Cooperated Moving Target Detection Approach for PA-FDA Dual-Mode Radar in Range-Ambiguous Clutter

Abstract

This paper proposes a cooperated range ambiguous clutter suppression method for moving target detection in the background of range-ambiguous clutter with a phased array (PA)–frequency diverse array (FDA) dual-mode radar. With the FDA mode, the range-ambiguous clutters are discriminated in the transmit spatial frequency domain, and thus the clutter covariance matrixes (CCMs) corresponding to unambiguous and ambiguous regions can be independently estimated. Therefore, the enhanced CCM can be reconstructed by using a linear combination of these distinguished CCMs from different range regions. With the PA mode, the enhanced CCM is applied, thus taking advantages of its high beampattern gain as well as alleviating the range ambiguous clutter suppression problem. Simulation results are presented to verify the effectiveness of the proposed method in range-ambiguous clutter scenarios.

1. Introduction

Ground moving target detection with an airborne radar system is the most significant task in civil traffic monitoring, as well as military surveillance and reconnaissance [1,2]. For the down-looking working mode, the clutter spectrum would broaden severely in the Doppler space due to the airborne platform motion, which causes moving targets, especially slow-moving targets, to be submerged by the spread clutter. Space-time adaptive processing (STAP) is a powerful means of improving moving target detection performance in nuisance clutter and jamming. However, the range dependence and range ambiguity of clutter poses a threat to the STAP performance. The clutter spectrum in the angle–Doppler domain is no longer linear and varies with range bins for the nonside-looking array configuration, which is regarded as range dependence [3]. In this case, it is challenging to obtain the adequate independent identical distributed (IID) training samples. Several algorithms have been developed to alleviate the range dependence in the STAP radar, such as Doppler warping [4], angle–Doppler compensation [5], and space-time interpolation [6]. Nevertheless, range ambiguity renders their compensation performance in vain.

For radar operating at medium or high pulse-repetition frequency (PRF) mode, range ambiguity occurs. To cope with the range-ambiguous clutter problem, many methods have been developed. The direct data domain (DDD) method can implement sliding windows operation within the single range cell under test to mitigate the range ambiguity problem in [7,8]. Unfortunately, the energy cancellation of the target signal and the loss of the space-time degrees-of-freedom (DOFs) result in the performance degradation of clutter suppression and target detection. In [9,10,11], the planar array with additional vertical DOFs was utilized to distinguish the clutter corresponding to the near region. However, the condition of sufficient vertical DOFs need to be satisfied, which would increase the system design complexity. The 3-dimensional (3D) STAP methods were proposed in [12,13,14], which can separate range-ambiguous clutter in the horizontal–vertical-Doppler domain. However, a large number of IID training samples are required, which is unavailable in the actual environment. Several advance reduced-dimension/rank STAP approaches were developed to alleviate the problem of limited IID training samples in [15,16,17]. The frequency diverse array with multiple-input multiple-output technique (FDA-MIMO) radar has demonstrated benefits in some applications, including joint range-angle determination [18,19], main-lobe deceptive jamming suppression [20,21], and high-resolution and wide-swath synthetic aperture radar imaging [22], since the beampattern function of FDA are modulated by the range, angle, and time compared to the traditional PA radar [23]. In [24,25], an adaptive range-angle-Doppler processing approach has been developed with MIMO technique to improve clutter suppression performance for FDA STAP radar. However, a large computational complexity and IID training samples are required. In [26], the frequency diverse array (FDA) was utilized as a transmit array, and the range-ambiguous clutter can be separated in the transmit spatial frequency domain. In [27,28], the pre-STAP filtering method is proposed to separate the range-ambiguous clutter in the vertical FDA-MIMO radar. An range-ambiguous clutter suppression method based on transmitted beam mainlobe correction for FDA-MIMO radar is proposed in [29]. Compared to the common FDA, the constraint of parameter design is relaxed. In [30], the range-ambiguous clutter is separated using transmit dimensional DOFs obtained by using a slow-time coding program. In [31], a range-ambiguous clutter suppression method based on multi-waveform technique is proposed, in which the secondary waveforms are transmitted to obtain a target-free covariance matrix in transmit–receive spatial dimension; thus, the range-ambiguous clutter is suppressed. However, this restricts the utilization of Doppler information. However, it is worth noting that the wide beampattern coverage of FDA results in power loss within the identical coherent processing interval (CPI) compared to PA.

In this paper, we propose a PA-FDA dual-mode radar framework which aims at improving the target signal-to-clutter-plus-noise ratio (SCNR) output in range-ambiguous clutter scenarios. The PA-FDA dual-mode radar is composed of two independent parts, i.e., PA and FDA. The FDA part is employed to discriminate the overlapped range-ambiguous clutter. As range ambiguity is resolved, the clutter covariance matrixes (CCMs) corresponding to unambiguous and ambiguous range regions can be estimated accurately. Under specific conditions, the PA and FDA counterpart share the same clutter spectrum structures in the spatial-temporal plane. Therefore, the enhanced CCM for the PA counterpart can be obtained with a linear combination of these separated CCMs estimated by the FDA counterpart. In the sequel, the adaptive weight corresponding to the PA part is calculated by using the enhanced CCM. As a result, SCNR output is improved due to a high beampattern gain and improved estimation accuracy of CCM with PA in range ambiguous clutter scenarios.

