- Research
- Open Access

# Target scattering estimation in clutter with polarization optimization

- Xu Cheng
^{1}Email author, - Longfei Shi
^{1}, - Yuliang Chang
^{1}, - Yongzhen Li
^{1}and - Xuesong Wang
^{1}

**2015**:105

https://doi.org/10.1186/s13634-015-0286-y

© Cheng et al. 2015

**Received:**28 March 2015**Accepted:**12 November 2015**Published:**18 December 2015

## Abstract

In this paper, we propose an adaptive waveform polarization method for the estimation of target scattering matrix in the presence of clutter. The proposed sequential algorithm, based on the concept of sequential minimum mean square error (MSE) estimation, to determine the coefficients of the scattering matrix, guarantees the convergence and the resulting computational complexity is linear with the number of iterations. The effectiveness of the proposed method is validated through numerical results, underlining the performance improvement given by joint transmission and reception (Tx/Rx) polarization optimization for the scalar system. Also, the results show that the vector system with transmission polarization optimization provides a comparative performance as the scalar measurement system employing joint Tx/Rx polarization optimization. Less computation burden highlights the advantage of the former mode.

## Keywords

- Radar polarization
- Minimum MSE estimation
- Sequential algorithm
- Waveform design
- Scattering matrix

## 1 Introduction

Polarization, together with the amplitude, time, frequency, phase, and bearing descriptor of radar signals, completes the information description of target returns from radars. The exploitation of information on the echo polarization can provide a significant improvement on radar performance [1]. To obtain the target polarimetric scattering information, conventional polarimetric radar systems alternately switch (or simultaneously transmit) the horizontal (H) and vertical (V) polarizations at the transmission side, and simultaneously receive both polarizations at the reception side, consequently resulting in four polarization combinations: HH, HV, VH, and VV polarizations.

Motivated in part by recent advances in the hardware and sensor information processing, such as solid state transmission and digital arbitrary waveform generators, in modern radar systems, any polarization on either transmission or reception can be synthesized by using the linear combination of the H and V components. Thus, besides the four types of transmitter/receiver combinations above, polarimetric radar can achieve any pair of transmitter/receiver polarizations. Such flexibility greatly enhances the polarimetric sensing capability of the radar system. For the mentioned reason, several papers concerning polarization optimization have appeared in the open literature during the last two decades aiming at the performance of polarimetric radars on target estimation [2, 3], detection [4–8], tracking [9], and identification [4, 10].

This paper will tackle with the problem of adaptively selecting waveform polarization to optimally estimate the target scattering matrix. As to this topic, the technique of optimizing transmission polarization for “vector measurement systems" [10] (with the term “vector measurement system”, the authors refer to the reception side of the radar that receives separately the horizontal and vertical polarization components to form a vector) has been addressed in [2]. The numerical results there show that, by optimally selecting waveform polarization, the performance of the target estimation can be significantly improved compared to the fixed polarization design. In most cases, polarimetric radar systems combine two received signals linearly and coherently at the receiver to give a scalar measurement [1]. Following [6], we call the radar employing such measurement mode as the “scalar measurement system”. As to this kind of system, in [3], the authors devise an optimization algorithm to jointly design Tx/Rx polarizations that minimize the MSE of estimating the target scattering vector. The original problem therein is formulated into a convex form which is solvable employing semi-definite programming (SDP). As a subfield of convex optimization, SDP concerns with the optimization of a linear objective function to be maximized or minimized over the intersection of the cone of positive semi-definite matrices with an affine space, i.e., a spectrahedron [11]. Such optimization problems can be solved via the well-known CVX toolbox for Matlab software [12]. But unfortunately, we find the problem formulation in [3] cannot be effectively solved by exploiting the devised algorithm (see Appendix 1 for details). In fact, the SDP formulation there is a relaxation version of the original problem to be handled.

