Optimization on the Polarization and Waveform of Radar for Better Target Detection Performance under Rainy Condition

Abstract

Under rainy conditions, atmospheric settling particles have significant depolarization effects on radar waves and thus affect the target detection performance. This paper establishes an extended target detection model for fully polarized radar under complex rainy conditions and proposes two polarization and waveform optimization methods by taking into account the transmission effect of rainfall. If the rainfall information is known, the polarization and waveform for transmitting and receiving are optimized by maximizing the system Signal-to-Clutter/Noise Ratio (SCNR). While the rainfall information is absent, the optimal transmitting polarization and waveform are obtained by directly compensating the transmission effect from non-rainfall optimization results using fully polarized radar observation parameters. Using measured data from the T-72 tank, The experimental results verify the effectiveness and robustness of the two methods in real scenarios and 4 dB on the improvement of SCNR can be achieved.

1. Introduction

The advancement of radar technology and the increasing complexity of application environments necessitate that radar systems maintain effective target detection capabilities even under adverse meteorological conditions. When radar targets are situated in complex weather environments, they are particularly affected by atmospheric particles, notably raindrop particles. The propagation characteristics of radar waves through the transmission medium differ substantially from those in clear sky conditions, significantly impacting radar detection performance [1,2,3,4,5]. Thus, it is crucial to study the scattering characteristics of raindrops and the propagation characteristics of electromagnetic waves in rainy conditions, as this research provides theoretical support for the effective detection of radar targets in complex rainy conditions.

In early studies, raindrops were generally viewed as spherical particles and their scattering properties were calculated using the classical MIE scattering method. Later, with the deepening of the study, the raindrops were modeled as flat ellipsoids closer to the actual situation. Various computational methods were generated, among which the point matching method, Fredholm integral equation method and T-matrix method [6,7] have been widely used. The wave propagation characteristics in the troposphere have been widely considered by scholars at home and abroad, and Oguchi et al. established a transmission matrix analysis method to obtain mathematical characterization of the propagation characteristics under a variety of conditions. This method has also been widely used in the modeling of radar target detection under rainy and foggy conditions.

Polarization, a fundamental characteristic of electromagnetic waves, provides critical physical information about the target and environment, including geometry, material and orientation. Utilizing polarization information can markedly enhance the detection capabilities of radar systems. Recent advancements in computing architectures and hardware have enabled the implementation of more advanced signal processing techniques in modern radar systems. These advancements have led to the development of fully polarized radars featuring transmitter/receiver-optimized and waveform-adaptive designs [8,9,10,11]. The SCNR can be significantly improved by jointly designing the transmit waveform and receive filter banks and is widely used in applications such as target identification and target detection [12,13,14,15].

Another attractive problem of research is the design of waveforms for “extended targets” [16,17,18]. This research is driven by the potential advancements in high-resolution radar (HRR) systems. Studies have demonstrated that significant performance gains can be achieved with HRR. In HRR, targets are finely segmented into multiple range cells, each corresponding to a scattering center. The echoes from these scattering centers can be individually resolved, enhancing the corresponding SCNR [19,20,21,22]. Moreover, it has been shown that improving SCNR can indirectly lead to better detection capabilities [23]. At the same time, the method can effectively detect multichannel signals in the presence of multiple noises and interferences [24,25,26].

The joint optimization design approach relies on the detected target impulse response (TIR) model and the statistical properties of the clutter. Among the existing studies, some papers explore deterministic TIR models [17,18,19], while others assume partially known TIRs or model them as stochastic processes. Additionally, it is essential to characterize clutter statistical features in the polarization domain [27,28]. Compared to conventional single polarization systems, fully polarized radars can achieve significant improvements in output SCNR through polarization diversity techniques [29,30,31,32,33]. Thus, in the literature [17], an algorithm for designing fully polarized transmit waveforms and receive filters with the objective of maximizing SCNR was proposed for enhanced performance in cluttered environments. Notably, the fully polarized scattering characteristics of extended targets were described by the TIR Matrix (TIRM). Specifically, it was noted that in scenarios with substantial clutter, an iterative approach can be employed to attain the maximum objective SCNR. However, convergence to an optimal solution is not guaranteed [34,35,36]. Hence, an improved algorithm is presented in [16] to ensure that SCNR does not decrease with each iteration and converges to a stable value. However, both studies [16,17] are constrained by the following assumptions:

• Assume that the TIRM is perfectly known;
• Only energy constraints are considered in the optimization process;
• Clutter is modeled as polarization-dependent but independent of the range bin.

Indeed, the TIRM exhibits high sensitivity to the target aspect angle (TAA), necessitating a robust design to address this uncertainty. Moreover, target detection in precipitation environments becomes challenging due to the unavoidable transmission effects. Particularly in the polarization domain, the depolarization effect induced by rainfall can significantly alter both radar signals and clutter statistical properties, thereby influencing the system SCNR.

In this paper, we revisit the approach to enhancing the SCNR of fully polarized radar systems, considering transmission effects from a broader perspective. When the rainfall information is known, we model the statistical properties of clutter caused by transmission effects. Subsequently, a filter bank is utilized to adjust each branch to a specific TAA, formulating a joint optimization problem with the objective of maximizing the system output SCNR. Alongside energy constraints, we introduce similarity constraints to regulate signal characteristics accordingly. In scenarios where rainfall information is unknown, we optimize the polarization and waveform of the transmitted signal based on observation parameters of the fully polarized radar. In the analysis phase, we evaluate the performance of the transmission effect-based SCNR enhancement method in complex rainfall and clutter environments using real-world data of T-72 tanks. Finally, we analyze the influence of transmission effects on SCNR to validate the scientific efficacy of the proposed method.

The rest of this paper is organized as follows. In Section 2, we introduce the rainfall transmission model and the fully polarized radar detection model. In Section 3, a joint transmitter/receiver optimization method based on the transmission effect to improve the SCNR and an optimal transmit signal design method based on the fully polarized radar observation covariates are proposed. Section 4 discusses the simulation experimental results. Finally, the conclusions of this paper and possible future research directions are described in Section 6.

Notation: We assume that the notation ${\mathbb{C}}^{N\times N}$ denotes N × N dimension complex matrices. I and 0 denote, respectively, the identity matrix and the matrix and the matrix of zero entries. The Euclidean norm of the vector x is donoted by $\left|\left|x\right|\right|$. Finally, the symbols ${(·)}^T$, ${\left(·\right)}^H$, $\mathrm{E}[·]$, $\mathrm{Re}[·]$, $\mathrm{Im}[·]$ and ⊗ denote the transpose, Hermitian, expectation, real part operators, imaginary part operators and Kronecker product, respectively.

2. System Model

We investigate a fully polarized radar system for precise detection of targets under rainy conditions, as illustrated in Figure 1. At the transmitting end, the radar alternately emits horizontally and vertically polarized signals, passing through a complex, non-uniform rain area during transmission, whereas at the radar receiver, the radar captures both horizontally and vertically polarized signals through the corresponding antenna. This provides comprehensive target information in the polarization domain.

