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Low-Grazing Angle Detection in Compound-Gaussian Clutter with Hybrid MIMO Radar
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.
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 , higher sensitivity to or detection of moving targets , and increased degrees of freedom for transmission beamforming . MIMO radar with widely separated antennas can capture the spatial diversity of the target’s radar cross section (RCS) . 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 , and tracking performance . 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  investigated the detection problem of the MIMO radar system with distributed apertures in Gaussian colored noise and partially correlated observation channels. Tang et al.  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  investigated detection performance of MIMO radar for Rician target. In , the optimal detector in the Neyman-Pearson sense was derived for the statistical MIMO radar using orthogonal waveforms. The authors in  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  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 . 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 .
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 from the receiver is considered. If the source is assumed to be a narrowband signal, it can be represented by where is the amplitude, is the angular frequency, and 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 while indirect signal is where is the complex reflection coefficient, is the wave number, is wavelength, target range can be obtained from the time delay, and is the total length of the indirect path. Thus, the total received signal is given by
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.
In (3), the term is the complex reflection coefficient. It generally consists of the Fresnel reflection coefficient divided into the vertical polarization and horizontal polarization , the divergence factor due to a curved surface, and the surface roughness factor; that is, . The vertical polarization and horizontal polarization Fresnel reflection coefficients are, respectively, as presented in . Consider the following: For horizontal polarization, is the grazing angle and is the complex dielectric constant which is given by where is the relative dielectric constant of the reflecting medium and 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 is given by
The surface roughness factor is given by and is the surface roughness factor given by and 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 and subarrays for transmitting and receiving, respectively. The th transmit and th receive subarrays have, respectively, and closely spaced antennas. , and , and 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  where is the steering matrix of receive subarrays and component is the steering vector of th receive subarray at direction . denotes the RCSs for different transmit and receive subarray pairs with component . denotes the transmit signal matrix, where is the transmit waveform matrix, for each transmit subarray, the component , and , is the probing waveform of subarray; is the steering matrix of transmit subarray and the component is the steering vector of th transmit subarray at direction . is the received data matrix and denotes received signal of the th subarray. is the clutter matrix, each column of which is modeled as spherically invariant random vectors (SIRV), and 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 , where and are the texture and speckle components of the compound model, respectively. The fast-changing is a realization of a stationary zero mean complex Gaussian process, and the slow-changing is modeled as a nonnegative real random process .
We rewrite the received signal (11) in vector form, given by where , , , , is the vector operator, symbol denotes the Kronecker product. Then and is the compound Gaussian random vector with covariance matrix where is considered deterministic matrix with unknown parameters 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 and reflected point in ground is located at , . 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
The directly-directly path echo signal is given by (12). The directly-reflected path echo signal is where is amplitude of reflect coefficient and symbol represents the Hadamard product.
The reflected-directly path echo signal is
The reflected-reflected path echo signal is
Thus, the received signal of MIMO radar with multipath is where and is an 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:
Standard GLRT is the following decision rule: where and denote the probability density functions (pdfs) of the data under and , respectively. And the pdfs can be written, respectively, as where , and where and denote the determinant and the trace of a matrix, respectively.
The log-likelihood function of (21) is where denotes the elementary matrix with component and zero for others. Then, it is easy to obtain the Maximum Likelihood (ML) estimator of under ; that is
Thus, the ML estimator of is
The estimator is 
5.2. GLRT-LQ Detector Design
We rewrite the detection problem as where ; , , are the amplitudes of reflect coefficient, because the grazing angles are different; the reflect coefficient .
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
If we assume that covariance matrix is known and according to , and 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 where is the covariance matrix for the transmit-receive pair.
According to , the probability of false alarm is given by where and .
The probability of detection is given by
For a given signal-to-clutter ratio (SCR), denoted by , the amplitude of is given by where is the clutter power. In this paper, we consider that is the same for all and .
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 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 , the element of the covariance matrix of the speckle component, is chosen as Here, we select , . We define the signal-to-clutter pulse noise ratio (SCNR) by 
Figure 3 depicts the detection performance using GLRT-LQ detectors, as a function of the SCNR. The probability of false alarm is fixed at . For the given SCNR, the detection performance with multipath outperforms the one without considering multipath effect.
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 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.
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 ; 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.
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 . 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.
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.
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.
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