# Antenna Polarization Optimization for Target Detection in Non-Gaussian Clutter

Authors: Xu Cheng; Yong-zhen Li; Xue-song Wang
Journal: International Journal of Antennas and Propagation
 (2015)

---

## Abstract

Adaptive polarization design of radar antenna has recently become the focus of attention in radar polarization community. A polarimetric detector against non-Gaussian clutter with transmitter polarization optimization has been proposed in this paper. First, the radar data model including the realistic dependence of the clutter on the transmitted polarization is introduced. Then the polarimetric detector with transmitter polarization optimization is developed. By employing the simulation, we demonstrate that the polarization waveform optimization can bring the significant performance gain on target detection as compared to the conventional full-polarization approach. Besides, jointly optimizing transmitter and receiver polarization to form a scalar measurement is confirmed not to achieve a better detection performance than vector measurement with only transmitter polarization optimization.


---

## Body

## 1. Introduction

The detection of static or slowly moving targets in non-Gaussian clutter is considered as a difficult problem for a long time. Polarization diversity is an effective way to improve the performance of target detection. It has been widely studied during the last decades and extensively exploited for both military and civilian applications. The early studies on this topic are stated in [1], concerning the design of polarization filters and adaptive polarization cancelers. The recent work on polarization detection starts from [2], in which Novak et al. introduce the optimal polarimetric detector (OPD), the best linear polarimetric detector, and the polarimetric matched filter (PMF). Also new polarimetric target and clutter models are described there to predict the performance of the detectors. Here we highlight the OPD because it provides the best possible detection performance from polarimetric radar data, so seen as the optimal performance bound of polarimetric detection. But the OPD is derived based on the assumption of the known knowledge on target and clutter scattering matrices, which is unrealistic for many real radar applications. Thus, detection algorithms that use training data to estimate the clutter covariance matrix based on the Gaussian and compound-Gaussian distributions have been developed [3–9].More specifically, the polarimetric detectors combining polarization and space-time processing, with unknown polarimetric characteristics, embedded in the Gaussian clutter with unknown covariance matrix, have been addressed in [3, 4]. The proposed receivers ensure the constant false alarm rate (CFAR) property with respect to the clutter covariance. However, experimental evidences have shown that the Gaussian assumption can no longer be met for clutter returns in high resolution radars. As a consequence, a significant performance loss will occur for the detectors in [3, 4]. Real measurements have shown that clutter in high range resolution can be modeled as a compound-Gaussian process which consists of a Gaussian speckle component modulated by the “slowing varying” nonnegative random texture component. This motivates that the polarization diversity detection of targets embedded in compound-Gaussian clutter was addressed in [6–9]. In [7] an adaptive receiver is designed according to the generalized likelihood ratio (GLR) criteria, which also ensures the CFAR property with respect to the texture statistics. The detection algorithm in [8] employs a different GLR procedure and the quoted detector is faster to implement than the detector in [7] when the number of polarimetric channels to be processed is greater than the receiver in [7]. Different design strategies which can lead to decision statistics achieving higher detection probabilities or stronger robustness are considered in [9], in which two detection algorithms based on the Rao and the Wald tests [10] have been devised. Remarkably, the two detectors also guarantee CFARness with respect to the texture statistics. Moreover, the performance analysis shows that the Wald test achieves generally a performance level higher than the Rao test in the presence of a small amount of training data. However, it is necessary to point out that the aforementioned detectors based on the compound-Gaussian clutter are designed for radar systems with two polarimetric channels. When the polarimetric channels increase, the detection statistics can no longer retain the closed-form expression [7] or do not support the CFAR property anymore [8]. This represents serious limitations because four polarimetric channels or three polarimetric channels (based on the reciprocity) need to be employed when the complete polarization information is used.To overcome the existing problems, in [11] the authors have developed a polarimetric detector using only primary data vectors and have shown that this test statistic has the standard F-distribution. It is worth mentioning that the detector in [11] employs the adaptive transmitter antenna polarization design. Hence, the performance of the polarimetric detector can be improved by optimally choosing the polarization of the transmitted pulses to maximize the noncentrality parameter. As the evidences of the advantages of adaptive polarization technology, in [12] the full-polarization matched-illumination for target detection and identification has been investigated based on the assumption of known target and clutter polarization characteristics and a significant performance improvement over that corresponding to chirped full-polarization transmission waveforms has been verified with simulated target data. Then in [11, 13, 14] adaptive scheduling of radar polarization for unknown target and clutter responses to optimize the performance in target detection, estimation, and tracking is investigated and the significant performance gain over conventional polarization radar and single polarization radar also has been confirmed therein. Overall, the adaptive antenna polarization design can provide a potential performance improvement on target detection, estimation, tracking, and identification with respect to the conventional full-polarization radar.In this paper, we still address the problem of polarization detection design combined with adaptive polarization optimization and extend the results of [11] in a number of ways. First, the system response matrix is replaced from the vector-sensor array to the common polarization radar model, namely, dual transmitter antennas and dual receiver antennas (notice that the transmitter and receiver employ the single platform for monostatic radar systems and separate platforms for bistatic or multistatic systems; also see [1] where the block diagrams of a number of polarization diversity radars are shown). Second, we recast the full-polarization target and clutter scattering vectors from a 3-dimensional vector form into the 4-dimensional one which occurs for bistatic or multistatic radar so it represents a more general case. Third, the performance of scalar measurement with joint transmitter and receiver polarization optimization has been investigated to highlight the benefit of vector measurement.The remainder of the paper is organized as follows. Section2 describes the signal model. Section 3 focuses on the problem formulation and the derivation of the detector. Section 4 deals with the performance assessment of the detector with respect to non-Gaussian clutter and conventional full-polarization radar design, respectively. Section 5 illustrates that employing vector measurement provides the similar detection performance with the approach that linearly combines both received signals to give a scalar measurement. Finally, concluding remarks are given in Section 6.