For a fully polarized radar transmit signal, its fast time domain N samples can be represented as the following overlapping of two polarization channel data:

$$\mathit{s}\stackrel{\mathsf{\Delta }}{=}{[(s_{H,0},s_{V,0}),\dots ,(s_{H,N-1},s_{V,N-1})]}^T\in {\mathbb{C}}^{2N\times 1}$$

(1)

where H and V represent the horizontal and vertical polarization channel data, respectively. Since the waveform of any polarization mode can be represented as a combination of two orthogonally linear polarizations, this representation is general.

Considering that the range cell number of the TIRM is Q, and that observations are collected over M discrete intervals, where $M=N+Q-1$, so that the entire echo of the extended target can be fully received in a single sample, a schematic diagram of a fully polarized radar detecting a tank target is given in Figure 2.

For a given TAA $\theta$ and range bin n, the target fully polarized scattering matrix ${\mathit{T}}_{\mathit{n}}\left(\theta \right)\in {\mathbb{C}}^{2\times 2}$ is denoted by

$$\begin{array}{cc}\hfill & {\mathit{T}}_{\mathit{n}}\left(\theta \right)=\left[\begin{array}{cc}{\mathit{T}}_{\mathit{HH},\mathit{n}}& {\mathit{T}}_{\mathit{HV},\mathit{n}}\\ {\mathit{T}}_{\mathit{VH},\mathit{n}}& {\mathit{T}}_{\mathit{VV},\mathit{n}}\end{array}\right]\hfill \\ \hfill & \theta \in [0,2\pi ),n\in \{0,1,\dots ,Q-1\}\hfill \end{array}$$

(2)

Similarly, as part of the clutter impulse response matrix (CIRM), the clutter scattering matrix for the nth range cell is expressed as

$$\begin{array}{cc}\hfill & {\mathit{C}}_{\mathit{n}}\left(\theta \right)=\left[\begin{array}{cc}{\mathit{C}}_{\mathit{HH},\mathit{n}}& {\mathit{C}}_{\mathit{HV},\mathit{n}}\\ {\mathit{C}}_{\mathit{VH},\mathit{n}}& {\mathit{C}}_{\mathit{VV},\mathit{n}}\end{array}\right]\hfill \\ \hfill & n\in \{-N+1,-N+2,\dots ,M-1\}\hfill \end{array}$$

(3)

The effect of rainfall can be characterized by the rain media transmission matrix (RTM), which physically means that a transmission effect is exerted on the electromagnetic signals at each range bin of the extended target. In practice, there exists a wide range of rain areas in various regions with different precipitation amounts, assuming that the radial range along a rain area is divided into n regions, and the transmission matrices of each section of a rain area are ${\mathit{R}}_i(i=1,\dots ,n)$, then the RTM of a non-uniform rain area can be expressed as

$$\begin{array}{c}\hfill {\mathit{R}}_{equ}={\mathit{R}}_n{\mathit{R}}_{n-1}\cdots {\mathit{R}}_1\end{array}$$

(4)

As a result, the forward and backward RTM can be expressed as, respectively,

$$\begin{array}{c}\hfill {\mathit{R}}_F=\left[\begin{array}{cccc}{\mathit{R}}_{equ}& 0& \cdots & \mathbf{0}\\ 0& {\mathit{R}}_{equ}& \cdots & \mathbf{0}\\ ⋮& ⋮& \ddots & ⋮\\ \mathbf{0}& \mathbf{0}& \cdots & {\mathit{R}}_{equ}\end{array}\right]\in {\mathbb{C}}^{2N\times 2N}\end{array}$$

(5)

and

$$\begin{array}{c}\hfill {\mathit{R}}_B={\left[\begin{array}{cccc}{\mathit{R}}_{equ}& \mathbf{0}& \cdots & \mathbf{0}\\ \mathbf{0}& {\mathit{R}}_{equ}& \cdots & \mathbf{0}\\ ⋮& ⋮& \ddots & ⋮\\ \mathbf{0}& \mathbf{0}& \cdots & {\mathit{R}}_{equ}\end{array}\right]}^T\in {\mathbb{C}}^{2M\times 2M}\end{array}$$

(6)

the procedure for solving the transmission matrix is shown in Appendix A.

A specific expression for the radar echo is obtained by superimposing the observations during a sampling period into a single vector

$$\begin{array}{c}\hfill \mathit{r}={\alpha }_T{\mathit{R}}_B\mathit{T}\left(\theta \right){\mathit{R}}_F\mathit{s}+{\mathit{R}}_B\mathit{C}{\mathit{R}}_F\mathit{s}+\mathit{v}\end{array}$$

(7)

where ${\alpha }_T$ is a complex coefficient related to the radar performance parameters, transmission factors, etc. and $\mathit{v}$ is the radar internal additive noise vector

$$\begin{array}{c}\hfill \mathit{v}\triangleq {[(v_{H,0},v_{V,0}),\dots (v_{H,M-1},v_{V,M-1})]}^T\in {\mathbb{C}}^{2M\times 1}\end{array}$$

(8)

The matrices

$$\begin{array}{c}\hfill \mathit{T}\left(\theta \right)\triangleq \sum _{n=0}^{Q-1}{\mathit{J}}_n\otimes {\mathit{T}}_n\left(\theta \right)\end{array}$$

(9)

and

$$\begin{array}{c}\hfill \mathit{C}\triangleq \sum _{n=-N+1}^{M-1}{\mathit{J}}_n\otimes {\mathit{C}}_n\end{array}$$

(10)

are the 2M × 2N-dimensional TIRM and CIRM, respectively, with ${\mathit{J}}_n$ as the M × N-dimensional shift matrix defined as follows:

$$\begin{array}{cc}\hfill & {\mathit{J}}_n({\ell }_1,{\ell }_2)\triangleq \left\{\begin{array}{cc}1,\hfill & \mathrm{if}{\ell }_1-{\ell }_2=n\hfill \\ 0,\hfill & \mathrm{if}{\ell }_1-{\ell }_2\ne n\hfill \end{array}\right.\hfill \\ \hfill & {\ell }_1\in \{1,\dots ,M\},{\ell }_2\in \{1,\dots ,N\}\hfill \end{array}$$

(11)