Hence, in this paper, we reconsider the problem of adaptive polarization design of polarimetric radars to optimally estimate the target scattering vector in clutter. Starting from scalar measurement systems, we design an optimization procedure for the Tx/Rx polarizations to sequentially minimize the MSE of estimating the target scattering vector. The proposed algorithm converges to a certain value, and the computational complexity is linear with the number of iterations. Furthermore, we generalize the algorithm to the vector measurement system to optimally design the transmission polarization. At the analysis stage, we validate the effectiveness of the proposed method through numerical results, highlighting the performance improvement of the joint Tx/Rx polarization optimization for the scalar system. Our results show that the vector system with transmission polarization optimization offers a comparative performance as the scalar system employing joint Tx/Rx polarization optimization.

The remainder of the paper is organized as follows. The measurement model of the scalar measurement system with jointly adaptive Tx/Rx polarization design is introduced in Section 2. In Section 3, the problem formulation is described. The sequential method to optimally select waveform parameter with affordable computation burden is proposed in Section 4. Then, in Section 5, the devised algorithm is exploited to the vector measurement system. Numerical results are presented in Section 6, demonstrating the improvement in performance achieved by using the joint Tx/Rx polarization in comparison with the two other scalar measurement systems, as well as highlighting the advantage of vector measurement systems with respect to scalar ones. Finally, the conclusions are provided in Section 7.

## 2 Measurement model

In this section, we formulate the parametric measurement model of target scattering vector in additive Gaussian distributed clutter for the scalar measurement system with adaptive Tx/Rx polarization designs.

*w*(

*t*) is the white noise,

*r*is the distance from the target to radar,

*s*(

*t*) is the transmission waveform,

*τ*is the delay resulted from the waveform forward and backward propagation, and

*g*is a constant depending on the radar system characteristics including operating frequency, permittivity, permeability of free space, antenna gain at the target illumination angle, radar reception power, etc. S

_{t}and S

_{c}are the target and clutter scattering matrices, respectively, with the following matrix representation:

Here, we remark that the linear assumption has been employed to linearize the electromagnetic scattering problem. From a more general point of view, the electromagnetic scattering problem is actually highly non-linear. But the linearity assumption is usually employed to simplify the problem at hand. In this paper, we make the linearity assumption following [3, 5, 9], to address the similar signal model.

_{t}and S

_{c}, we could convert (3) into the observation model

*N*pulses with different polarizations used to estimate the fully polarimetric target information X

_{t}, the observation from these

*N*pulses can be written as

*i*) and η(

*i*) denote, respectively, the transmission and reception polarizations of the

*i*th pulse, i.e., \( \boldsymbol{a}\left(\boldsymbol{\xi} (i),\boldsymbol{\eta} (i)\right)={\left[{\xi}_{\mathrm{h}}^{(i)}{\eta}_{\mathrm{h}}^{(i)}\ {\xi}_{\mathrm{h}}^{(i)}{\eta}_{\mathrm{v}}^{(i)}\ {\xi}_{\mathrm{v}}^{(i)}{\eta}_{\mathrm{h}}^{(i)}\ {\xi}_{\mathrm{v}}^{(i)}{\eta}_{\mathrm{v}}^{(i)}\right]}^{\mathrm{T}} \). For notational simplicity, we define a(

*i*) ≜ a(ξ(

*i*), η(

*i*)). Introducing further the following vector notations

**A**here is an

*N*× 4 complex matrix; so we assume that

*N*> 4 and ensure rank(

**A**) = 4 to estimate four-dimensional complex vector X

_{t}[3, 5].