## 2. Signal Model

Let us consider one point-like target illuminated by the radar system. The polarized waveform transmitted by the radar can be written as(1)st=ξst=ξh,ξvTst,
where ·T is the transpose operation, ξ is the transmitter polarization vector, and st is the complex envelope of the transmitted signal. Assume that ξF=1 since ξ contains only the transmitter polarization property without the transmitter power, where ·F is the Frobenius norm of the matrix.As the returns consist of not only the target echoes but also the undesired reflections from the environment, the recorded data corresponding to the range cell can be represented as(2)yt=gr2st+scξst-τ+nt,
where nt is the white noise vector, τ is the delay that resulted from waveform forward and backward propagation, r is the distance from the target to the radar, and g is a constant depending on the radar system characteristic such as operating frequency and antenna gain at the target illumination angle. st and sc are the target and clutter scattering matrices, respectively, completely describing the polarization transforming properties of the target and clutter. Generally, they are expressed in the following matrices:(3)st=shhtshvtsvhtsvvt,sc=shhcshvcsvhcsvvc.After performing matched filtering on (2) and then the normalization by absorbing the constant g/r2 into nt, the observation model is obtained as
(4)y=st+scξ+n,
where n is the new white noise vector. For notational convenience, in our problem we convert (4) into a linear observation model by the vectorization of st and sc. That is,
(5)xt=shhtshvtsvhtsvvtT,xc=shhcshvcsvhcsvvcT.
Let the system response matrix be
(6)H≝ξhξv0000ξhξv,
and (4) is reformulated as
(7)y=Hxt+Hxc+n.Then, to detect the target, multiple pulses of different polarizations need be transmitted to obtain the multiple measurementsy. Supposing there are M pulses during the radar dwell, the observation model from these M pulses can be written as
(8)ym=Hmxtm+Hmxcm+nm,m=1,2,…,M.As stated before, the target is considered to be stationary or slow-moving. Hence, it is reasonable to suppose that the scattering coefficient of the target is unchanged during the radar illumination. Thusxt is a deterministic vector. However, the clutter in the range cell is a large collection of point scatters producing incoherent reflections of the radar signal. So xc can be seen as a zero-mean complex Gaussian random vector with the covariance matrix Σ. The noise n can be considered to be a zero-mean complex Gaussian random vector with the covariance matrix σ2IM, where IM is the M×M identity matrix. In addition, we assume that the clutter reflections and the thermal noise are statistically independent. As a consequence, the distribution of each snapshot is
(9)ym~CNHxt,HΣHH+σ2IM,m=1,2,…,M,
where CN denotes a complex Gaussian distribution. ·H is the conjugate transpose operation. In (9)  H is known since the waveform and the direction in which it has been transmitted are known for the active radar systems. Meanwhile, the power of the thermal noise σ2 can be easily estimated from the recorded data when no signal has been transmitted. However, we have no prior knowledge of the target and clutter. Therefore, the vector xt and the matrix Σ are the unknown parameters of the statistical data model. Thus the GLR test for the detection algorithm design is employed.