The remainder of this letter is organized as follows. The signal model of the PA-FDA dual-mode radar is introduced in Section 2, followed by range-ambiguous clutter suppression in Section 3. In Section 4, simulation results are provided to validate the effectiveness of the proposed method. Finally, conclusions are drawn in Section 5.

2. Signal Model of the PA-FDA Dual-Mode Radar

In this paper, we consider an airborne forward-looking uniform linear array (ULA) in the Cartesian coordinate system, as shown in Figure 1. The height of the platform is H, with a velocity V. A CPI includes K pulses. L indicates the number of range bins. $\theta$ and $\phi$ denote azimuth and elevation angles of an arbitrary scattering patch, respectively. $f_{PRF}=1/T$ denotes the $PRF$, with T being the pulse-repetition interval. The transmit array of the PA-FDA dual-mode radar consists of two parts, i.e., PA and FDA. As shown in Figure 2, the PA part is colored in red while the FDA part is colored in blue. In the receiver, there are $N=2M$ receive elements with a ULA.

PA part: For the PA part of the dual-mode radar, the coherent waveforms are transmitted at the identical carrier frequency $f_0$, which can provide a high directional spatial gain that contributes to detecting and tracking targets. This transmit beampattern gain of PA in the far-field is expressed as

$$G\left({\psi }_0\right)=\sum _{m=1}^M{e}^{j2\pi \frac{d\left(m-1\right)\left(cos\phi sin\theta -cos{\phi }_{0}sin{\theta }_0\right)}{{\lambda }_0}}$$

(1)

where $\psi$ is the conic angle, $cos\left(\psi \right)=cos\left(\theta \right)cos\left(\phi \right)$, and ${\psi }_0$ is the direction of beamforming. ${\lambda }_0$ is the radar wavelength.

FDA part: For the FDA part of the dual-mode radar, M orthogonal waveforms are transmitted at different carrier frequency and it provides wide beam coverage in space as well as range ambiguity resolution. The transmit carrier frequencies vary among elements, which are written as

$$f_m=f_0+(m-1)\Delta f m=1,2,\cdots M$$

(2)

where $\Delta f$ is frequency increment between two adjacent elements, which is much smaller than the bandwidth of the transmitted waveform and the reference carrier frequency. $f_0$ is the reference carrier frequency. Note that the Direct Digital Synthesizer (DDS) is used to generate a multi-frequency signal in the transmit system. The waveform of the PA part is denoted as ${\Phi }_0\left(t\right)$, and the mth waveform of the FDA part is denoted as ${\Phi }_m\left(t\right)$. This orthogonal assumption among these waveforms can be expressed as

$${\int }_{T_p}{\Phi }_m\left(t\right){\Phi }_n^{\ast }\left(t-\tau \right)e^{j2\pi \Delta f\left(m-n\right)t}dt={\delta }_{mn}\left(\tau \right) , m,n= 0,1,\cdots ,M$$

(3)

where $T_p$ and $\tau$ are the pulse duration and time delay, respectively. ${\left(\right)}^{\ast }$ denotes the conjugate operation. Notice that ideal orthogonality is unrealistic, and the orthogonal waveform design is a research topic in the radar community [32,33]. In this paper, similar to [24,25,26,27,34], we still take the orthogonality assumption for simplicity.

Considering the qth clutter scatter in the pth ambiguous range region within the lth range bin, the signal received at the nth receive element from the kth pulse can be expressed as

$$\begin{array}{c}{r}_{n,k}\left(t\right)={\displaystyle \sum _{m=1}^M}{\xi }_{l,p,q}{\Phi }_0\left(t-{\tau }_{m,n}^{PA}\right)e^{j2\pi f_0\left(t-{\tau }_{m,n}^{PA}\right)}{e}^{j2\pi f_{d0}\left({\psi }_{l,p,q}\right)\left(k-1\right)}\hfill \\ +{\displaystyle \sum _{m=1}^M}{\xi }_{l,p,q} {\Phi }_m\left(t-{\tau }_{m,n}^{FDA}\right)e^{j2\pi f_m\left(t-{\tau }_{m,n}^{FDA}\right)}{e}^{j2\pi f_d^m\left({\psi }_{l,p,q}\right)\left(k-1\right)}\hfill \end{array}$$

(4)

where ${\tau }_{m,n}^{PA}$ and ${\tau }_{m,n}^{FDA}$ are the round-trip time delay of PA and FDA, respectively.