As to the target and clutter statistical characterizations, we do not make restrictive model assumption on the multivariate statistical characterization of X
_{t} and X
_{c}. However, we assume that we have access to the first two moments of their probability density function (PDF). This assumption is reasonable, especially in a knowledge-aided (possibly cognitive) radar scenario. In such case, clutter statistical parameters can be obtained jointly using geographical information, meteorological measurements, and statistical (possibly empirical) models for the clutter [14]. The statistical parameters of the illuminated target can be obtained or roughly estimated by pre-scan procedures and employing cognitive methods [14–16]. Precisely, we assume X
_{t} is a four-dimensional random vector of parameters whose realization is to be estimated and has mean *E*(X
_{t}) and covariance matrix C
_{t}, and X
_{c} is a four-dimensional random vector with zero mean and covariance matrix C
_{c} and is uncorrelated with X
_{t}, where *E*(⋅) denotes the expectation operation. As to the statistical characterization of the noise vector, we assume that it is Gaussian white with covariance matrix \( {\sigma}_{\nu}^2{\mathbf{I}}_4 \), where **I**
_{4} is the identity matrix with a 4 × 4 size. Finally, we assume that the target, clutter, and noise are uncorrelated, so that the joint PDF \( p\left({\boldsymbol{X}}_{\mathrm{t}},{\boldsymbol{X}}_{\mathrm{c}},\mathbf{v}\right) \) is arbitrary.

## 3 Problem formulation

_{t}from the observations. By inspection of the measurement model in the previous section, it matches the Bayesian linear model form. By exploiting the Bayesian Gauss-Markov Theorem [[17], Theorem 12.1], we obtain the minimum MSE estimator of X

_{t}as

**I**

_{ N }is the identity matrix with a

*N*×

*N*size and (·)

^{H}denotes the conjugate transpose. The performance of the estimator is measured by the error \( \boldsymbol{e}={\boldsymbol{X}}_{\mathrm{t}}-{\widehat{\boldsymbol{X}}}_{\mathrm{t}} \) whose mean value is zero and covariance matrix is

*i*th diagonal element of D as the minimum Bayesian MSE of X

_{t}’s

*i*th element [Kay [17], Theorem 12.1]. Hence, the trace of D, i.e., Tr(D), represents the sum of the minimum Bayesian MSEs of all X

_{t}’s four elements. As a consequence, we define the MSE of X

_{t}as Tr(D). As done in [3], we consider Tr(D) as the relevant figure of merit and minimize it to optimize transmission and reception polarizations.

_{t}with

*N*diversely polarized pulses, can be formulated as

*Remarks* Actually, there exist several potential optimality criteria for the problem at hand, e.g., minimizing the determinant of D, minimizing the maximum eigenvalue of D and the aforementioned Tr(D), etc. [19]. In this paper, we employ Tr(D) because of the following: firstly, it is reasonable, as stated before and secondly, we exploit the same problem formulation as in [3] and then highlight our design.

## 4 Optimal waveform selection

In the sequel, we will deal with the non-convex (the objective is non-convex and ‖ξ(*i*)‖ = ‖η(*i*)‖ = 1 defines a non-convex set) optimization problem \( \mathcal{P} \). By reformulating the problem using trigonometric parameters and the inspection of its computational complexity, we exploit the sequential minimum MSE estimation scheme to compute the optimal Tx/Rx polarizations instead of directly utilizing the minimum MSE estimation.

### 4.1 Problem reformulation

^{ jφ }is the complex envelope of the source signal,

*α*denotes the rotation angle between the system coordinates and the electric ellipse axes, and

*β*determines the ellipse’s eccentricity, respectively, with the definition spaces of these trigonometric parameters being

*φ*∈ (−

*π*,

*π*],

*α*∈ [−

*π*/2,

*π*/2], and

*β*∈ [−

*π*/4,

*π*/4]. Since the transmitted signal is power limited, i.e., ‖ξ‖ = 1, by replacing

*Q*and w with their explicit forms, we can rewrite (13) as

*ϕ*∈ (−

*π*,

*π*],

*θ*∈ [−

*π*/2,

*π*/2], and

*ϑ*∈ [−

*π*/4,

*π*/4].

We can observe that, once ξ and η are written into (14) and (15), respectively, Tr(D) can be uniquely determined by trigonometric parameters {*φ*, *α*, *β*, *ϕ*, *θ*, *ϑ*}. Before proceeding further, herein we introduce an interesting observation of Tr(D) with respect to such parameters, as summarized in following property.