## 3. Problem Formulation and Design Issues

In this section, the target detection algorithm is derived. Firstly, the test model could be written as(10)H0:xt=0,Σ,H1:xt≠0,Σ.
Based on the Neyman-Pearson test criterion [10], H1 is decided when
(11)ln⁡L=ln⁡f1y1,…,yM;x^t,Σ^1-ln⁡f0y1,…,yM;Σ^0>ln⁡γ,
where f0 and f1 are the likelihood functions under H0 and H1, respectively. Σ^0 and Σ^1 are the MLEs of Σ under H0 and H1. x^t is the MLE of xt under H1. γ is the detection threshold. For notational convenience, we notate ln⁡f0y1,…,yM;Σ^0 and f1y1,…,yM;x^t,Σ^1 as ln⁡f0Σ^0 and ln⁡f1x^t,Σ^1 in the following.Under hypothesisH0, assuming that x^t=0 the likelihood function is
(12)ln⁡f0Σ=-Mln⁡π+ln⁡C+tr⁡C-1s0,
where · denotes the determinant of the matrix and C=HΣHH+σ2IM is the theoretical covariance matrix of the data, defined in (9). s0 is the sample covariance matrix with
(13)s0=1M∑m=1MymymH.The MLE ofΣ is obtained as (see the derivation in Appendix A of [11])
(14)Σ^0=H+s0H+H-σ2HHH-1,
where H+=HHH-1HH is the pseudo-inverse matrix.Thus, the logarithmic likelihood function concentrated with respect toΣ is
(15)ln⁡f0Σ^0=-M4+ln⁡π-3ln⁡σ2+ln⁡HHH+σ-2tr⁡Π⊥s0+ln⁡H+s0H+H,
where Π⊥=1-HH+.Under hypothesisH1, the logarithmic likelihood function is
(16)ln⁡f1xt,Σ=-Mln⁡π+ln⁡C+tr⁡C-1C~1,
where
(17)C~1=1M∑m=1Mym-Hxtym-HxtH.The unknown parametersx^t and Σ^1 are (see the derivation in Appendix A of [11])
(18)x^t=H+y-Σ^1=H+s1H+H-σ2HHH-1,
where y- is the sample mean vector
(19)y-=1M∑m=1Mym
and s1 is the sample covariance matrix
(20)s1=1M∑m=1Mym-y-ym-y-*,
where ·* is the conjunction of complex number ·. Then the logarithmic likelihood function concentrated with respect to xt and Σ is
(21)ln⁡f1x^t,Σ^1=-M4+ln⁡π-3ln⁡σ2+ln⁡HHH+σ-2tr⁡Π⊥s0+ln⁡H+s1H+H.
Substituting the concentrated likelihood functions (21) and (15) into (11), with applying the following equality which is valid for any matrix s of dimension M×M(22)ln⁡H+sH+H=ln⁡HHsH-2ln⁡HHH,
the logarithmic GLR test statistic is obtained as
(23)ln⁡L=-Mln⁡HHs1H-ln⁡HHs0H.Since the following equality holds for the above equation [15]
(24)HHs0H=HHs1H1+y-HHHHs1H-1HHy-
then
(25)ln⁡L=Mln⁡1+y-*HHHs1H-1HHy-
is a monotonously increasing function of the term y-*HHHs1H-1HHy-. Therefore, the equivalent detection test statistic is written as
(26)T=y-*HHHs1H-1HHy-.
To this point, we obtain the test statistic for the signal model.