For the noise vector $\mathit{v}$, it is assumed to have a mean of 0 and a covariance matrix of ${\sum }_v={\sigma }_v^2\mathit{I}$, where ${\sigma }_v^2$ is the variance of a single noise sample. Regarding the statistical properties of the clutter, due to the reciprocity of single-base radars, we assume $C_{HV,n}=C_{VH,n}$. And, for the clutter in a single range cell, we assume that it is zero-mean, i.e., $E\left[\mathit{C}\right]=0$. Also, we consider not only the clutter correlation in the polarization domain, but also the influence caused by transmission effects on the clutter characteristics. Regarding the polarization correlation of the clutter, we utilize the polarization clutter model in literature  [37]. Furthermore, we observe that the CIRM can be rewritten in the following form:

$$\begin{array}{cc}\hfill \mathit{C}& =\sum _{n=-N+1}^{M-1}{\mathit{J}}_n\otimes {\mathit{C}}_n\hfill \\ & =\sum _{n=-N+1}^{M-1}[C_{HH,n}{\mathit{J}}_n\otimes {\mathit{A}}_1+C_{HV,n}{\mathit{J}}_n\otimes {\mathit{A}}_2\hfill \\ & +C_{VV,n}{\mathit{J}}_n\otimes {\mathit{A}}_3]\hfill \end{array}$$

(12)

where ${\mathit{A}}_1\triangleq \left[\begin{array}{cc}1& 0\\ 0& 0\end{array}\right]$, ${\mathit{A}}_2\triangleq \left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]$ and ${\mathit{A}}_3\triangleq \left[\begin{array}{cc}0& 0\\ 0& 1\end{array}\right]$. Then, considering the transmission effect of the rain medium, we get the 2M × 2M-dimensional clutter covariance matrix under rainy conditions as follows:

$$\begin{array}{cc}\hfill & {\sum }_{\mathit{C}}(\mathit{s},\mathit{R})=E[{\mathit{R}}_B\mathit{C}{\mathit{R}}_F\mathit{s}·{\left({\mathit{R}}_B\mathit{C}{\mathit{R}}_F\mathit{s}\right)}^H]\hfill \\ \hfill & =\sum _{n=-N+1}^{M-1}[{\mathit{R}}_B{\sigma }_n({\mathit{J}}_n\otimes {\mathit{A}}_1){\mathit{R}}_F\mathit{s}{\mathit{s}}^H{\mathit{R}}_F^H({\mathit{J}}_n^T\otimes {\mathit{A}}_1){\mathit{R}}_B^H\hfill \\ \hfill & +{\mathit{R}}_B{\sigma }_n\sqrt{{\chi }_n}{\rho }_n({\mathit{J}}_n\otimes {\mathit{A}}_1){\mathit{R}}_F\mathit{s}{\mathit{s}}^H{\mathit{R}}_F^H({\mathit{J}}_n^T\otimes {\mathit{A}}_3){\mathit{R}}_B^H\hfill \\ \hfill & +{\mathit{R}}_B{\sigma }_n{\epsilon }_n({\mathit{J}}_n\otimes {\mathit{A}}_2){\mathit{R}}_F\mathit{s}{\mathit{s}}^H{\mathit{R}}_F^H({\mathit{J}}_n^T\otimes {\mathit{A}}_2){\mathit{R}}_B^H\hfill \\ \hfill & +{\mathit{R}}_B{\sigma }_n\sqrt{{\chi }_n}{\rho }_n^{*}({\mathit{J}}_n\otimes {\mathit{A}}_3){\mathit{R}}_F\mathit{s}{\mathit{s}}^H{\mathit{R}}_F^H({\mathit{J}}_n^T\otimes {\mathit{A}}_1){\mathit{R}}_B^H\hfill \\ \hfill & +{\mathit{R}}_B{\sigma }_n{\chi }_n(J_n\otimes {\mathit{A}}_3){\mathit{R}}_F\mathit{s}{\mathit{s}}^H{\mathit{R}}_F^H({\mathit{J}}_n^T\otimes {\mathit{A}}_3){\mathit{R}}_B^H]\hfill \end{array}$$

(13)

where

$$\begin{array}{cc}\hfill & {\sigma }_n\triangleq \mathrm{E}\left[{\left|C_{HH,n}\right|}^2\right],{\epsilon }_n\triangleq \frac{\mathrm{E}\left[{\left|C_{HV,n}\right|}^2\right]}{\mathrm{E}\left[{\left|C_{HH,n}\right|}^2\right]},{\chi }_n\triangleq \frac{\mathrm{E}\left[{\left|C_{VV,n}\right|}^2\right]}{\mathrm{E}\left[{\left|C_{HH,n}\right|}^2\right]}\hfill \\ \hfill & {\rho }_n=\frac{\mathrm{E}\left[C_{HH,n}{C}_{VV,n}^{*}\right]}{\sqrt{\mathrm{E}\left[{\left|C_{HH,n}\right|}^2\right]\mathrm{E}\left[{\left|C_{VV,n}\right|}^2\right]}}\hfill \end{array}$$

(14)

3. Materials and Methods

Our goal is to optimize the transmit polarization and waveform of the fully polarized radar to improve target detection performance in different application situations. In this section, we present SCNR improvement methods based on the transmission effect for scenarios where rainfall information is both known and unknown, respectively. Subsequently, we introduce the enhancement method for the case where RTM is known.

3.1. Joint Optimization of Transmit Signals and Receive Filters

In Section 2 we obtained an expression for the echo observation vector, and if the echo is passed through the filter bank $\mathit{w}\triangleq {\left[w\left(0\right),\cdots ,w(2M-1)\right]}^T\in {\mathbb{C}}^{2M}$, the system output can be written as

$$\begin{array}{c}\hfill y={\mathit{w}}^H\mathit{r}={\alpha }_T{\mathit{w}}^H{\mathit{R}}_B\mathit{T}\left(\theta \right){\mathit{R}}_F\mathit{s}+{\mathit{w}}^H{\mathit{R}}_B\mathit{C}{\mathit{R}}_F\mathit{s}+{\mathit{w}}^H\mathit{v}\end{array}$$

(15)

The corresponding output SCNR is

$$\begin{array}{c}\hfill {\mathrm{SCNR}}_{\theta }(\mathit{s},\mathit{w})\triangleq \frac{{\left|{\alpha }_T\right|}^2{\left|{\mathit{w}}^H{\mathit{R}}_B\mathit{T}\left(\theta \right){\mathit{R}}_{F}s\right|}^2}{\mathrm{E}\left[{\left|{\mathit{w}}^H{\mathit{R}}_B\mathit{C}{\mathit{R}}_F\mathit{s}\right|}^2\right]+\mathrm{E}\left[{\left|{\mathit{w}}^H\mathit{v}\right|}^2\right]}\end{array}$$

(16)

In general, the exact TAA is unknown, leading to a lack of precise information of the TIRM. To overcome the algorithmic performance perturbation due to target scattering uncertainty, in the optimization process, we assume that the complete TIRM is known and that each branch of the filter bank corresponds to a specific TAA$\theta$, hereby jointly optimize the transmit signal and receive filters. Specifically, let the observation vector $\mathit{r}$ pass through the filter bank ${\left\{{\mathit{w}}_i\right\}}_{i=1}^K$, with each branch corresponding to a given TAA ${\theta }_i,i=1,2,\dots ,K$. Thus, the output ${\mathrm{SCNR}}_i,i=1,2,\dots ,K$ of the $\mathit{i}$th branch of the filter bank is

$$\begin{array}{c}\hfill {\mathrm{SCNR}}_i(\mathit{s},{\mathit{w}}_i)=\frac{{\left|{\alpha }_T\right|}^2{\left|{\mathit{w}}^H{\mathit{R}}_B\mathit{T}\left(\theta \right){\mathit{R}}_F\mathit{s}\right|}^2}{{\mathit{w}}_i^H{\sum }_{\mathit{C}}(\mathit{s},\mathit{R}){\mathit{w}}_i+{\sigma }_{\mathit{v}}^2{\mathit{w}}_i^H{\mathit{w}}_i}\end{array}$$

(17)

where the numerator of the above equation is the target power and the denominator is the sum of clutter and noise power.