**Property 1**
*The initial signal phases φ and ϕ do not affect the value of*
D.

*Proof* See Appendix 2.

*α*,

*β*,

*θ*,

*ϑ*} instead of six trigonometric parameters {

*φ*,

*α*,

*β*,

*ϕ*,

*θ*,

*ϑ*}. Thus, the problem \( \mathcal{P} \) can be equivalently recast as

We can see that \( {\mathcal{P}}_1 \) is still a non-convex optimization problem because the objective function is the same non-convex function as in \( \mathcal{P} \). Nevertheless, as opposed to \( \mathcal{P} \), the equivalent formulation provided by \( {\mathcal{P}}_1 \) shows that lattice search along the trigonometric parameters {*α*, *β*, *θ*, *ϑ*} can be employed to calculate the optimal polarization, as done in [2] and [5]. But since lattice search does not explore any optimization property, it requires a high computational burden. Precisely, the computational complexity is firstly exponential to the observation number *N*. Also, it highly depends on the maximum point search algorithm used for {*α*, *β*, *θ*, *ϑ*}. If we use lattice search with *l*
_{
α
}, *l*
_{
β
}, *l*
_{
θ
} and *l*
_{
ϑ
} points in each dimension of the domain space {*α*, *β*, *θ*, *ϑ*}, respectively, the relevant complexity burden is *O*((*l*
_{
α
}
*l*
_{
β
}
*l*
_{
θ
}
*l*
_{
ϑ
})^{
N
}).

### 4.2 Sequential minimum MSE estimation

Since to solve \( {\mathcal{P}}_1 \) straightforward requires a high computational burden, we exploit the sequential minimum MSE estimation to tackle it. By insight of \( {\mathcal{P}}_1 \), we can observe that it works at the case of block signal polarization design, namely designing the polarizations of *N* consecutive temporal signal samples at a time. However, in real radar signal processing applications, the returns are ongoing as time progresses. So it is reasonable to process the data sequentially in time. As to the problem at hand, the resulting solution belongs to the sequential minimum MSE estimation. Thus, by following the sequential minimum MSE estimation procedure in [Kay [17], Eq. (12.47)–(12.49)], we develop the sequential minimum MSE estimation to handle \( {\mathcal{P}}_1 \) as follows.

*y*(1), …,

*y*(

*n*)]

^{T}and D

_{ n }be the corresponding minimum MSE matrix (just the sequential version of (10) and (11)). Then, when the new sample

*y*(

*n*+ 1) is available, the estimator is updated as

**A**

_{ n + 1}= [a(1), …, a(

*n*+ 1)]

^{T}and a(

*i*) having the same definition as before and [·]

_{ n + 1,n + 1}denoting the

*n*+ 1th diagonal element of matrix [·]. In the meantime, the minimum MSE matrix is updated as

As to D
_{
n + 1}, based on [Kay [17], p. 393], with *n* increasing, [D
_{
n + 1}]_{
ii
}, *i* = 1, 2, 3, 4, decreases and converges to a certain value. Hence, tr(D
_{
n + 1}) is also a monotonic decreasing sequence and converges to a certain value.

### 4.3 Optimal waveform parameter selection

_{ n }). As to each iteration, we still employ lattice search to compute the optimal Tx/Rx polarizations. Precisely, the sequential procedure to update the optimal polarizations is summarized as Algorithm 1. Also, Fig. 1 exhibits a pictorial representation of the sequential estimation processing.

Now let us examine the computational complexity of Algorithm 1. Let us still consider that there are *N* observation samples. Different from solving \( {\mathcal{P}}_1 \), Algorithm 1 is linear to the observation number *N*, with requiring the lattice search for the optimal polarizations in the domain space {*α*, *β*, *θ*, *ϑ*} in each iteration. Thus, the overall complexity is *O*(*l*
_{
α
}
*l*
_{
β
}
*l*
_{
θ
}
*l*
_{
ϑ
}
*N*). In comparison with the direct lattice search to solve \( {\mathcal{P}}_1 \), the computation burden is greatly reduced. It is not surprising since the proposed algorithm is sequential, namely it does not optimize for all observations and thus, it is clearly less expensive than the one proposed in (16).