## 4. Performance Assessment

In this section, we will take the performance assessment of the proposed detection algorithm. First, the analytical expression of the detection performance is derived. After that the performance against non-Gaussian clutter is examined, and subsequently the gain of the transmitter polarization optimization over the approach with only horizontal polarization (H polarization) and the approach with only vertical polarization (V polarization) is verified to demonstrate the advantage of the proposed detector.Letzm=HHymm=1,…,M, and the test statistic (26) can be written as(27)T=z-Hsz-1z-,
where z- and sz are the samples mean and covariance, respectively, formed from a random sample of size M of the distribution CNHHHxt,HHHΣHHH+σ2HHH and
(28)z-=1M∑m=1Mzm,sz=1M∑m=1Mzm-z-zm-z-H.Applying Corollary5.2.1 from [16], the detection statistic is distributed as follows:
(29)TM-44~F8,2M-4underH0F8,2M-4′λunderH1,
where Fυ1,υ2 denotes an F-distribution with υ1 and υ2 degrees of freedom and Fυ1,υ2′λ denotes a noncentral F-distribution with υ1 and υ2 degrees of freedom and noncentrality parameter λ. λ is given by
(30)λ=2MxtHHHHHHHΣHH+σ2H-1HHHxt=2MxtHH+HΣHH+σ2H+H-1xt=2MxtHΣ-1xt-xtHΣ+ΣHHHΣσ2-1xt.
Finally, the detection performance is obtained as
(31)pfa=QF8,2M-4γ′,pd=QF8,2M-4′λγ′,
where Q is the right-tail probability function and γ′ is the detection threshold for the required probability of false alarm. In particular, note that the expression for pfa does not depend on the covariance of clutter and thermal noise, nor on the transmitted signal; thus (31) is a CFAR test.


### 4.1. The Performance against Non-Gaussian Clutter

The test statistic of the proposed detector is computed with only the data of the test cell and without the secondary cells. Therefore, it should have good performance against the non-Gaussian clutter environment. To demonstrate the robustness of the proposed detector against non-Gaussian clutter, the OPD [2], polarization-space-time GLR detector (PST-GLR) [3], and texture-free GLR (TF-GLR) detector [5] are chosen as its counterparts to evaluate the detection performance. As the PST-GLR and TF-GLR detectors employ secondary data but have no analytical expression in non-Gaussian clutter, the Monte Carlo simulation is employed. The compound-Gaussian distribution is chosen as the clutter model with the covariance matrix τΣ, for which τ is the texture and meets the generalized Gamma probability distribution
(32)pτ=1Γννδντν-1e-ν/δτ,
where ν is the order and δ is the average power. Note that when ν=+∞, (32) corresponds to clutter with Gaussian distribution; when ν decreases, (32) deviates from Gaussian.SetξH=2/2 and ξV=2/2 for H in (6) because conventional full-polarization radar systems alternatively or simultaneously transmit H and V polarization. Also let δ=50, Σ=10000.10001. For different value of ν we generate the corresponding clutter data with the length of test range cell N=1, the guard cells H=2 and secondary cells K=32 in both sides, respectively, and the total number of snapshots Mt=104. But as for each trial, 10 snapshots of the clutter data are employed, which means M=10 in (9). The Monte Carlo simulation is run for 105 times. Figure 1 shows the result of the values of pfa as a function of the order ν, which clearly shows that the proposed detector and the OPD keep their false alarm rates when the clutter becomes non-Gaussian. As the counterparts, the false alarm rates of TF-GLRT and PST-GLRT detectors increase obviously when the clutter departs from Gaussian. This confirms that the proposed approach has the robustness in non-Gaussian clutter.Figure 1
The values ofpfa versus the order parameter ν of the clutter texture distribution for the proposed detector, OPD, and PST-GLR and TF-GLR detectors.