$$\left\{\begin{array}{c}{\tau }_{m,n}^{PA}=\left(2R_{l,p}-\left(m-1\right)dsin\left({\theta }_{l,p,q}\right)cos\left({\phi }_{l,p}\right)-\left(n-1\right)dsin\left({\theta }_{l,p,q}\right)cos\left({\phi }_{l,p}\right)\right)/c\hfill \\ {\tau }_{m,n}^{FDA}=\left(2R_{l,p}-\left(M+m-1\right)dsin\left({\theta }_{l,p,q}\right)cos\left({\phi }_{l,p}\right)-\left(n-1\right)dsin\left({\theta }_{l,p,q}\right)cos\left({\phi }_{l,p}\right)\right)/c\hfill \end{array}\right.$$

(5)

where $f_{d0}\left({\psi }_{l,p,q}\right)=2VTcos\left({\psi }_{l,p,q}\right)/{\lambda }_0$ and $f_d^m\left({\psi }_{l,p,q}\right)=2VT{f}_{m}cos\left({\psi }_{l,p,q}\right)/c$ are the Doppler frequencies of the PA and FDA parts, respectively. ${\xi }_{l,p,q}$ is the complex coefficient. By performing down converting and matched filtering with $h\left(t\right)={\Phi }_0^{\ast }\left(t\right)e^{-j2\pi f_{0}t}$, the signal corresponding to the PA part can be extracted from the originally aggregated echoes, that is:

$$\begin{array}{l}{r}_{n,k}^{PA}=\left[r_{n,k}\left(t\right)e^{-j2\pi f_{0}t}\right]⊛{\Phi }_0^{\ast }\left(t\right)={\stackrel{⏜}{r}}_{n,k}\left(t\right)⊛{\Phi }_0^{\ast }\left(t\right)\\ ={\int \stackrel{⏜}{r}}_{n,k}\left(\tau -{\tau }_{m,n}^{PA}\right){\Phi }_0^{\ast }\left(\tau -t\right)d\tau \\ ={\displaystyle \sum _{m=1}^M}{W}_0^H{\zeta }_{l,p,q}\delta \left(t-{\tau }_0\right)e^{-j2\pi f_0{\tau }_{m,n}^{PA}}{e}^{j2\pi f_{d0}\left({\psi }_{l,p,q}\right)\left(k-1\right)}\\ ={\zeta }_{l,p,q}G\left({\psi }_0\right)e^{-j2\pi \frac{2R_{l,p}}{{\lambda }_0}}{e}^{j2\pi \left(f_{d0}\left({\psi }_{l,p,q}\right)\right)\left(k-1\right)}{e}^{-j2\pi \frac{\left(n-1\right)d\sin \left({\theta }_{l,p,q}\right)\cos \left({\phi }_{l,p}\right)}{{\lambda }_0}}\end{array}$$

(6)

where ⊛ denotes the convolution operator. $W_0=e^{j2\pi d\left(m-1\right)cos\left({\phi }_0\right)sin\left({\theta }_0\right)}$ is the beamforming weight generated by phase shifter. The narrow approximation is used, i.e., ${\Phi }_0\left(t-{\tau }_{m,n}^{PA}\right)\approx {\Phi }_0\left(t-{\tau }_0\right)={\Phi }_0\left(t-2R_{l,p}/c\right)$. Notice that the signal received by the FDA part is eliminated by orthogonal matched filtering:

$$\sum _{m=1}^M{\xi }_{l,p,q}{e}^{-j2\pi f_m{\tau }_{m,n}^{FDA}}{e}^{j2\pi \Delta ft}{e}^{j2\pi f_d^m\left({\psi }_{l,p,q}\right)\left(k-1\right)}\int {\Phi }_m\left(\tau -{\tau }_{m,n}^{FDA}\right){\Phi }_0^{\ast }\left(\tau -t\right)d\tau =0$$

(7)

By straight mathematical derivation, (6) can be further simplified as

$$r_{n,k}^{PA}\left(l,p,q\right)={\xi }_{l,p,q}G\left({\psi }_0\right)e^{j2\pi \left[f_h\left({\phi }_{l,p},{\theta }_{l,p,q}\right)\left(n-1\right)+f_{d0}\left({\psi }_{l,p,q}\right)\left(k-1\right)\right]}$$

(8)

where $f_h\left({\phi }_{l,p},{\theta }_{l,p,q}\right)=dsin\left({\theta }_{l,p,q}\right)cos\left({\phi }_{l,p}\right)/{\lambda }_0$ is the receive spatial frequency. Stacking $r_{n,k}^{PA}\left(l,p,q\right)$ into the $NK\times 1$ vector, we can obtain the space-time snapshot of the PA part as

$$\begin{array}{c}{\mathit{y}}^{PA}\left(l,p,q\right)={\left[\begin{array}{ccc}{r}_{1,1}^{PA}\left(l,p,q\right)& \cdots & r_{N,K}^{PA}\left(l,p,q\right)\end{array}\right]}^T\hfill \\ ={\xi }_{l,p,q}G\left({\psi }_{l,p,q}\right){\mathit{\alpha }}_h\left(f_h\left({\phi }_{l,p},{\theta }_{l,p,q}\right)\right)\otimes \mathit{\beta }\left(f_{d0}\left({\psi }_{l,p,q}\right)\right)\hfill \end{array}$$