## 5 Vector measurement system

Vector sensors, which employ two-dimensional sensors to measure both horizontal and vertical components of the electric field at each of the receivers, provide a significant improvement in performance over scalar sensors for a variety of applications [10, 18]. For such mentioned reason, in this section, we develop the approach to optimize transmission polarization of vector measurement systems by following the similar way of designing the optimal Tx/Rx polarizations for the aforementioned scalar system. In general, the derivation of such system model and the corresponding adaptive design process follow the similar way as the model in Section 2, so it is simplified.

### 5.1 Vector measurement model

*N*-dimensional vectors

**y**′ and \( {\mathbf{v}}^{\mathbf{\prime}} \), i.e., \( {\mathbf{y}}^{\mathbf{\prime}}={\left[{y}_{\mathrm{h}}(1),{y}_{\mathrm{v}}(1),\dots, {y}_{\mathrm{h}}(N),{y}_{\mathrm{v}}(N)\right]}^{\mathrm{T}} \) and \( {\mathbf{v}}^{\mathbf{\prime}}={\left[{\nu}_{\mathrm{h}}(1),{\nu}_{\mathrm{v}}(1),\dots, {\nu}_{\mathrm{h}}(N),{\nu}_{\mathrm{v}}(N)\right]}^{\mathrm{T}} \), respectively. Vectors X

_{t}and X

_{c}remain the same as defined earlier.

*i*= 1, …,

*N*have the same power with \( \frac{\sigma^2}{2} \) and are uncorrelated with each other. Then, the covariance matrix D′ of X

_{t}’s minimum MSE estimation from

**y**′ is

*φ*does not affect the value of D′. Since the relevant proof can be obtained similarly as that given for Property 1, it is omitted for the sake of brevity. Hence, the optimization problem of seeking the optimal transmission polarization to minimize the MSE of estimating X

_{t}for vector systems can be formulated as

Similar in solving \( {\mathcal{P}}_1 \), the computational complexity of solving \( {\mathcal{P}}_2 \) depends exponentially on the observation number *N*, as well as the complexity burden required at each observation, i.e., the maximum point search algorithm used for parameters {*α*, *β*}. Considering employing lattice search with *l*
_{
α
} and *l*
_{
β
} points in each dimension of the domain space {*α*, *β*}, the required complexity is *O*((*l*
_{
α
}
*l*
_{
β
})^{
N
}).

### 5.2 Sequential estimation algorithm for vector systems

*n*+ 1), the estimator is updated as

**T**

_{ n + 1}= [t(1), …, t(

*n*+ 1)]

^{T}and [·]

_{2(n + 1),2(n + 1)}denoting the 2(

*n*+ 1)th diagonal element of matrix [·]. In the meantime, the covariance matrix is updated as

*n*increasing, \( {\left[{\boldsymbol{D}}_{n+1}^{\boldsymbol{\prime}}\right]}_{ii} \) decreases and converges to a certain value. As a consequence, \( \mathrm{t}\mathrm{r}\left({\boldsymbol{D}}_{n+1}^{\boldsymbol{\prime}}\right) \) is also a monotonic decreasing sequence and converges to a certain value. Finally, the devised sequential optimization procedure is summarized in Algorithm 2.

It is worth noticing that the computational complexity of Algorithm 2 is linear to the observation number *N*, with requiring the lattice search to seek the optimal polarization in the domain space {*α*, *β*} for every observation. Thus, the overall complexity is *O*(*l*
_{
α
}
*l*
_{
β
}
*N*). Compared to handling \( {\mathcal{P}}_2 \) straightly with lattice search (the corresponding computational complexity is *O*((*l*
_{
α
}
*l*
_{
β
})^{
N
})), the computation burden is greatly reduced. The reason why the proposed algorithm is less expensive than solving \( {\mathcal{P}}_2 \) is similar as that in Algorithm 1, namely it is sequential and does not require optimization for all observations.

