝐲^𝑁𝐿×1=Vec(𝐘), 𝐓^𝑁𝐿×𝑀𝑁=(𝐒^𝑇⊗𝐀), 𝜷𝑀𝑁_×1=Vec(𝐁_𝜙), 𝐧^𝑁𝐿×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 𝐂_𝐧^{𝑁𝐿×𝑁𝐿}=𝐸𝐧𝐧^†𝑠=𝐏⊗𝐂,𝐏=diag²₁,𝑠²₂,…,𝑠²_𝐿=diag𝜆₁,𝜆₂,…,𝜆_𝐿, (13) where 𝐏 is considered deterministic matrix with unknown parameters {𝜆_ℓ}^𝐿_ℓ=1 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 at 𝑋₀=(𝑥₀,𝑦₀) and reflected point in ground is located at 𝑋_𝑖=(𝑥_𝑖,𝑦_𝑖),  𝑖=1,2. Figure 2 illustrates a four-way MIMO radar propagation model with multipath.

(a) Directly-directly path

(b) Directly-reflected path

(c) Reflected-directly path

(d) Reflected-reflected path

Figure 2 

Multipath MIMO radar.

The directly-directly path echo signal is given by (12). The directly-reflected path echo signal is 𝐲_dr=𝐓𝐤^dr⊙𝜷+𝐧, (14) where 𝐊^dr is 𝑀𝑁×1 amplitude of reflect coefficient and symbol ⊙ represents the Hadamard product.

The reflected-directly path echo signal is 𝐲_rd=𝐓𝐤^rd⊙𝜷+𝐧. (15)

The reflected-reflected path echo signal is 𝐲_rr=𝐓𝐤^rr⊙𝜷+𝐧. (16)

Thus, the received signal of MIMO radar with multipath is 𝐲_mp=𝐲_dd+𝐲_dr+𝐲_rd+𝐲_rr=𝐓𝟏+𝐤^dr+𝐤^rd+𝐤^rr()⊙𝜷+𝐧=𝐓𝐤⊙𝜷+𝐧, (17) where 𝐤=𝟏+𝐤^dr+𝐤^rd+𝐤^rr and 𝟏𝑀𝑁 is an 𝑀𝑁×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: 𝐻₀∶𝐲_mp𝐻=𝐧,₁∶𝐲_mp=𝐓(𝐤⊙𝜷)+𝐧. (18)

Standard GLRT is the following decision rule: max_{𝜷,𝜆1,…,𝜆𝐿}𝑓𝐘∣𝐻₁,𝜆₁,…,𝜆_𝐿,𝜷max_{𝜆1,…,𝜆𝐿}𝑓𝐘∣𝐻₀,𝜆₁,…,𝜆_𝐿^𝐻1>_𝐻<₀𝛾_𝐺, (19) where 𝑓(𝐘∣𝐻₁,𝜆₁,…,𝜆_𝐿,𝜷) and 𝑓(𝐘∣𝐻₁,𝜆₁,…,𝜆_𝐿) denote the probability density functions (pdfs) of the data under 𝐻₁ and 𝐻₀, respectively. And the pdfs can be written, respectively, as 𝑓𝐘∣𝐻₁,𝜆₁,…,𝜆_𝐿=1,𝜷𝜋^𝑁𝐿det𝐂_𝐧exp{−tr(ℑ)}, (20) where ℑ=𝐂_𝐧⁻¹(𝐲−𝐓(𝐤⊙𝜷))×(𝐲−𝐓(𝐊⊙𝜷))^†, and 𝑓𝐘∣𝐻₀,𝜆₁,…,𝜆_𝐿=1𝜋^𝑁𝐿det𝐂_𝐧exp−tr𝐂_𝐧⁻¹𝐲𝐲^†, (21) where det(•) and tr denote the determinant and the trace of a matrix, respectively.

The log-likelihood function of (21) is ln𝑓𝐘∣𝐻₀,𝜆₁,…,𝜆_𝐿=−𝑁𝐿ln𝜋−𝐿lndet(𝐂)−𝑁^𝐿_ℓ=1ln𝜆_ℓ−^𝐿_ℓ=1𝐲^†𝐸_ℓℓ⊗𝐂⁻¹𝐲𝜆_ℓ, (22) where 𝐸_ℓℓ 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 𝜆_ℓ=𝐲^†𝐸_ℓℓ⊗𝐂⁻¹𝐲𝑁. (23)

According to [21], we rewrite the log-likelihood function of (20) as ln𝑓𝐘∣𝐻₁,𝜆₁,…,𝜆_𝐿(𝐂),𝜷=−𝑁𝐿ln𝜋−𝐿lndet−𝑁^𝐿_ℓ=1ln𝜆_ℓ−^𝐿_ℓ=1(𝐲−𝐓(𝐤⊙𝜷))^†𝐸_ℓℓ⊗𝐂⁻¹𝑦−𝐓(𝐤⊙𝜷)𝜆_ℓ. (24)

Thus, the ML estimator of {𝜆_ℓ}^𝐿_ℓ=1 is 𝜆_ℓ=(𝐲−𝐓(𝐤⊙𝜷))^†𝐸_ℓℓ⊗𝐂⁻¹(𝐲−𝐓(𝐤⊙𝜷))𝑁. (25)