(9)

where ${\mathit{\alpha }}_h\left(f_h\left({\phi }_{l,p},{\theta }_{l,p,q}\right)\right)$ and $\mathit{\beta }\left(f_{d0}\left({\psi }_{l,p,q}\right)\right)$ are the receive spatial and temporal steering vector, respectively. They are written as

$${\mathit{\alpha }}_h\left(f_h\left({\phi }_{l,p},{\theta }_{l,p,q}\right)\right)={\left[\begin{array}{cccc}1& e^{j2\pi f_h\left({\phi }_{l,p},{\theta }_{l,p,q}\right)}& \cdots & e^{j2\pi f_h\left({\phi }_{l,p},{\theta }_{l,p,q}\right)\left(N-1\right)}\end{array}\right]}^T$$

(10)

$$\mathit{\beta }\left(f_{d0}\left({\psi }_{l,p,q}\right)\right)={\left[\begin{array}{cccc}1& e^{j2\pi f_{d0}\left({\psi }_{l,p,q}\right)}& \cdots & e^{j2\pi f_{d0}\left({\psi }_{l,p,q}\right)\left(K-1\right)}\end{array}\right]}^T$$

(11)

Similar to the matched processing in (6), by performing orthogonal matched filtering with ${\Phi }_m^{\ast }\left(t\right)e^{-j2\pi f_{0}t}$ and straightforward mathematical manipulation for the FDA counterpart, the signal of the FDA part corresponding to the mth transmit element, the nth received element, and the kth pulse is written as

$$\begin{array}{c}{r}_{m,n,k}^{FDA}\left(l,p,q\right)={\xi }_{l,p,q}{e}^{j2\pi f_t\left(R_{l,p},{\phi }_{l,p},{\theta }_{l,p,q}\right)\left(m-1\right) }\hfill \\ \times e^{j2\pi f_h\left({\phi }_{l,p},{\theta }_{l,p,q}\right)\left(n-1\right)}{e}^{j2\pi f_{d0}\left({\psi }_{l,p,q}\right)\left(k-1\right)}\hfill \end{array}$$

(12)

where $f_t\left(R_{l,p},{\phi }_{l,p},{\theta }_{l,p,q}\right)=f_r\left(R_{l,p}\right)+f_v\left({\phi }_{l,p},{\theta }_{l,p,q}\right)$ is the transmit spatial frequency, and $f_r\left(R_{l,p}\right)=-2\Delta f\left[R_l+\left(p-1\right)R_u\right]/c$ and $f_v\left({\phi }_{l,p},{\theta }_{l,p,q}\right)=dsin\left({\theta }_{l,p,q}\right)cos\left({\phi }_{l,p}\right)/{\lambda }_0$ are the range frequency and spatial frequency, respectively. Note that the additional second phase terms are negligible for (12) under the condition of $\Delta f\ll f_0$ [27], and the constant exponential phase terms are merged to ${\xi }_{l,p,q}$. The space-time snapshot of the FDA part can be expressed as

$$\begin{array}{c}{\mathit{y}}^{FDA}\left(l,p,q\right) ={\left[\begin{array}{ccc}{r}_{1,1,1}^{FDA}\left(l,p,q\right)& \cdots & r_{M,N,K}^{FDA}\left(l,p,q\right)\end{array}\right]}^T\hfill \\ ={\xi }_{l,p,q}{\mathit{\alpha }}_v\left(f_t\left(R_{l,p},{\phi }_{l,p},{\theta }_{l,p,q}\right)\right)\otimes {\mathit{\alpha }}_h\left(f_h\left({\phi }_{l,p},{\theta }_{l,p,q}\right)\right)\otimes \mathit{\beta }\left(f_{d0}\left({\psi }_{l,p,q}\right)\right) \hfill \end{array}$$

(13)

where ${\mathit{\alpha }}_v\left(f_t\left(R_{l,p},{\phi }_{l,p},{\theta }_{l,p,q}\right)\right)$ represents the transmit spatial steering vector, which can be expressed as

$$\begin{array}{c}\hfill {\mathit{\alpha }}_v\left(f_t\left(R_{l,p},{\phi }_{l,p},{\theta }_{l,p,q}\right)\right) ={\left[\begin{array}{cccc}1& e^{j2\pi f_t\left(R_{l,p},{\phi }_{l,p},{\theta }_{l,p,q}\right)}& \cdots & e^{j2\pi f_t\left(R_{l,p},{\phi }_{l,p},{\theta }_{l,p,q}\right)\left(M-1\right)}\end{array}\right]}^T\end{array}$$

(14)