## 6 Numerical examples

In this section, the performance analysis of the proposed algorithms is presented. As to scalar measurement systems with the joint Tx/Rx polarization optimization, we provide numerical examples to demonstrate the effectiveness of the devised algorithms, in comparison with two other polarization approaches. Additionally, we also compare their performances with that of the vector measurement system with transmission polarization optimization and highlight the advantage of the latter design.

_{t}is an arbitrary unitary matrix constructed with the left singular vectors of a 4 × 4 matrix M with independent and identical distribution complex Gaussian entries, i.e., M = U

_{t}

**Λ**

_{ M }U

_{r},

**Λ**

_{t}= diag(rand(1, 4)) with rand(1, 4) denoting a random four-dimensional real vector, and diag(·) being the diagonalization of the vector (·). Note that we normalize rand(1, 4) to keep the target power to be constant.

*a*is the factor that controls the signal-to-clutter-plus-noise ratio (SCNR) with the definition [5]:

**Λ**

_{c}= diag([0.25, 0.25, 0.25, 0.25]) and U

_{c}being chosen with the similar operation to generate U

_{t}. Throughout the simulations, we assume the noise power \( {\sigma}_{\nu}^2 \) = 0 dB.

Furthermore, by inspection of Eqs. (19) and (29), we can observe that D
_{
n + 1} and \( {\boldsymbol{D}}_{n+1}^{\boldsymbol{\prime}} \) do not depend on the observations \( {\left\{{y}_{i+1}\right\}}_{i=0}^n \) and \( {\left\{{\boldsymbol{y}}_{i+1}\right\}}_{i=0}^n \), respectively, even though in Algorithm 1 and Algorithm 2, both of them are the input parameters that update the estimation of target scattering vector \( {\widehat{\boldsymbol{X}}}_{\mathrm{t}} \). Since the MSE of \( {\widehat{\boldsymbol{X}}}_{\mathrm{t}} \) is the figure of merit for the optimization problem and Tr(D
_{
n + 1}), \( \mathrm{T}\mathrm{r}\left({\boldsymbol{D}}_{n+1}^{\boldsymbol{\prime}}\right) \) are the objectives to be optimized for each algorithm, respectively, in this section, we will investigate the changes of Tr(D
_{
n + 1}) and \( \mathrm{T}\mathrm{r}\left({\boldsymbol{D}}_{n+1}^{\boldsymbol{\prime}}\right) \) without considering updating of \( {\widehat{\boldsymbol{X}}}_{\mathrm{t}} \). As a consequence, we do not use \( {\left\{{y}_{i+1}\right\}}_{i=0}^n \) and \( {\left\{{\boldsymbol{y}}_{i+1}\right\}}_{i=0}^n \) in the simulation. Also notice that *n* ≥ 4 is required for both the proposed algorithms. So we randomly choose \( {\left\{{\alpha}_i\right\}}_{i=1}^4={\left\{{\beta}_i\right\}}_{i=1}^4={\left\{{\theta}_i\right\}}_{i=1}^4={\left\{{\vartheta}_i\right\}}_{i=1}^4=\pi /8 \) (some other values within the definition ranges can be also used) as the initial trigonometric parameters to generate **A**
_{4} and **T**
_{4} for Algorithm 1 and Algorithm 2, respectively.

Finally, regarding the lattice search used to seek the optimal polarizations for Algorithm 1 and Algorithm 2, set *l*
_{
α
} = *l*
_{
θ
} = 10^{3} and *l*
_{
β
} = *l*
_{
ϑ
} = 500. By using the Monte Carlo method, the MSEs are calculated by averaging 10^{5} independent realizations of C
_{t} and C
_{c}, respectively.