### 4.2. Performance Comparison with the Conventional Approach

In this section, we will demonstrate the advantage of the transmitter polarization optimization over the approach with fixedH or V polarization. In only H or V polarization systems, the system response matrix of transmitter polarization is H=10000010 or H=01000001, respectively. For adaptive transmitter polarization systems, suppose xt and Σ are well estimated based on (18); then H can be designed to maximize λ in (30) and subsequently the probability of detection. Here the clutter covariance matrix in the following form is chosen so that the covariance matrix is a unitary matrix with the full rank:
(33)Σ=UΛΛHUH,
where a unitary matrix is randomly picked for U, which is constructed with the left singular vectors of a 4 × 4 matrix M with i.i.d. complex Gaussian entries; that is, the singular value decomposition of M can be written as M=UΛMUr. Λ is a diagonal matrix and its diagonal elements are positive and are chosen randomly. Meanwhile, the target vector is randomly chosen, but the signal-to-clutter ratio (SCR) is kept in the certain level. The definition of the SCR and clutter-to-noise ratio (CNR) is given in the Appendix.The grid search method is used to seek the optimal transmitter polarizationH for the transmitter adaptive system. The power of noise is fixed to be σ2=0.1 and the probability of false alarm is fixed to be pfa=10-3, and the value of SCR varied. For each SCR, 105 Monte Carlo runs are taken to obtain the mean value of λ and then the probability of detection pd. In Figure 2, the curves of pd as a function of the SCR for the detectors with the transmitter polarization optimization, fixed H transmitter polarization, and fixed V transmitter polarization are plotted. As shown in the figure, a significant improvement in performance is visible with using the optimal waveform polarization.Figure 2
Detection performance comparison as a function of the SCR for the detectors with the transmitter polarization optimization,H transmit polarization, and V transmit polarization, respectively; pfa=10-3 and σ2=0.1.Furthermore, let SCR be equal to 0 dB withσ2=0.1 and then vary the value of pf. For each pf take 105 Monte Carlo runs to obtain the mean value of λ and then the probability of detection pd. pds as a function of pfs are plotted for the detectors with the transmitter polarization optimization and also fixed H transmitter polarization and fixed V transmitter polarization. Also it can be seen from Figure 3 that the striking improvement in performance is offered by the polarization optimization.Figure 3
Detection performance comparison as a function of the probability of false alarmpf for the detectors with optimal transmitter polarization, fixed H transmitter polarization, and V transmitter polarization, respectively; SCR=0dB and σ2=0.1.Besides, the influence of CNR on the performance of the proposed detector is investigated. That is, to fix SCR and the false alarm rate and then check the change of probability of detection with CNR, letSCR=10dB, let pfa=10-3, and vary the value of σ2. With the simulation approach being the same as the numerical examples before, the probabilities of detection for three detectors employing optimal transmitter polarization, fixed H transmitter polarization, and V transmitter polarization, respectively, as a function of the probability of false alarm have been plotted in Figure 4. The numerical results show that when CNR increases, namely, σ2 is reduced, the probabilities of detection are higher than before. This coincides with (30), in which the increase of σ2 means the decrease of λ, finally resulting in the loss of the probability of detection.Figure 4
Detection performance comparison as a function of CNR for the detectors with the transmitter polarization optimization,H transmit polarization, and V transmit polarization, respectively; pfa=10-3 and SNR=10dB.To this point, it has been demonstrated that, by optimizing transmitter polarization design, radar systems will obtain the significant improvement on the performance of target detection. Then one could think that joint transmitter and receiver polarization optimization will certainly have a better performance than only transmitter polarization optimization. However, the fact could oppose the judgment. Detailed discussion on this topic will be provided in the next section.