The clutter corresponding to the lth range bin results from the coherent summation of lots of scattering patches from unambiguous and ambiguous range regions. The clutter of the PA-FDA dual-mode radar can be expressed as

$$\begin{array}{cc}\hfill {\mathit{c}}_l& =\sum _{p=1}^{N_a}\sum _{q=1}^{N_c}\left[\begin{array}{c}{\mathit{y}}^{PA}\left(l,p,q\right)\hfill \\ {\mathit{y}}^{FDA}\left(l,p,q\right)\hfill \end{array}\right]\hfill \\ \hfill & =\sum _{p=1}^{N_a}\sum _{q=1}^{N_c}{\xi }_{l,p,q}\left[\begin{array}{c}G\left({\psi }_0\right)\\ {\mathit{\alpha }}_v\left(f_t\left(R_{l,p},{\phi }_{l,p},{\theta }_{l,p,q}\right)\right)\end{array}\right]\otimes {\mathit{\alpha }}_h\left(f_h\left({\phi }_{l,p},{\theta }_{l,p,q}\right)\right)\otimes \mathit{\beta }\left(f_{d0}\left({\psi }_{l,p,q}\right)\right)\hfill \end{array}$$

(15)

The total echoes usually consist of clutter, target and noise, which is expressed as:

$${\mathit{x}}_l={\mathit{c}}_l+{\mathit{s}}_l+{\mathit{n}}_l$$

(16)

where ${\mathit{s}}_l$ and ${\mathit{n}}_l$ represent the target and additive Gaussian white noise, respectively.

3. Cooperated Range Ambiguous Clutter Suppression with PA-FDA Dual-Mode Radar

In this section, a cooperated range ambiguous clutter suppression scheme of PA-FDA dual-mode radar is proposed, which contains two key procedures: (i) range-ambiguous separation using the FDA part; (ii) the enhanced CCM reconstructed for cutter suppression of the PA counterpart, resulting in the improvement of SCNR output.

3.1. Range-Ambiguous Clutter Separation with FDA

The range-ambiguous clutters are separated in the transmit–Doppler domain for the FDA counterpart. Therefore, the receive beamforming is performed to synthesize receive dimensional data, and the weight is designed as

$${\mathit{\omega }}_r={\left[\begin{array}{cccc}1& e^{j2\pi dsin\left({\theta }_0\right)cos\left({\phi }_0\right)/{\lambda }_0}& \cdots & e^{j2\pi dsin\left({\theta }_0\right)cos\left({\phi }_0\right)/{\lambda }_0}\end{array}\right]}^T$$

(17)

where the transmit spatial frequency is expressed as $f_t\left(R_{l,p},{\phi }_{l,p},{\theta }_{l,p,q}\right)=f_r\left(R_{l,p}\right)+f_v\left({\phi }_{l,p},{\theta }_{l,p,q}\right)$ in (12) that is

$$f_t\left(R_{l,p},{\phi }_{l,p},{\theta }_{l,p,q}\right){|}_{p=1.2}=-2\Delta f\left[R_l+\left(p-1\right)R_u\right]/c+dsin\left({\theta }_{l,p,q}\right)cos\left({\phi }_{l,p}\right)/{\lambda }_0$$

(18)

It is observed that $f_r\left(R_{l,p}\right)$ is introduced into the transmit spatial frequency, which offers the capability of range-ambiguous clutter separation in the transmit spatial frequency domain. To obtain a better separation scheme, a secondary range dependence compensation (SRDC) [17] vector is devised as

$${\mathit{h}}_c=\left[\begin{array}{cccc}1& e^{j2\pi f_c^{\alpha }}& \cdots & e^{j2\pi f_c^{\alpha }\left(M-1\right)}\end{array}\right]$$

(19)

where $f_c^{\alpha }$ represents the compensated spatial frequency with the superscript $\alpha$ denoting the unambiguous and ambiguous regions, that is

$$f_c^{\alpha }=\left\{\begin{array}{c}\frac{2\Delta f{R}_1}{c} \alpha =1\\ \frac{2\Delta f{R}_2}{c} \alpha =2 \end{array}\right.$$

(20)

where $R_1=R_l$ and $R_2=R_l+R_u$. After performing SRDC for the transmit spatial frequency in (18)

$${\stackrel{⏜}{f}}_t\left(R_{l,p},{\phi }_{l,p},{\theta }_{l,p,q}\right)\left|{}_{\alpha =1}\right.=\left\{\begin{array}{l}\frac{d\sin \left({\theta }_{l,p,q}\right)\cos \left({\phi }_{l,p}\right)}{{\lambda }_0} \mathit{unambiguous}\mathit{region}\\ \frac{-2\Delta f{R}_u}{c}+\frac{d\sin \left({\theta }_{l,p,q}\right)\cos \left({\phi }_{l,p}\right)}{{\lambda }_0} \mathit{ambiguous}\mathit{region}\end{array}\right.$$