In order to improve the robustness of the algorithm and reduce the impact of TAA angle uncertainty on the performance of the algorithm, we consider the SCNR of the outputs of all the branches of the filter bank.The optimization problem takes the maximization of the worst output of the filter as the objective function and imposes the following two constraints:

• The transmit signal power constraint, which requires ${∥\mathit{s}∥}^2=1$ in general.
• the similarity constraint, which requires ${∥\mathit{s}-{\mathit{s}}_0∥}^2\le \gamma$ in order to control the correlation characteristics of the signals not to be too cluttered. Where the parameter $0\le \gamma \le 1$ restricts the feasible domain of similarity and ${\mathit{s}}_0$ is a preset waveform [32].

As a result, with the above optimization objectives and constraints, the joint transmitter/receiver optimization problem of fully polarized radar under rainfall conditions can be formulated as

$$\begin{array}{c}\hfill \mathcal{P}\left\{\begin{array}{c}{\displaystyle \underset{\mathit{s},{\left\{{\mathit{w}}_i\right\}}_{i=1}^K}{max}}{\displaystyle \underset{i=1,\dots ,K}{min}}\frac{{\left|{{\mathit{w}}_i}^H{\mathit{R}}_B\mathit{T}\left({\theta }_i\right){\mathit{R}}_F\mathit{s}\right|}^2}{{\mathit{w}}_i^H{\sum }_{\mathit{C}}(\mathit{s},\mathit{R}){\mathit{w}}_i+{\sigma }_{\mathit{v}}^2{\mathit{w}}_i^H{\mathit{w}}_i}\hfill \\ s.t.{∥\mathit{s}∥}^2=1,{∥\mathit{s}-{\mathit{s}}_0∥}^2\le \gamma \hfill \end{array}\right.\end{array}$$

(18)

The non-convex optimization problem $\mathcal{P}$ can be transformed into one in which the feasible domain is a convex set and the objective function is the ratio of two convex functions. After mathematical analysis and derivation, $\mathcal{P}$ can be deconstructed into two optimization problems $\mathcal{P}{\mathit{s}}^{\left(m\right)}$ and $\mathcal{P}{\mathit{w}}^{\left(m\right)}$ that solve ${\mathit{s}}^{\left(m\right)}$ and ${\left\{{\mathit{w}}_i\right\}}_{i=1}^K$, respectively, and solve for the optimal value by alternating iterative optimization.

$$\begin{array}{c}\hfill \mathcal{P}{\mathit{s}}^{\left(m\right)}\left\{\begin{array}{c}{\displaystyle \underset{s}{max}}{\displaystyle \underset{i=1,\cdots ,K}{min}}\frac{\mathrm{Re}\left({{\mathit{w}}_i}^{H(m-1)}{\mathit{R}}_B\mathit{T}\left({\theta }_i\right){\mathit{R}}_F\mathit{s}\right)}{\sqrt{{\mathit{w}}_i^{H(m-1)}({\sum }_{\mathit{C}}(\mathit{s},\mathit{R})+{\sigma }_v^2{∥\mathit{s}∥}^2\mathit{I}){\mathit{w}}_i^{(m-1)}}}\hfill \\ \mathrm{s}.\mathrm{t}.{∥\mathit{s}∥}^2\le 1\hfill \\ \mathrm{Re}\left({\mathit{s}}_0^H\mathit{s}\right)\ge 1-\frac{\gamma }{2}\hfill \\ \mathrm{Re}\left({\mathit{s}}_0^H\mathit{s}\right)\ge (1-\frac{\gamma }{2})∥\mathit{s}∥\hfill \\ \mathrm{Re}\left({{\mathit{w}}_i}^{H(m-1)}{\mathit{R}}_B\mathit{T}\left({\theta }_i\right){\mathit{R}}_F\mathit{s}\right)\ge 0,i=1,2,\cdots K\hfill \end{array}\right.\end{array}$$

(19)

and

$$\begin{array}{c}\hfill \mathcal{P}{\mathit{w}}^{\left(m\right)}\left\{\begin{array}{c}{\displaystyle \underset{{\left\{{\mathit{w}}_i\right\}}_{i=1}^K}{max}}{\displaystyle \underset{i=1,\cdots ,K}{min}}\frac{\mathrm{Re}\left({{\mathit{w}}_i}^H{\mathit{R}}_B\mathit{T}\left({\theta }_i\right){\mathit{R}}_F{\mathit{s}}^{\left(m\right)}\right)}{\sqrt{{\mathit{w}}_i^H{\sum }_{\mathit{C}}(\mathit{s},\mathit{R}){\mathit{w}}_i+{\sigma }_{\mathit{v}}^2{\mathit{w}}_i^H{\mathit{w}}_i{∥s∥}^2}}\hfill \\ \mathrm{s}.\mathrm{t}.\mathrm{Re}\left({{\mathit{w}}_i}^H{\mathit{R}}_B\mathit{T}\left({\theta }_i\right){\mathit{R}}_F{\mathit{s}}^{\left(m\right)}\right)\ge 0,i=1,2,\cdots K\hfill \end{array}\right.\end{array}$$

(20)

According to the literature [32], the optimal solution of the filter can be equated to the form of a Capon filter. Equation (21) gives the explicit expression for the ${\mathit{w}}^m$.

$$\begin{array}{c}\hfill {\mathit{w}}_i^{\left(m\right)}=\frac{{\left({\mathbf{\Sigma }}_{\mathit{C}}\left({\mathit{s}}^{\left(m\right)},\mathit{R}\right)+{\sigma }_v^2{\parallel {\mathit{s}}^{\left(m\right)}\parallel }^2\mathit{I}\right)}^{-1}\mathit{T}\left(\theta \right){\mathit{s}}^{\left(m\right)}}{{∥{\left({\mathbf{\Sigma }}_{\mathit{C}}\left({\mathit{s}}^{\left(m\right)},\mathit{R}\right)+{\sigma }_v^2{\parallel {\mathit{s}}^{\left(m\right)}\parallel }^2\mathit{I}\right)}^{-1/2}\mathit{T}\left(\theta \right){\mathit{s}}^{\left(m\right)}∥}^2}\end{array}$$

(21)

The optimal transmit waveform can be solved based on the generalized fractional programming (GFP) theory [38,39] by transforming the objective function (19) into the linear form of the generalized Dinklelbach algorithm, which is solved using the CVX toolbox of MATLAB R2021b. The complete joint transmitter/receiver alternating iterative optimization flow is shown in Algorithm 1.