The estimator 𝜷is [23] 𝜷=(𝐓𝐤)^†𝐂_𝐧^−𝟏(𝐓𝐤)(𝐓𝐤)^†𝐂_𝐧^−𝟏𝐲. (26)

Substituting the estimator {𝜆_ℓ}^𝐿_ℓ=1, 𝜷under 𝐻₁ and 𝐻₀ into (19)–(21), the final GLRT becomes ^𝐿_ℓ=1𝐲^†𝐸_ℓℓ⊗𝐂⁻¹𝐲𝐲−𝐓𝐤⊙𝜷^†𝐸_ℓℓ⊗𝐂⁻¹𝐲−𝐓𝐤⊙𝜷^𝐻1>_𝐻<₀𝛾_𝐺. (27)

5.2. GLRT-LQ Detector Design

We rewrite the detection problem as 𝐻₀∶𝑦_𝑚,𝑛=𝑧_𝑚,𝑛,𝐻₁∶𝑦_𝑚,𝑛=𝜌_{𝑀𝑚𝑁𝑛}𝛽(𝑚,𝑛)𝑇_𝑚,𝑛+𝑧_𝑚,𝑛, (28) where 𝜌_{𝑀𝑚𝑁𝑛}=(1+𝜌^(dr)_{𝑀𝑚𝑁𝑛}+𝜌_𝑀^(rd)_𝑚𝑁𝑛+𝜌^(rr)_{𝑀𝑚𝑁𝑛}); 𝜌^(dr)_{𝑀𝑚𝑁𝑛}, 𝜌_𝑀^(rd)_𝑚𝑁𝑛, 𝜌^(rr)_{𝑀𝑚𝑁𝑛} are the amplitudes of reflect coefficient, because the grazing angles are different; the reflect coefficient 𝜌^(dr)_{𝑀𝑚𝑁𝑛}≠𝜌_𝑀^(rd)_𝑚𝑁𝑛≠𝜌^(rr)_{𝑀𝑚𝑁𝑛}.

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 𝑝log𝑦∣𝐻₀𝑝𝑦∣𝐻₁^𝐻1_𝐻>₀<𝛾_{𝐺−𝐿𝑄}. (29)

If we assume that covariance matrix 𝐂_𝑐 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 Λ(𝑌)=𝑝𝑌∣𝐻₁𝑝𝑌∣𝐻₀=∏_𝑚,𝑛𝑝𝑦_𝑚,𝑛∣𝐻₁∏_𝑚,𝑛𝑝𝑦_𝑚,𝑛∣𝐻₀=_𝑚,𝑛𝑇1×1−^†𝑚,𝑛𝐂⁻¹_𝑚,𝑛𝑦_𝑚,𝑛²𝑇^†_𝑚,𝑛𝐂⁻¹_𝑚,𝑛𝑇_𝑚,𝑛𝑦^†_𝑚,𝑛𝑀⁻¹_𝑚,𝑛𝑦_𝑚,𝑛⁻¹_𝐻1>_𝐻<₀𝛾_{𝐺−𝐿𝑄}, (30) where 𝐂_𝑚,𝑛 is the covariance matrix for the 𝑚−𝑛 transmit-receive pair.

According to [24], the probability of false alarm 𝑃_fa is given by 𝑃_fa=𝑃^𝐼_𝑖=1Λ_𝑖𝑦_𝑖>𝛾_{𝐺−𝐿𝑄}∣𝐻₀=𝛾^−𝐿+1_{𝐺−𝐿𝑄}^𝐼−1_𝑖=0(𝑊−𝐼)^𝑖𝑖!ln𝛾_{𝐺−𝐿𝑄}^𝑖, (31) where 𝐼=𝑀𝑁 and 𝑊=𝑀_𝑚𝑁_𝑛.

The probability of detection 𝑃_𝑑 is given by 𝑃_𝑑=𝑃^𝐼_𝑖=1Λ_𝑖𝑦_𝑖>𝛾_{𝐺−𝐿𝑄}∣𝐻₀. (32)

For a given signal-to-clutter ratio (SCR), denoted by SCR_𝑚,𝑛, the amplitude of 𝛽(𝑚,𝑛) is given by 𝛽(𝑚,𝑛)=SCR_𝑚,𝑛𝜎², (33) 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 wellknown 𝐾 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 𝐂_𝑝,𝑞=𝜎²×0.9^{|𝑝−𝑞|}𝑗𝜋×exp2𝑝−𝑞. (34) Here, we select 𝜎²=10.17, 𝐿=16. We define the signal-to-clutter pulse noise ratio (SCNR) by [25] 1SCNR=𝐿∑^𝐿_ℓ=1[𝐓(𝐤⊙𝜷)]^†[𝐓(𝐤⊙𝜷)]𝐸{𝑢}tr𝐶_𝑛. (35)

Figure 3 depicts the detection performance using GLRT-LQ detectors, as a function of the SCNR. The probability of false alarm is fixed at 𝑃_fa=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.

Figure 4 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.

Figure 5 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.

Figures 6 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 𝑃_fa=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.

Figures 8 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 𝑃_fa=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.

Acknowledgments

The authors wish to thank the anonymous reviewers for their efforts in providing comments that have helped to significantly enhance the quality of this paper. This work was supported in part by the National Science Foundation of China under Grant no. 61302142.