### 6.1 Scalar measurement systems

We choose the conventional polarimetric radar and the polarimetric radar with adaptive transmission polarization as the counterparts to compare their performances with the polarimetric radar with joint Tx/Rx optimization. As to conventional polarimetric systems, the transmission side alternatively transmits horizontal and vertical polarizations, i.e., ξ = [1, 0]^{T} in the current pulse and ξ = [0, 1]^{T} in next pulse. In the reception side, the horizontal and vertical polarizations are simultaneously received, so the reception vector satisfies \( \boldsymbol{\eta} ={\left[\sqrt{2}/2,\sqrt{2}/2\right]}^{\mathrm{T}} \). Therefore, the relevant MSE can be calculated by substituting the waveform parameters {*α* = 0, *β* = 0, *θ* = ‐ *π*/4, *ϑ* = 0} and {*α* = − *π*/2, *β* = 0, *θ* = − *π*/4, *ϑ* = 0}, alternatively, into Algorithm 1 with the increase of observation samples and without optimal polarization search. Then, as to the scalar system with transmission polarization optimization, the transmission polarization \( {\left[{\xi}_{\mathrm{h}},{\xi}_{\mathrm{v}}\right]}^{\mathrm{T}} \) is allowed to be chosen freely while the reception polarization is fixed as \( {\left[\sqrt{2}/2,\sqrt{2}/2\right]}^{\mathrm{T}} \). As such, we can employ Algorithm 1 to optimally select transmission polarization by fixing *θ* = − *π*/4 and *ϑ* = 0.

_{t}versus the number of observation samples for the aforementioned three scalar systems, with SCNR = 0 dB. As expected, the devised method monotonically reduces the MSE and converges to a certain value. On the other hand, the plots clearly show that the scalar system with joint Tx\Rx polarization optimization performs better than that with transmission polarization optimization as well as the scalar system with transmission polarization optimization showing a significant performance gain with respect to conventional polarization radars.

_{t}versus the SCNR, with a fixed 50 observation samples. It can be observed that, the scalar system with optimally designed Tx/Rx polarizations leads to a power gain of 4–6 dB with respect to that with only transmission polarization optimization, as well as 6–8 dB gain with respect to the conventional polarization radar.

### 6.2 Scalar measurement systems versus vector measurement systems

## 7 Conclusions

Adaptive polarization design of polarimetric radars, for the purpose of optimally estimating the target polarization scattering vector, is studied. Starting from the problem of designing the optimal polarizations for the scalar measurement system with joint Tx/Rx polarization optimization, under considering the minimum MSE of the estimator as the figure of merit, a sequential method of selecting the optimal Tx/Rx polarizations for such system is developed by owing to a suitable reformulation of the considered non-convex design problem. Furthermore, by employing a similar deviation procedure, a sequential method to select the optimal transmission polarization for the vector system with transmission polarization optimization is also devised. Since both the proposed algorithms make use of the sequential minimum MSE estimation, they monotonically decrease the MSE of the estimation and converge to a stationary point. The complexity of the proposed methods is linear with the number of outer iterations whereas at each iteration, it mainly requires the lattice search along four and two trigonometric parameters, respectively.

Several numerical examples have been provided to assess the effectiveness of the proposed methods. Precisely, the performance of the scalar measurement system with joint Tx\Rx polarization optimization, the one with transmission polarization optimization and the one with conventional polarimetric design, is compared. The system with joint Tx\Rx polarization optimization shows a significant performance improvement with respect to the one with transmission polarization optimization, as well as the conventional polarization approach. Moreover, the numerical results also show that the vector measurement system with transmission polarization optimization provides the comparative estimation performance with the scalar measurement system with joint Tx/Rx optimization. This is because, in the latter case, the receiver polarization optimization is implicitly performed. But the vector measurement system requires a lower computation burden, highlighting the advantage of such system design.

## Declarations

### Acknowledgements

The authors would like to thank Mr. John Lucynski for providing some comments on how to correct the language. This work was also supported by the National Natural Science Foundation of China (Nos. 61401488, 61501475, and 61490692).

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

## Authors’ Affiliations

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