(21)

$${\stackrel{⏜}{f}}_t\left(R_{l,p},{\phi }_{l,p},{\theta }_{l,p,q}\right)\left|{}_{\alpha =2}\right.=\left\{\begin{array}{l}\frac{-2\Delta f{R}_u}{c}+\frac{d\sin \left({\theta }_{l,p,q}\right)\cos \left({\phi }_{l,p}\right)}{{\lambda }_0} \mathit{unambiguous}\mathit{region}\\ \frac{d\sin \left({\theta }_{l,p,q}\right)\cos \left({\phi }_{l,p}\right)}{{\lambda }_0} \mathit{ambiguous}\mathit{region}\end{array}\right.$$

(22)

It is observed from (21) and (22) that this generates the constant frequency difference $\frac{2\Delta f{R}_u}{c}$ between the clutters from ambiguous and unambiguous regions in the transmitted spatial frequency domain. Figure 3a–d show the spatial and Doppler frequencies relationship in the traditional PA radar, the FDA radar before performing SRDC, and the FDA radar after SRDC is implemented, when $\alpha =1$ and $\alpha =2$, respectively. It is observed that unambiguous and ambiguous clutter overlap together in the traditional PA radar. As shown in Figure 3b, the unambiguous and ambiguous clutter have been discriminated in the FDA radar framework, but this separation result should be enhanced. For Figure 3c, the unambiguous clutter is consistent with the PA radar, while the clutter of the ambiguous region is moved to both sides of the transmitted spatial frequency domain when $\alpha =1$. For Figure 3d, the ambiguous clutter is consistent with the PA radar, while the clutter of the unambiguous region is moved to both sides of the transmitted spatial frequency domain when $\alpha =2$.

After performing received beamforming and SDRC, the clutter corresponding to the $\alpha$th ambiguous range region is expressed as

$$\begin{array}{c}\hfill \begin{array}{c}{\mathit{c}}_l^{\alpha }={\displaystyle \sum _{p=1}^{N_a}}{\displaystyle \sum _{q=1}^{N_c}}{\xi }_{l,p,q}\left[{\mathit{\alpha }}_v\left(f_t\left(R_{l,p},{\phi }_{l,p},{\theta }_{l,p,q}\right)\right)\odot {\mathit{h}}_c\left(f_c^{\alpha }\right)\right]\\ \otimes \left[{\mathit{\omega }}_r^H{\mathit{\alpha }}_h\left(f_h\left({\phi }_{l,p},{\theta }_{l,p,q}\right)\right)\right]\otimes \mathit{\beta }\left(f_{d0}\left({\psi }_{l,p,q}\right)\right)\end{array}\end{array}$$

(23)

The CCMs corresponding to the unambiguous and ambiguous range regions can be estimated independently with maximum likelihood approach, that is,

$$\begin{array}{c}\hfill {\mathit{R}}_{\alpha }^{FDA}=E\left[\left({\mathit{c}}_l^{\alpha }\right){\left({\mathit{c}}_l^{\alpha }\right)}^H\right]\approx \frac{1}{L-1}\sum _{i=1,i\ne l_0}^L\left({\mathit{c}}_l^{\alpha }\right){\left({\mathit{c}}_l^{\alpha }\right)}^H\end{array}$$

(24)

Notice that the clutter range dependence needs to be compensated for training samples ${\mathit{c}}_l^{\alpha }$ before estimating CCM [5,6].

3.2. Range-Ambiguous Clutter Suppression with PA

Recall that CCM estimation for the PA counterpart is difficult due to the range dependence and range ambiguity of clutter. Fortunately, the range-ambiguous clutters have been separated by using the FDA, and the distinct CCMs can be calculated with (24). It is worth noting that spatial and temporal clutter spectrum structures of the FDA part in the transmit-Doppler domain are consistent with that of the PA part in the receive-Doppler domain under specific conditions. Therefore, the CCMs corresponding to each range bin of PA can be represented with that of the FDA part. In other words, the CCMs estimated by the FDA part can be employed in the PA counterpart to resolve the CCM estimation problem. The reconstructed CCM of the PA part is expressed as

$$\begin{array}{c}\hfill {\mathit{R}}_l^{PA}=\sum _{\alpha =1}^{N_a}{\mathit{R}}_l^{\alpha }\end{array}$$

(25)

The adaptive weight in minimum variance distortionless response (MVDR) is calculated as

$$\begin{array}{c}\hfill {\mathit{\omega }}_l^1=\frac{{\left({\mathit{R}}_l^{PA}\right)}^{-1}\mathit{b}\left({\phi }_0,{\theta }_0,v_0\right)}{{\mathit{b}}^H\left({\phi }_0,{\theta }_0,v_0\right){\left({\mathit{R}}_l^{PA}\right)}^{-1}\mathit{b}\left({\phi }_0,{\theta }_0,v_0\right)}\end{array}$$