Algorithm 1 Joint Transmitter/Receiver Design Under Rainy Condition.

Input: ${\sigma }_v^2$, ${\sigma }_n,{\epsilon }_n,{\chi }_n,{\rho }_n$, ${\mathit{R}}_{equ}$, ${\mathit{s}}_0$, $\gamma$ Output: The solution $\left({\mathit{s}}^{*},{\left\{{\mathit{w}}_i^{*}\right\}}_{i=1}^K\right)$ to $\mathcal{P}\mathit{s}$ and $\mathcal{P}\mathit{w}$   1: set $m:=0$, ${\mathit{s}}^{\left(m\right)}={\mathit{s}}_0$, then the filter optimizer is given in (21), and ${\mathrm{SCNR}}^{\left(m\right)}={\left(f({\mathit{s}}^{\left(m\right)},{\left\{{\mathit{w}}_i^{\left(m\right)}\right\}}_{i=1}^K)\right)}^2$;   2: repeat   3:    $m:=m+1$;   4:    constructing $({\sum }_{\mathit{C}}(\mathit{s},\mathit{R})+{\sigma }_v^2{∥\mathit{s}∥}^2\mathit{I})$ using singular value decomposition(SVD) method;   5:    solve $\mathcal{P}{\mathit{s}}^{\left(m\right)}$ to get an optimal radar code ${\mathit{s}}^{\left(m\right)}$;   6:    construct the matrix ${\mathbf{\Sigma }}_{\mathit{C}}\left({\mathit{s}}^{\left(m\right)},\mathit{R}\right)$;   7:    solve $\mathcal{P}{\mathit{w}}^{\left(m\right)}$ to get an optimal receive filter using Equation (21);   8:    let ${\mathrm{SCNR}}^{\left(m\right)}={\left(f({\mathit{s}}^{\left(m\right)},{\left\{{\mathit{w}}_i^{\left(m\right)}\right\}}_{i=1}^K)\right)}^2$   9: until$|{\mathrm{SCNR}}^{\left(m\right)}-{\mathrm{SCNR}}^{(m-1)}|\le \zeta$ 10: output ${\mathit{s}}^{*}={\mathit{s}}^{\left(m\right)}$, ${\mathit{w}}^{*}={\mathit{w}}^{\left(m\right)}$, ${\mathrm{SCNR}}_{\max }={\mathrm{SCNR}}^{\left(m\right)}$.

3.2. Polarization and Waveform Optimization Design Based on Fully Polarized Radar Observation Parameters

From Equation (16), we observe that the system output SCNR is primarily influenced by TIRM, CIRM and RTM. In practical scenarios, targets typically include tanks, vehicles, ships, airplanes and other scatterers with identifiable characteristics and extensive databases. The clutter background generally remains relatively stable or exhibits gradual variations, allowing for the acquisition of its statistical characteristics prior to radar operation. However, real-time acquisition of RTM is challenging due to the dynamic nature of rainy conditions, rendering the implementation of the joint optimization algorithm in Section 3.1 difficult. Furthermore, the joint optimization algorithm imposes stringent requirements on radar hardware performance. Therefore, it is necessary to study the radar polarization and waveform optimization methods adapted to the actual situation, which are specifically analyzed below.

The most important aspect of improving radar detection performance under rainy conditions is that the radar wave extracts as much target information as possible and carries it back to the receiving end, and in the process also reduces the transmission effects caused by the rain medium. Therefore, as depicted in the flowchart in Figure 3, this challenge is decomposed into three sub-problems:

(i)

Finding the radar signal ${\left[\begin{array}{c}{\mathit{E}}_H^i,{\mathit{E}}_V^i\end{array}\right]}^T$ that optimally extracts the target information at the target.

(ii)

Designing the optimal radar transmit signal and receive filter based on the transmission effect to reduce the influence of the rain medium.

(iii)

Designing the optimal transmit signal ${\left[\begin{array}{c}{\mathit{E}}_H^{opt},{\mathit{E}}_V^{opt}\end{array}\right]}^T$ without the information of rainfall.

The literature [32] investigates the optimal transmit signal ${\left[{\mathit{E}}_H^s,{\mathit{E}}_V^s\right]}^T$ and receive filter ${\mathit{w}}^{opt}$ design for fully polarized radar under clear sky conditions with the goal of maximum output SCNR, where ${\left[{\mathit{E}}_H^s,{\mathit{E}}_V^s\right]}^T$ can be used as a solution to sub-problem (i). For subproblem (ii), from the physical process, the rain medium has a depolarizing effect on the radar wave before and after it reaches the target, respectively. From the mathematical point of view, the depolarizing effect of the rain medium is characterized by the transmission matrix. Therefore, applying the inverse of the RTM to the ${\left[{\mathit{E}}_H^s,{\mathit{E}}_V^s\right]}^T$ can counteract the influence of the rain area on the radar wave before it reaches the target, so that the signal irradiated to the target can maximize the extraction of target information. Since the rain medium only changes the polarization state of the signal but not its waveform, the radar wave reflected from the target passes through the rain area again without loss of target information stored in the waveform, although the polarization state changes again. Therefore, ${\mathit{w}}^{opt}$ is still the optimal matched filter to extract the target information carried by the echo. In summary, the optimal radar transmission signal under rainfall conditions is obtained:

$$\begin{array}{c}\hfill \left[\begin{array}{c}{\mathit{E}}_H^{opt}\\ {\mathit{E}}_V^{opt}\end{array}\right]={\mathit{R}}_F^{-1}\left[\begin{array}{c}{\mathit{E}}_H^s\\ {\mathit{E}}_V^s\end{array}\right]\end{array}$$

(22)

Finally we investigate how to solve sub-problem (iii). We know that a polarized electromagnetic wave propagating in dielectric space can be expressed as follows

$$\begin{array}{c}\hfill E=V{\mathrm{e}}^{j\mathsf{\Phi }}\end{array}$$

(23)

where V denotes the electromagnetic wave amplitude and $\mathsf{\Phi }$ denotes its phase. Assuming that ${\mathit{E}}^r$ and ${\mathit{M}}^i$ are the electromagnetic waves at the receiver and transmiter ends of the radar, respectively, the relationship between the two and the point target (extended target has similar conclusions) backward scattering matrix ${\mathit{S}}_{BSA}$ under the transmission medium is:

$$\left[\begin{array}{c}{\mathit{E}}_H^r\\ {\mathit{E}}_V^r\end{array}\right]=\sqrt{\frac{Z_{0}G}{2\pi }}\frac{1}{r^2}\left[\begin{array}{cc}{R}_{HH}& R_{HV}\\ R_{VH}& R_{VV}\end{array}\right]{\left[\begin{array}{cc}{S}_{HH}& S_{HV}\\ S_{VH}& S_{VV}\end{array}\right]}_{BSA}\left[\begin{array}{cc}{R}_{HH}& R_{HV}\\ R_{VH}& R_{VV}\end{array}\right]\left[\begin{array}{c}{\mathit{M}}_H^i\\ {\mathit{M}}_V^i\end{array}\right]$$

