Clutter Rank Estimation Method for Bistatic Radar Systems Based on Prolate Spheroidal Wave Functions

Abstract

Bistatic radar exhibits spatial isomerism and diverse configurations, leading to unique clutter characteristics distinct from those of monostatic radar. The clutter rank serves as a pivotal indicator of clutter characteristics, enabling the quantification of clutter severity. Space-time adaptive processing (STAP) is a critical technique to detect moving targets, and clutter rank determines the number of independent and identically distributed (IID) training samples and the degree of freedom (DOF) for effective suppression of clutter that STAP requires. Therefore, the accurate estimation of clutter rank for bistatic radar can provide a crucial indicator for designing and constructing STAP processors, thereby facilitating fast and efficient clutter suppression in bistatic radar systems. This study is based on the idea that clutter rank is the number of prolate spheroidal wave function (PSWF) orthogonal bases utilized for approximating the clutter signal. Firstly, the challenge of utilizing PSWF orthogonal bases for approximating the clutter signal in bistatic radar is elucidated. This pertains to the fact that, unlike monostatic radar clutter, bistatic radar clutter is not capable of being expressed as a single-frequency signal. The clutter rank estimation for bistatic radar is thus derived as the frequency bandwidth estimation. Secondly, to achieve this estimation, the frequency distribution of each individual scattering unit is investigated, thereby determining their extending frequency broadening (EFB) as compared to that of single-frequency. Subsequently, the integral average of EFB across the entire range bin is computed, ultimately enabling the acquisition of bistatic radar’s frequency bandwidth. Finally, the estimation method is extended to non-side-looking mode and limited observation areas with pattern modulation. Simulation experiments confirm that our proposed method provides accurate clutter rank estimations, surpassing 99% proportions of large eigenvalues across various bistatic configurations, observation modes, and areas.

1. Introduction

The earliest concept of early warning radar involved placing the transmitter and receiver on the same platforms, such as airborne and spaceborne radar [1,2]. However, the survivability of monostatic radar is threated by its inherent vulnerability. This vulnerability arises from the similar position of the transmitter and receiver in monostatic radar systems, which enables easy location of both the transmitter and receiver once the transmitted signal is captured by an enemy [3]. Subsequently, the platform is susceptible to deliberate interference. In response, the concept of bistatic radar emerged [4], wherein the transmitter and receiver are situated on separate platforms. Even if the enemy manages to locate the position of the transmitter, the receiver’s position differs from that of the transmitter. The receiver remains in a silent state by solely receiving signals without any transmission, which ensures that the position of the receiving platform remains undetectable, thereby endowing bistatic systems with inherent anti-jamming capabilities compared to monostatic radar systems. Additionally, the radar cross sections (RCS) of bistatic targets increase, thereby elevating the signal-to-noise ratio (SNR) and enhancing the anti-stealth capabilities of bistatic radar. Consequently, bistatic radar presents substantial research value [5,6,7].

Airborne and spaceborne bistatic radars, while advantageous, suffer from significant ground/sea clutter interference due to their looking-down operational state [8]. This becomes particularly problematic when the signal-to-clutter-plus-noise ratio (SCNR) is low, as moving targets are obscured by clutter and cannot be directly detected, necessitating clutter suppression [9,10]. Space-time adaptive processing (STAP) is an effective technique for suppressing ground/sea clutter to detect moving targets, but it requires a large number of independent and identically distributed (IID) training samples, as per adaptive processing theory [11,12,13,14,15]. The separation of transmitter and receiver in bistatic radars results in clutter characteristics that are substantially different from those of monostatic radars. Klemm categorized airborne bistatic configurations into four typical configurations based on the flight velocity directions of the two platforms in 2000: heading consistent, parallel, orthogonal, and cross configurations [16,17]. Formulas for calculating the Doppler and spatial frequencies of bistatic radar were provided, which subsequent scholars have used to analyse clutter space-time characteristics under different configurations of airborne [18,19,20] and space–air based bistatic radar systems [21] through simulation. The complex clutter space-time characteristics of bistatic radar may exacerbate the degradation of the number of IID training samples, consequently leading to a deterioration in the performance of STAP for bistatic radar.