## 4.1. The Performance against Non-Gaussian Clutter

The test statistic of the proposed detector is computed with only the data of the test cell and without the secondary cells. Therefore, it should have good performance against the non-Gaussian clutter environment. To demonstrate the robustness of the proposed detector against non-Gaussian clutter, the OPD [2], polarization-space-time GLR detector (PST-GLR) [3], and texture-free GLR (TF-GLR) detector [5] are chosen as its counterparts to evaluate the detection performance. As the PST-GLR and TF-GLR detectors employ secondary data but have no analytical expression in non-Gaussian clutter, the Monte Carlo simulation is employed. The compound-Gaussian distribution is chosen as the clutter model with the covariance matrix τΣ, for which τ is the texture and meets the generalized Gamma probability distribution
(32)pτ=1Γννδντν-1e-ν/δτ,
where ν is the order and δ is the average power. Note that when ν=+∞, (32) corresponds to clutter with Gaussian distribution; when ν decreases, (32) deviates from Gaussian.SetξH=2/2 and ξV=2/2 for H in (6) because conventional full-polarization radar systems alternatively or simultaneously transmit H and V polarization. Also let δ=50, Σ=10000.10001. For different value of ν we generate the corresponding clutter data with the length of test range cell N=1, the guard cells H=2 and secondary cells K=32 in both sides, respectively, and the total number of snapshots Mt=104. But as for each trial, 10 snapshots of the clutter data are employed, which means M=10 in (9). The Monte Carlo simulation is run for 105 times. Figure 1 shows the result of the values of pfa as a function of the order ν, which clearly shows that the proposed detector and the OPD keep their false alarm rates when the clutter becomes non-Gaussian. As the counterparts, the false alarm rates of TF-GLRT and PST-GLRT detectors increase obviously when the clutter departs from Gaussian. This confirms that the proposed approach has the robustness in non-Gaussian clutter.Figure 1
The values ofpfa versus the order parameter ν of the clutter texture distribution for the proposed detector, OPD, and PST-GLR and TF-GLR detectors.


## 4.2. Performance Comparison with the Conventional Approach

In this section, we will demonstrate the advantage of the transmitter polarization optimization over the approach with fixedH or V polarization. In only H or V polarization systems, the system response matrix of transmitter polarization is H=10000010 or H=01000001, respectively. For adaptive transmitter polarization systems, suppose xt and Σ are well estimated based on (18); then H can be designed to maximize λ in (30) and subsequently the probability of detection. Here the clutter covariance matrix in the following form is chosen so that the covariance matrix is a unitary matrix with the full rank:
(33)Σ=UΛΛHUH,
where a unitary matrix is randomly picked for U, which is constructed with the left singular vectors of a 4 × 4 matrix M with i.i.d. complex Gaussian entries; that is, the singular value decomposition of M can be written as M=UΛMUr. Λ is a diagonal matrix and its diagonal elements are positive and are chosen randomly. Meanwhile, the target vector is randomly chosen, but the signal-to-clutter ratio (SCR) is kept in the certain level. The definition of the SCR and clutter-to-noise ratio (CNR) is given in the Appendix.The grid search method is used to seek the optimal transmitter polarizationH for the transmitter adaptive system. The power of noise is fixed to be σ2=0.1 and the probability of false alarm is fixed to be pfa=10-3, and the value of SCR varied. For each SCR, 105 Monte Carlo runs are taken to obtain the mean value of λ and then the probability of detection pd. In Figure 2, the curves of pd as a function of the SCR for the detectors with the transmitter polarization optimization, fixed H transmitter polarization, and fixed V transmitter polarization are plotted. As shown in the figure, a significant improvement in performance is visible with using the optimal waveform polarization.Figure 2
Detection performance comparison as a function of the SCR for the detectors with the transmitter polarization optimization,H transmit polarization, and V transmit polarization, respectively; pfa=10-3 and σ2=0.1.Furthermore, let SCR be equal to 0 dB withσ2=0.1 and then vary the value of pf. For each pf take 105 Monte Carlo runs to obtain the mean value of λ and then the probability of detection pd. pds as a function of pfs are plotted for the detectors with the transmitter polarization optimization and also fixed H transmitter polarization and fixed V transmitter polarization. Also it can be seen from Figure 3 that the striking improvement in performance is offered by the polarization optimization.Figure 3
Detection performance comparison as a function of the probability of false alarmpf for the detectors with optimal transmitter polarization, fixed H transmitter polarization, and V transmitter polarization, respectively; SCR=0dB and σ2=0.1.Besides, the influence of CNR on the performance of the proposed detector is investigated. That is, to fix SCR and the false alarm rate and then check the change of probability of detection with CNR, letSCR=10dB, let pfa=10-3, and vary the value of σ2. With the simulation approach being the same as the numerical examples before, the probabilities of detection for three detectors employing optimal transmitter polarization, fixed H transmitter polarization, and V transmitter polarization, respectively, as a function of the probability of false alarm have been plotted in Figure 4. The numerical results show that when CNR increases, namely, σ2 is reduced, the probabilities of detection are higher than before. This coincides with (30), in which the increase of σ2 means the decrease of λ, finally resulting in the loss of the probability of detection.Figure 4
Detection performance comparison as a function of CNR for the detectors with the transmitter polarization optimization,H transmit polarization, and V transmit polarization, respectively; pfa=10-3 and SNR=10dB.To this point, it has been demonstrated that, by optimizing transmitter polarization design, radar systems will obtain the significant improvement on the performance of target detection. Then one could think that joint transmitter and receiver polarization optimization will certainly have a better performance than only transmitter polarization optimization. However, the fact could oppose the judgment. Detailed discussion on this topic will be provided in the next section.