(24)

where G is the antenna gain, $Z_0$ is the intrinsic impedance of the vacuum and r is the distance between the radar and the target. For the axial plane perpendicular to the electromagnetic wave propagation direction ($\mathsf{\Phi }=0$) of the directional ellipsoidal particles, the above equation is simplified as follows according to Appendix A.

$$\begin{array}{c}\hfill \left[\begin{array}{c}{\mathit{E}}_H^r\\ {\mathit{E}}_V^r\end{array}\right]=\sqrt{\frac{Z_{0}G}{2\pi r^4}}\left[\begin{array}{cc}{S}_{HH}{e}^{2{\lambda }_{1}r}& S_{HV}{e}^{({\lambda }_1+{\lambda }_2)r}\\ S_{VH}{e}^{({\lambda }_1+{\lambda }_2)r}& S_{VV}{e}^{2{\lambda }_{2}r}\end{array}\right]\left[\begin{array}{c}{\mathit{M}}_H^i\\ {\mathit{M}}_V^i\end{array}\right]\end{array}$$

(25)

For horizontal and vertical polarization the complex wave numbers $k_{eff}^H$, $k_{eff}^V$ can be expressed in complex form:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{k}_{eff}^H=k_{re}^H-j{k}_{im}^H\hfill \\ k_{eff}^V=k_{re}^V-j{k}_{im}^V\hfill \end{array}\right.\end{array}$$

(26)

Combining Equation (A6), its real and imaginary parts can be expressed as, respectively,

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{k}_{re}^H=k_0+Re\left[\frac{j}{2}(P_{HH}+P_{VV}+\gamma )\right]=k_0+k_{p,re}^H\hfill \\ k_{im}^H=-Im\left[\frac{j}{2}(P_{HH}+P_{VV}+\gamma )\right]=k_{p,im}^H\hfill \\ k_{re}^V=k_0+Re\left[\frac{j}{2}(P_{HH}+P_{VV}-\gamma )\right]=k_0+k_{p,re}^V\hfill \\ k_{im}^H=-Im\left[\frac{j}{2}(P_{HH}+P_{VV}-\gamma )\right]=k_{p,im}^V\hfill \end{array}\right.\end{array}$$

(27)

where $k_p$ is the “perturbation” component due to the change in the number of polarization waves caused by the rain media, and thus the echo can be further expressed as Equation (28):

$$\left[\begin{array}{c}{\mathit{E}}_H^r\\ {\mathit{E}}_V^r\end{array}\right]=\sqrt{\frac{Z_{0}G}{2\pi }}\frac{1}{r^2}\left[\begin{array}{cc}{S}_{HH}{e}^{-2k_{p,im}^{H}r}{e}^{-2j{k}_{p,re}^{H}r}& S_{HV}{e}^{-(k_{p,im}^H+k_{p,im}^V)r}{e}^{-j(k_{p,re}^H+k_{p,re}^V)r}\\ S_{VH}{e}^{-(k_{p,im}^H+k_{p,im}^V)r}{e}^{-j(k_{p,re}^H+k_{p,re}^V)r}& S_{VV}{e}^{-2k_{p,im}^{V}r}{e}^{-2j{k}_{p,re}^{V}r}\end{array}\right]\left[\begin{array}{c}{\mathit{M}}_H^i\\ {\mathit{M}}_V^i\end{array}\right]$$

(28)

The Equation (28) demonstrates that the real part of $k_p$ induces a phase shift in the signal, while the imaginary part affects its amplitude. Consequently, we establish that the fundamental nature of the transmission effect lies in the modulation of both amplitude and phase between the two channels of the polarized wave. Hence, in designing the actual transmit signal, it suffices to fix the polarization state of one channel and then adjust the amplitude and phase characteristics of the other polarized channel as needed.

In order to eliminate the depolarization effect brought by the rain medium, this section proposes an optimal polarization design method of the transmit signal based on the fully polarized radar observation parameters to correct the amplitude and phase variations brought by $k_p$. For fully polarized radar, the radar parameters that we can directly obtain include the specific differential attenuation $A_{dp}(\mathrm{dB}/\mathrm{km})$ and the specific differential phase $K_{dp}(\mathrm{rad}/\mathrm{km})$, which are defined as the differentiation of amplitude and phase with distance, respectively,

$$\begin{array}{c}\hfill A_{dp}=\frac{\mathrm{d}\left(20{log}_{10}\left|V_{dp}\left(r\right)\right|\right)}{\mathrm{d}r}=\frac{\mathrm{d}\left(20{log}_{10}\left|\frac{V_H\left(r\right)}{V_V\left(r\right)}\right|\right)}{\mathrm{d}r}\end{array}$$

(29)

and

$$\begin{array}{c}\hfill K_{dp}=\frac{\mathrm{d}{\mathsf{\Phi }}_{dp}\left(r\right)}{\mathrm{d}r}=\frac{\mathrm{d}[{\mathsf{\Phi }}_H\left(r\right)-{\mathsf{\Phi }}_V\left(r\right)]}{\mathrm{d}r}\end{array}$$

(30)

where $V_{dp}$ and ${\mathsf{\Phi }}_{dp}$ denote the amplitude ratio and phase difference of the H and V polarization channels per unit distance, respectively, and r denotes the radial distance of the rain area. According to Equations (29) and (30), $V_{dp}$ and ${\mathsf{\Phi }}_{dp}$ of the transmit signal can be solved to obtain the solution of sub-problem (iii).

$$\begin{array}{c}\hfill V_{dp}=V_{dp}^{opt}-\frac{{\int }_0^r{A}_{dp}\mathrm{d}r}{20}\end{array}$$

(31)

and

$$\begin{array}{c}\hfill {\mathsf{\Phi }}_{dp}={\mathsf{\Phi }}_{dp}^{opt}-{\int }_0^r{K}_{dp}\mathrm{d}r\end{array}$$

(32)

The $V_{dp}^{opt}$ and ${\mathsf{\Phi }}_{dp}^{opt}$ represent the amplitude ratio and phase difference of the ${\left[\begin{array}{c}{\mathit{E}}_H^s,{\mathit{E}}_V^s\end{array}\right]}^T$ horizontal and vertical polarization channels, respectively. Assuming that the radar transmits a known horizontally polarized signal of ${\mathit{s}}_H=V_H{e}^{j\mathsf{\Phi }}$, the vertically polarized signal can be solved by the above equation:

$$\begin{array}{c}\hfill {\mathit{s}}_V=\frac{V_H}{V_{dp}}{e}^{j({\mathsf{\Phi }}_H-{\mathsf{\Phi }}_{dp})}\end{array}$$

(33)

In practice, fully polarized radar requires different echo processing for different detection scenarios, so different transmit signal and receive filters are used for each coherent processing interval(CPI). L pulse acquisition times constitutes a CPI, and a single pulse transmit signal is sampled by M times for digital processing at the receiving end. The specific process is shown in Figure 4. Figure 4a represents the radar processing in different detection scenarios. Each column of Figure 4b represents the sampling data of one pulse and the L pulse forms a CPI.