In order to reduce the demand for IID training samples and maintain STAP performance, numerous STAP methods have been extensively studied. Clutter rank plays a pivotal role in quantifying the severity of clutter and evaluating the performance of STAP. It determines the required number of IID training samples and the degree of freedom (DOF) of STAP processors for effective suppression of clutter [22]. For example, reduced rank STAP (RR-STAP) algorithms [23,24,25,26] utilize a fixed processing structure through narrowband filtering to achieve adaptive processing, thereby significantly reducing the requirement for IID training samples. However, the RR-STAP is sensitive to the accurate estimation of clutter rank, and overestimating the clutter rank does not impact the performance, but increases computational processing. Conversely, underestimating the clutter rank results in a significant decline in processing performance. Furthermore, sparse recovery STAP (SR-STAP) [27,28,29,30] is a class of methods that utilize the sparse properties of clutter power spectrum in the angle-Doppler domain and employ sparse recovery algorithms to reconstruct the clutter covariance matrix. The sparsity of clutter is closely associated with the clutter rank, and this sparsity information plays a crucial role in sparse recovery, which can enhance the precision of sparse recovery [31]. Therefore, accurate clutter rank estimation for bistatic radar holds great research value for improving the performance of STAP methods applied to bistatic radar. To the best of our knowledge, the clutter rank for bistatic radar in scenarios where both aircraft maintain the same altitude and horizontal flight can be accurately estimated [32]. However, the clutter rank estimation for other arbitrary bistatic configurations remains unexplored.

The aperture-bandwidth product theory [1], typically used to estimate clutter rank in monostatic radar, indicates that when the array orientation aligns with the velocity direction, the Doppler domain of echo data can be projected to the array element domain, thus obtaining the virtual aperture. However, this linear relationship between Doppler frequency and spatial frequency does not exist when array orientation diverges from the velocity direction [33]. Expanding the virtual aperture to two dimensions—along and perpendicular to the velocity direction—allows the clutter rank to be considered as the sum of the aperture-bandwidth products in these two directions. In bistatic radar systems, this approach has been explored in scenarios where both aircraft maintain the same altitude and horizontal flight, allowing the clutter rank estimation to mirror the two-dimensional aperture-bandwidth product problem of monostatic radar [32].

Another method for clutter rank estimation considers a limited time-frequency region characteristic in the clutter signal. The clutter signal can be approximated using a complete set of orthogonal function bases [34,35], among which the prolate spheroidal wave functions (PSWF) [36,37,38] are completely orthogonal within this region. The limited time-frequency region signal can be approximated by a linear combination of PSWF orthogonal bases with significant eigenvalues. To fulfil the accuracy requirements, the number of selected PSWF orthogonal bases corresponds to the time-bandwidth product, which is the clutter rank. In bistatic radar, the presence of transmitting Doppler frequencies introduces a non-linear relationship between the Doppler and spatial frequencies, resulting in the following expression for the Doppler frequency: $f_{di}=\gamma f_{si}+f_{di,T}$. The space–time coupling of bistatic radar results in the inability to express the space–time steering vector as a single-frequency signal, leading to the signal’s frequency values varying with time values and rendering it difficult to estimate the frequency bandwidth. Furthermore, the spatial variability of bistatic range resolution may lead to the aggregation of diverse ranges within the same range bin in certain areas, thereby resulting in a further extended frequency region. Currently, no research has been conducted on the estimation of clutter rank based on PSWF for bistatic radar.

Therefore, clutter rank estimation methods suitable for arbitrary bistatic radar configurations are lacking. Given these challenges, this study explores the direct correlation between bistatic radar’s space-time coupling and the inconsistency of the frequency region relative to the clutter signal’s PSWF representation. The frequency values’ distribution regularity over time is analysed, and a method is proposed to determine the equivalent frequency bandwidth of bistatic radar, enabling accurate clutter rank estimation. This study establishes a relationship among clutter rank, clutter Doppler bandwidth, and azimuth resolution, elucidating the variation rule of clutter rank. The main contributions of this paper are as follows:

(1)

Addressing the gap in existing research through proposing a bistatic clutter rank estimation method based on PSWF, providing crucial design indicators for reduced dimension and rank STAP in bistatic radar.

(2)

The proposed method enables the estimation of clutter rank for either the entire range bin or limited observation areas, whether in side-looking or non-side-looking mode, rapidly evaluating clutter rank for diverse relationships between the relative velocity and antenna of two platforms within different observation areas, thus supporting bistatic configuration design and observation area optimization.

(3)

Establishing the corresponding relationship between clutter rank, clutter Doppler bandwidth, and azimuth resolution, and elucidating the variation rule of clutter rank. This finding serves as a valuable theoretical foundation for optimizing the configuration of bistatic radar.

This paper is organized as follows: Section 2 introduces the geometry structure and signal model of bistatic radar. Section 3 discusses the challenges of employing PSWF-based techniques for estimating clutter rank in bistatic radar and introduces the proposed solution, along with a comprehensive analysis exploring the method’s extended application potential. Section 4 presents the simulation results verifying the method’s effectiveness. Finally, the conclusions are drawn in Section 5.

2. Signal Model

In early warning detection, the coherent processing interval (CPI) is typically brief, ranging from tens to hundreds of milliseconds. During this period, the relative positions and velocities of the transmitting and receiving platforms in a bistatic radar system can be considered stable. Consequently, the construction of the space-time steering vector is based on the receiving platform. To represent the positions and velocities of both the transmitter and receiver within a unified coordinate system, the receiving platform velocity coordinate system (RVCS) is established. As shown in Figure 1, $T$ and $R$ are the position of the transmitting and receiving platform. The RVCS considers the receiving platform as the origin. The velocity direction of the receiving platform defines the $x_{vR}$ axis, the direction pointing from the receiving platform toward the centre of the Earth establishes the $y_{vR}$ axis, and the $z_{vR}$ axis is determined by the right-hand spiral rule.