## 5. Scalar Measurement with Adaptive Transmitter and Receiver Polarization Design

Many conventional polarization radar systems unite the two received signals linearly and coherently at the receiver to form a scalar measurement [1]. For such systems, the output at the receiver is an inner product of the returns and the receiver antenna polarization. The receiver polarization vector is optimally chosen along with the transmitter polarization to achieve improved performance. In this section the signal model for such systems is briefly provided as the main derivation is similar to the approach in Section 2, and then the performance of the detector with joint transmitter and receiver optimization is discussed.Letη=ηh,ηvT be the receiver polarization vector, where ηF=1. The rest of the variables are defined the same as in Section 2. Then the scalar measurement is represented as
(34)yt=gr2ηTst+scξst-τ+nt.
After the signal passes through a series of matched filters and is then normalized to move the effect of g/r2 into the noise term, the output at the receiver is
(35)yt=ηTst+scξ+n.
Let
(36)H=ηhξh,ηhξv,ηvξh,ηvξv.
Then convert (35) into a linear observation model as follows:
(37)y=Hxt+Hxc+n.Equation (37) is the signal model for scalar measurement systems. The only difference with vector measurement in Section 2 is on H. The detector is proven to have the same form as the earlier model. Also the performance formulation can be expressed in the same formulation except the difference of H. As done in Section 4.2, the optimal system response matrix H of jointly optimizing transmitter and receiver polarization in order to make the probability of detection maximum is computed with grid search method. Fix the power of noise to be σ2=0.1 and pfa=10-3 and vary the value of SCR. For each SCR 105 Monte Carlo runs are taken to obtain the mean value of λ and then pd. The curves of probability of detection as a function of the SCR for the detectors with the transmitter polarization optimization and joint transmitter and receiver polarization are provided in Figure 5, which clearly shows that the detection performances of the two optimization approaches are almost the same.Figure 5
Detection performance comparison as a function of the SCR for the detectors with the transmitter polarization optimization and joint transmitter and receiver polarization optimization, withpfa=10-3 and σ2=0.1.Then, let SCR be equal to 0 dB, letσ2=0.1, and vary the value of pf. For each pf 105 Monte Carlo runs are taken to obtain the mean value of λ and then the probability of detection pd. pds as a function of pf for the detectors with joint transmitter and receiver polarization optimization and the transmitter polarization optimization, respectively, are plotted in Figure 6. The similar performance offered by the different polarization optimization is clear from Figure 6. However, the optimization for the joint optimization system will include not only optimization over the transmitter polarization but also the receiver polarization, so more onerous computational burden is faced as compared to the approach of only transmitter polarization optimization, which highlights the advantage of vector measurement.Figure 6
Detection performance comparison as a function of the probability of false alarmpf for the detectors with the transmitter polarization optimization and joint transmitter and receiver polarization optimization. SCR=0 dB and σ2=0.1.


## 6. Conclusion

In this paper, we investigated the transmitter polarization optimization for the detection of a target in a non-Gaussian clutter environment. We derive a detector with transmitter waveform design. Then we demonstrate its better performance against non-Gaussian clutter comparing with three other detectors by using the numerical example. Our numerical results also demonstrate that, by carefully choosing the transmitter polarizations, the detection performance could be sufficiently improved compared to the conventional way. Meanwhile, we also point out and verify that joint transmitter and receiver optimization to form a scalar measurement has not shown a better performance than only transmitter polarization optimization. In the future, we will conduct research on the computational burden reduction on seeking for the optimal polarization, also the polarization adaptive detection of extended target in non-Gaussian clutter.



---

*Source: 640730-2015-02-12.xml*