In summary, we have solved the problem of target detection in practical applications without the rainfall information. It should be noted that in this section, only the influence of transmission effects on target detection is targeted and the fine design of transmit signals with integrated clutter background needs to be further explored.

4. Results

In this section, we analyze the performance of the two methods proposed in Section 3.1 and Section 3.2. The computing resources used in the simulation experiments are shown in

Table 1. Simulation platform configuration.

Parameter	Content
Operating system	Windows10
CPU type	i7-12700 H
GPU type	RTX 3060
RAM	16 GB
MATLAB	2021b

.

4.1. Experimental Setup Considered

We utilize a publicly available dataset [40] from the Georgia Institute of Technology as a priori knowledge of the target. The dataset consists of turntable phase data for the T-72 tank measured by a fully polarized radar over a set of azimuth and pitch ranges, where the transmit signal is a stepped-frequency pulsed signal with a center frequency of $f_0=9.6$ $\mathrm{GHz}$, 221 stepping units and a stepping interval of 3 MHz. The entire database covers the pitch range of $[{28}^{\circ },{32}^{\circ }]$, with an equal interval of $0.{14}^{\circ }$ and an azimuth range of $[0^{\circ },{360}^{\circ }]$, with an equal interval of $4.{25}^{\circ }$ as shown in Figure 5.

Throughout the simulations, we fix the radar observation angle to ${30}^{\circ }$ and set the TIRMs range bin as $Q=37$. Unless otherwise stated, we specify ${\left|{\alpha }_T\right|}^2=1$. The H, V polarization channel lengths of the initial transmit signal are $N=40$ and the following LFM signal is taken as the initial transmit signal

$$\begin{array}{c}\hfill {\mathit{s}}_{H0}\left(n\right)={\mathit{s}}_{V0}\left(n\right)=\frac{1}{\sqrt{2N}}{e}^{j\pi \frac{n^2}{2N}},n=0,\dots ,N-1\end{array}$$

(34)

Additionally, the clutter parameters reported in the literature [37] are used, assuming that the clutter is uniformly distributed, ${\epsilon }_n=0.19$, ${\chi }_n=1.03$ and ${\rho }_n=0.52$. In the simulation, $\mathrm{SNR}=5\mathrm{dB}$, $\mathrm{CNR}=10\mathrm{dB}$ and the distance between the clutter is taken to be uncorrelated.

Rainfall information is based on empirical data and actual observations to give the characteristic parameters shown in

Table 2. Characteristic parameters of rain area.

Parameter	Value
Temperature	$25°C$
Longitudinal distance of rainy area	1 km
Radar operating frequency	10 GHz
Radar wave incidence angle ${\theta }_i$	${90}^{\circ }$
Raindrop axial inclination ${\theta }_b$ and ${\varphi }_b$	${60}^{\circ }$, ${30}^{\circ }$
Normalized intercept parameter $N_w$	3000
Raindrop shape parameter $\mu$	11.5
Median diameter $D_0$	(2, 3, 4, 5) mm
Axial ratio of raindrop particles	1.25

.

The probability of raindrop rupture during motion increases exponentially with volume, as indicated by previous studies. Raindrops with diameters exceeding 8 mm tend to rapidly rupture into smaller droplets. The raindrop spectral distribution serves as a crucial parameter for characterizing the overall features of the rain area, and selecting an appropriate median diameter can enhance the realism of rainfall simulation. Figure 6 illustrates the probability distributions of different raindrop sizes under various median diameters. It is evident that for small $D_0$, raindrop sizes are approximately uniformly distributed around $D_0$, whereas for $D_0$ larger than 4 mm, the distribution deviates significantly from reality. Hence, according to common practice, $D_0=3$ mm is adopted in subsequent simulations and the RTM can be derived using Equations (5) and (6) and Appendix A.

4.2. Effectiveness of the Proposed Method in Section 3.1

The transmission effect-based joint transmitter/receiver optimization algorithm proposed in this paper ensures that the optimization result of SCNR monotonically increases and converges to a fixed value with increasing number of iterations at any TAA. To verify this performance, two sets of TAAs are randomly selected for simulation: ${\varphi }_T^{\left(a\right)}=[25.5^{\circ },27.1^{\circ }]$ and ${\varphi }_T^{\left(b\right)}=[115.5^{\circ },117.1^{\circ }]$. Figure 7 gives the variation of the filter bank output SCNR with the number of iterations under the two sets of TAAs. Each of these figures include two sets of similarity parameters $\gamma \in \{0.1,1.0\}$, and when the parameter $\gamma$ is increased, the optimized SCNR increases accordingly due to the reduced feasible-domain constraints on the transmit signal.

The variation of the maximum system output SCNR under different ranges of rainfall areas is given in Figure 8. The blue and red lines denote the maximum system output SCNR of the literature [32] and the method in Section 3.1 under rainy conditions, respectively. it can be seen that the performance of Section 3.1 is not only significantly better than that of [32], but also maintains a good robustness to different ranges of rain areas. The main reason is that the polarization state of the electromagnetic wave changes during propagation under the influence of $A_{dp}$ and $K_{dp}$, resulting in the fluctuation of SCNR. However, the optimization method that takes into account the transmission effect can adaptively counteract this effect and always keeps the SCNR at a high level.

4.3. Effects of the Filter Bank Size in Method of Section 3.1

Define ${\mathrm{SCNR}}_i\left(\theta \right),i=1,\dots ,K$ as the output SCNR of the ith filter in the filter bank when TAA is $\theta$, i.e.,

$$\begin{array}{c}\hfill {\mathrm{SCNR}}_i\left(\theta \right)=\frac{{\left|{\alpha }_T\right|}^2{\left|{\mathit{w}}^H{\mathit{R}}_B\mathit{T}\left(\theta \right){\mathit{R}}_F\mathit{s}\right|}^2}{{\mathit{w}}_i^H{\sum }_{\mathit{C}}(\mathit{s},\mathit{R}){\mathit{w}}_i+{\sigma }_v^2{\mathit{w}}_i^H{\mathit{w}}_i},i=1,\dots ,K\end{array}$$

(35)

In this subsection, we evaluate the effect of TAA uncertainty and filter bank size on the performance of the method in Section 3.1. The TAA uncertainty range is chosen to be ${\varphi }_T=[25.5^{\circ },27.1^{\circ }]$, and the three filter bank sizes $K=4,8,16$ are considered separately. The ${\mathrm{SCNR}}_i\left(\theta \right),i=1,\dots ,K$ under $K=4$ is given in Figure 9a. From the figure, it can be seen that due to the finite number of filters, a single filter can only match part of the angle, which brings about a SCNR loss. For further analysis, Figure 9b gives the variation of ${\mathrm{SCNR}}_{\max }\left(\theta \right)$ with $\theta$ for the $K=8$ and $K=16$ conditions. The larger the K is, the smaller the SCNR loss, which is due to the fact that the increase in the number of filters improves the robustness of the system to the uncertainty of the TAA.