(26)

where $\mathit{b}\left({\phi }_0,{\theta }_0,v_0\right)={\mathit{\alpha }}_h\left(f_h\left({\phi }_0,{\theta }_0\right)\right)\otimes \mathit{\beta }\left(f_{d0}\left({\psi }_0,v_0\right)\right)$ represents the target steering vector. By utilizing cell under test (CUT) of the PA counterpart with a high beampattern gain, it is possible to improve SCNR output. Notice that the dimension of weight ${\mathit{\omega }}_l^1$ is $MK$, whereas the space-time snapshot of CUT in the PA part is $2MK$. Obviously, this will cause array gain loss if only half of the receive array data are used. To match the weight vector, an augmented weight vector is generated to further improve the SCNR; that is

$$\begin{array}{c}\hfill {\mathit{\omega }}_l^2=\frac{{\left({\mathit{R}}_l^{PA}\right)}^{-1}\mathit{\gamma }\left({\phi }_0,{\theta }_0,v_0\right)}{{\mathit{\gamma }}^H\left({\phi }_0,{\theta }_0,v_0\right){\left({\mathit{R}}_l^{PA}\right)}^{-1}\mathit{\gamma }\left({\phi }_0,{\theta }_0,v_0\right)}\end{array}$$

(27)

where $\mathit{\gamma }\left({\phi }_0,{\theta }_0,v_0\right)=\left[{\mathit{\gamma }}_r\left({\phi }_0,{\theta }_0\right)\right]\otimes \mathit{\beta }\left(f_{d0}\left({\psi }_0,v_0\right)\right)$ is the target matched steering vector. ${\mathit{\gamma }}_r\left({\phi }_0,{\theta }_0\right)$ can be expressed as

$$\begin{array}{c}\hfill {\mathit{\gamma }}_r\left({\phi }_0,{\theta }_0\right)={\left[\begin{array}{cccc}{e}^{j2\pi \frac{Mdcos{\phi }_{0}sin{\theta }_0}{{\lambda }_0}}& e^{j2\pi \frac{\left(M+1\right)dcos{\phi }_{0}sin{\theta }_0}{{\lambda }_0}}& \cdots & e^{j2\pi \frac{\left(2M-1\right)dcos{\phi }_{0}sin{\theta }_0}{{\lambda }_0}}\end{array}\right]}^T\end{array}$$

(28)

The augmented vector corresponding to full received array is written as

$$\begin{array}{c}\hfill {\mathit{\omega }}_l={\left[\begin{array}{cc}{\mathit{\omega }}_l^1& {\mathit{\omega }}_l^2\end{array}\right]}^T\end{array}$$

(29)

The SCNR output can be expressed as

$$\begin{array}{c}\hfill y_{out}={\mathit{\omega }}_l^H{\mathit{x}}_l^{PA}\end{array}$$

(30)

where ${\mathit{x}}_l^{PA}$ is the PA part data under test.

The main procedures of the proposed moving target detection method with the PA-FDA dual-mode radar in range-ambiguous clutter scenario are summarized in Figure 4. The proposed PA-FDA dual-mode radar can fully combine the advantages of PA and FDA radar, which improves the target detection performance in range-ambiguous clutter. Firstly, the range-ambiguous clutter is separated by using FDA in the transmitted spatial frequency domain, and thus the CCMs of the unambiguous and ambiguous range regions can be estimated independently. Secondly, an enhanced CCM corresponding to the PA part is reconstructed by combining the CCMs estimated by the FDA part. Thirdly, the adaptive weight is calculated to suppress clutter in the PA part, which can maximize the advantage of high beampattern gain for target detection and tracking. Moreover, it is noted that the FDA part can contribute to target searching by using wide-beam coverage property.

4. Simulation Result

In this section, the simulation results are shown to verify the effectiveness of the proposed method. The simulated parameters are listed in

Table 1. Radar parameters for the simulated data.

Parameter	Value	Parameter	Value
CNR	50	PRF	15 kHz
Platform height	6000 m	Platform speed	100 m/s
Element spacing	0.0075 m	Frequency increment	22.5 kHz
Carrier frequency	10 GHz	Pulse number	16
Number of range bins	800	Rx element number	34
Tx element number of PA	17	Tx element number of FDA	17

.

4.1. Comparison of Clutter Spectra

Figure 5a shows the clutter spectra for the traditional PA radar without range ambiguity after DW is implemented; the “spectral centers” are well-aligned. However, as seen in Figure 5b, the clutter spectra spreads seriously after performing DW, due to the existence of range ambiguity. Figure 6a shows separated clutter spectra after SRDC is implemented when $\alpha =1$. We can observe that the unambiguous and ambiguous clutter have been separated in the transmitted spatial frequency domain, and the transmitted spatial frequency of clutter corresponding to the ambiguous range region occupies $\left[-0.5,-0.25\right]\cup \left[0.25,0.5\right]$, while that of clutter from the unambiguous range region occupies $\left[-0.25,0.25\right]$. The clutter spectrum corresponding to the unambiguous range region that belongs to the range of spatial frequency $\left[-0.25,0.25\right]$ is seriously spread due to clutter range dependence. The DW compensation method is applied to compensate clutter range dependence, as shown in Figure 6b. The “spectral centers” of clutter from the unambiguous region are well-aligned since the range-ambiguous clutter has been separated. Figure 6c shows separated clutter spectra after SRDC is implemented when $\alpha =2$. It is seen that the transmitted spatial frequency of clutter corresponding to the ambiguous range region occupies $\left[-0.25,0.25\right]$, while that of clutter from the unambiguous range region occupies$\left[-0.5,-0.25\right]\cup \left[0.25,0.5\right]$. By compensating clutter range dependence with DW, the clutter spectrum corresponding to the ambiguous region is further focused, as shown in Figure 6d.

Moreover, an ideal range-ambiguous clutter spectrum corresponding to PA is shown as a benchmark in Figure 7a. The clutter spectrum for the proposed PA-FDA dual-mode radar is presented in Figure 7b. It is observed that the clutter spectrum estimated by the proposed method is effective compared to the ideal case.

4.2. Comparison of SCNR Loss

It is common practice to use the signal-to-clutter-plus-noise ratio (SCNR) loss factor to assess the STAP radar’s performance in terms of detection. The SCNR loss factor is calculated as the ratio between noise-limited output SNR and clutter-limited output SCNR, i.e.,

$$\begin{array}{c}\hfill SCN{R}_{\mathit{loss}}=\frac{SCN{R}_{\mathit{out}}}{SN{R}_{\mathit{out}}}=\frac{\frac{{\mathit{\omega }}^H{\mathit{R}}_s\mathit{\omega }}{{\mathit{\omega }}^H{\mathit{R}}_{c+n}\mathit{\omega }}}{\frac{{\mathit{\omega }}^H{\mathit{R}}_s\mathit{\omega }}{{\mathit{\omega }}^H{\mathit{R}}_n\mathit{\omega }}}=\frac{{\mathit{\omega }}^H{\mathit{R}}_s\mathit{\omega }}{{\mathit{\omega }}^H{\mathit{R}}_{c+n}\mathit{\omega }}\frac{{\sigma }_n^2}{{\sigma }_s^{2}NK}\end{array}$$

(31)

where ${\mathit{R}}_s$, ${\mathit{R}}_{c+n}$ and ${\mathit{R}}_n$ stand for the signal covariance matrix, clutter-plus-noise covariance matrix and noise covariance matrix, respectively. $\mathit{\omega }$ is the space-time adaptive weight vector. ${\sigma }_n^2$ and ${\sigma }_s^2$ represent the noise and signal power, respectively.

Figure 8 depicts the SCNR loss of different methods, including the STAP method in traditional PA radar with and without the DW compensation, and the proposed method in the PA-FDA dual-mode radar. Moreover, the ideal curves are also presented as a benchmark. For the traditional single-mode PA STAP radar, the clutter suppression performance degrades substantially due to range dependence and range ambiguity. For the proposed PA-FDA dual-mode radar, the performance of range-ambiguous clutter suppression is well maintained compared to the ideal case.

4.3. Analysis of Target Detection Probability

Figure 9 illustrates the SCNR output of various radar modes when the target speed and input SNR are set as 10 m/s and 0; the specific parameters are shown in

Table 2. Simulation parameters of target.

Parameter	Value
Target speed	10 m/s
Target azimuth	0°
SNR	0 dB
False probability	${10}^{-5}$

. As can be seen, the SCNR output of the proposed dual-mode radar when received with a full array elements is superior to that of other traditional mode radars. Figure 10 depicts the detection probability of different mode radars when false probability is set as ${10}^{-5}$. It is observed that the target detection probability of the proposed PA-FDA dual mode radar is superior to that of traditional single mode PA and FDA radar. The reason for this is that the proposed dual-mode radar overcomes the problems of range-ambiguous clutter suppression in PA radar and low target gain in FDA radar simultaneously. Furthermore, the entire receive array data are employed to further improve the array gain.

5. Conclusions

In this paper, a PA-FDA dual-mode radar framework is established to address the problems of range-ambiguous clutter suppression in PA radar and low target gain in FDA radar, simultaneously. By using the FDA part, the CCMs corresponding to unambiguous and ambiguous range regions can be independently estimated. According to characteristics of sharing clutter spectrum structures between PA and FDA parts, the enhanced CCM of the PA part is reconstructed by linearly combining these CCMs estimated by the FDA part, which maximizes the high beampattern gain of PA and range ambiguity resolvability of FDA. Compared to the traditional PA radar, the SCNR loss and minimum detectable velocity (MDV) of the proposed method based on the PA-FDA dual-mode radar was significantly improved. Moreover, in a range-ambiguous clutter scenario, the SCNR output of the proposed dual-mode radar outperforms the single-mode PA and FDA radars by about 23 dB and 16 dB, respectively. The detection probability performance of the proposed method outperforms the single-mode FDA radar by about 15 dB, using a high beampattern gain of the PA part when a full array is used to receive. For the proposed PA-FDA dual-mode radar, applications for future consideration should include wide-area surveillance and target tracking.