4.4. Effectiveness of the Proposed Method in Section 3.2

When the rainfall information is unknown, we make $K_{dp}=8.2^{\circ }/\mathrm{km}$ and $A_{dp}=0.02$$\mathrm{dB}/\mathrm{km}$ in this section based on the rain area parameters set in Section 4.1 and the empirical data. The method’s performance is initially validated from the standpoint of SCNR. Following a similar analysis method as depicted in Figure 8, comparison between the method in Section 3.2 and that in the literature [32] of Figure 10 demonstrates that the method proposed in Section 3.2 significantly enhances SCNR while ensuring good robustness.

Both methods in Section 3.1 and Section 3.2 are able to improve the system output SCNR through transmit signal design. Figure 11 compares the average waveforms designed by the two methods at different distances from the rain area. It is noteworthy that the transmit signals designed by the two methods exhibit strong correlation, and the correlation coefficient coeff = 0.9642 is calculated. Combined with Figure 10, it proves the effectiveness of the methods in Section 3.2 in improving the SCNR for the radar detection if the rainfall information is not provided.

4.5. Detection Probability Analysis

It has been verified above that both methods significantly improve the output SCNR of the fully polarized radar system under rainy conditions, and the target detection probability is analyzed in this subsection. Figure 12 gives the variation of the detection probability $p_d$ with the false alarm probability $p_{fa}$ for the initial transmit signal and transmit signal designed by two methods with signal-to-noise ratios (SNR) of 3 dB, 6 dB, 9 dB and 12 dB, respectively. Set the Monte Carlo number to 1000 and set $p_{fa}\in ({10}^{-3},1)$, when fixed $p_{fa}$, the detection threshold h is determined by Equation (36):

$$\begin{array}{c}\hfill h=p_{fa}^{-\frac{1}{m}}-1\end{array}$$

(36)

where m is the number of training units and $m=20$ is taken in the simulation.

From Figure 12, it is easy to see that, compared with the unoptimized initial transmit waveform, both methods proposed in this paper improve the detection probability significantly, and the method in Section 3.1 is more advantageous. This is because the jointly optimized transmitting waveform has better performance against background noise perturbation and a stronger detection capability for weak target signals. Moreover, as the initial SNR increases, the resolution of the signal relative to the noise increases, and the detection probability under small false alarms also increases.

4.6. Signal Ambiguity Analysis

The ambiguity function is an important measure of signal characteristics. The ambiguity plots of the transmit signals of Section 3.1 and Section 3.2 methods are given in Figure 13, where Figure 13a,b, Figure 13c,d and Figure 13e,f correspond to the results when the similarity parameter is 0.2, 0.5 and 1.0, respectively. As $\gamma$ increases, the signal ambiguity function becomes progressively worse, the amplitude of the main peak decreases, the purity becomes lower and lower while the spurious component becomes more and more. This is because larger $\gamma$ implies looser similarity constraints. At the same SCNR boost, the optimal transmit signal ambiguity designed by method in Section 3.2 is better than that in Section 3.1. This is because the method in Section 3.2 only changes the polarization characteristics of the transmit signal, whereas the method in Section 3.1 performs global optimization of the waveform and polarization, which can only guarantee the overall characteristics but not satisfy the local requirements.

In particular, we note that in terms of algorithmic complexity, the method in Section 3.1 is determined by the complexity of solving the SOCP problem, i.e., $O({\left(2M\right)}^{3.5}log(1/\eta ))$[41], where $\eta$ is a preset accuracy value. In contrast, the method in Section 3.2 does not need to iterate and solve the optimization problem, but only needs to solve the transmit signal based on the polarized radar parameter with the complexity of $O\left(M\right)$. Obviously, the method in Section 3.2 has a lower complexity, requires and consumes less radar hardware and has a smaller time cost.

In summary, we validate the performance of the two methods in real scenarios. The advantages and disadvantages of the two methods proposed in this paper are compared in different aspects and the following conclusions are given for the comprehensive applicable scenarios. When the rainfall information is known and the radar hardware performance is high, the method of Section 3.1 can be used to achieve higher target detection accuracy and anti-jamming capability, but at the expense of time efficiency. When the rainfall information is unknown, the method in Section 3.2 is needed to design the transmit signal, which can greatly improve the time efficiency with small performance loss.

5. Discussion

Through the analysis of real targets, it can be seen that both methods proposed in this paper have significantly improved the detection performance of fully polarized radar targets under rainy conditions. In particular, the optimal polarization design method based on the observation parameters of the fully polarized radar can improve the detection performance under the condition of unknown rainfall information by using lower computational resources. Of course, the method in this paper still has some room for improvement. For example, it can be combined with deep learning algorithms to carry out fine inversion of rainfall scenarios [42] to improve the credibility of the algorithm effect. At the same time, combining the one-dimensional distance image of the target and the polarization information, and using deep convolutional neural network for learning, deep feature extraction can be done on the polarization distance matrix of the target, which can effectively improve the detection ability of the target [43]. All these works will be our subsequent research direction.

6. Conclusions

In this paper, an optimization method for the polarization and waveform of fully polarized radar transmit signals under rainy conditions was investigated. Firstly, we established a system detection model for complex rainy conditions by integrating the propagation effects of polarized electromagnetic waves in the rain medium. Subsequently, we proposed two methods to achieve improved target detection performance in fully polarized radars: a joint optimization method for transmission and reception based on transmission effects and an optimal transmit signal design method leveraging the observation coefficients of fully polarized radar, with or without prior knowledge of the rainfall area, respectively.

The problem of joint optimization is a complex non-convex problem, and in this paper, the objective is to maximize the output SCNR of the system by imposing dual constraints of energy and similarity. This problem is then transformed into an alternating iterative optimization process, where each iteration involves a convex problem and a GFP problem. With the rainfall information unknown, we explored the nature of the transmission effect and design the optimal transmit signal using two known radar observation covariates $A_{dp}$ and $k_{dp}$. We eliminated the depolarization effect of the rain medium and improve the system SCNR by changing the amplitude–phase characteristics between the horizontal and vertical polarization channels.

We provided some simulation examples to evaluate and validate the effectiveness of the proposed methods in a variety of experimental scenarios. The results show that both methods under arbitrary TAA can ensure significant SCNR and detection probability enhancement, and better overcome the negative impact of rainfall. In addition, the differences between the two methods in terms of ambiguity characteristics and algorithm complexity were compared, and method selection strategies for practical applications were given.