The distance between the two platforms is $R_{rt}$. $T_{\perp }$ is the projection point of $T$ in the $x_{vR}o{z}_{vR}$ plane. $x_{vR//}o{z}_{vR//}$ is the plane in parallel to plane $x_{vR}o{z}_{vR}$ over point $T$. The aircraft in airborne bistatic radar typically maintains a level flight at a low altitude, where the negligible curvature of the earth and its velocity confined to the $x_{vR//}o{z}_{vR//}$ plane are considered. Conversely, in spaceborne bistatic radar, the transmitting satellite moves along its orbit with deviations from the $x_{vR//}o{z}_{vR//}$ plane.

To accommodate both airborne and spaceborne bistatic radar systems within the same signal model, ${\delta }_{t\beta }$ and ${\delta }_{t\alpha 0}$ are used to denote the directions of transmitting velocity $v_T$. Specifically, ${\delta }_{t\beta }$ is the angle between vector $v_T$ and plane $x_{vR//}o{z}_{vR//}$, and ${\delta }_{t\alpha 0}$ is the angle between vector $v_{T\perp }$ (the projection vector of $v_T$ in plane $x_{vR//}o{z}_{vR//}$) and the positive $x_{vR//}$ axis. These angles are measured in a positive counterclockwise direction.

Using the specified angles and the distance between the two platforms, the velocity of the receiving platform and the position and velocity of the transmitting platform can be uniquely determined in RVCS as follows:

$$v_R={\left[‖v_R‖,0,0\right]}^T,$$

(1)

$$p_T={\left[R_{rt}\cos {\beta }_{rt}\cos {\delta }_{r\alpha },-R_{rt}\sin {\beta }_{rt},R_{rt}\cos {\beta }_{rt}\sin {\delta }_{r\alpha }\right]}^T,$$

(2)

$$v_T={\left[‖v_T‖\cos {\delta }_{t\beta }\cos {\delta }_{t\alpha 0},-‖v_T‖\sin {\delta }_{t\beta },‖v_T‖\cos {\delta }_{t\beta }\sin {\delta }_{t\alpha 0}\right]}^T.$$

(3)

In an airborne bistatic radar system with specific configurations, i.e., two aircraft flying at the same altitude and maintaining steady horizontal flight, the transmitting aircraft’s velocity vector can be considered to exhibit ${\delta }_{t\beta }=0$.

In spaceborne bistatic radar systems, the rotation of the Earth is a critical factor that must be considered. Assuming the velocity vectors of two satellites, $v_{T0}$ and $v_{R0}$ can be derived from Kepler’s equations based on the orbital parameters of the satellites. To simplify analysis and calculations, the Earth can be considered stationary, and the effects of Earth’s rotation are transferred to the satellite movements [39]. This approach allows the additional velocity vector caused by Earth’s rotation to be expressed as the cross product of the Earth’s rotation angular velocity ${\omega }_e$ and the position vector of the satellite. Consequently, under the assumption that the Earth is stationary, the equivalent velocity vectors of the transmitting and receiving satellites can be expressed as follows [39]:

$$v_T=v_{T0}-{\omega }_e\times \overrightarrow{O_{e}T},$$

(4)

$$v_R=v_{R0}-{\omega }_e\times \overrightarrow{O_{e}R},$$

(5)

where ${\omega }_e$ represents the earth’s rotation angle velocity vector. The geometric depictions of $\overrightarrow{O_{e}T}$ and $\overrightarrow{O_{e}R}$ are shown in Figure 2. $\overrightarrow{O_{e}T}$ and $\overrightarrow{O_{e}R}$ represent the vectors directed from the centre of the earth to the transmitting and receiving satellites, respectively.

According to the finite element division theory [1], the ground can be segmented into a grid-like structure. Each grid element is treated as a discrete scattering unit, and the scene clutter echo data results from the cumulative superposition of the echo signals from all the scattering units within it. During the operation of the radar system, the transmitting platform emits a series of pulses at a specified pulse repetition frequency (PRF) throughout the CPI ($K$). Each scattering unit reflects the transmitted signal, which is then captured by the receiving platform.

The Doppler frequency, a crucial parameter in radar signal analysis, is defined as the ratio of the rate of change of the distance between the transmitting and receiving platforms to the signal wavelength $\lambda$. The formula for calculating the Doppler frequency is

$$f_{d0,i}={\displaystyle \frac{1}{\lambda }}\left({\displaystyle \frac{d{R}_{T,i}\left(t\right)}{dt}}+{\displaystyle \frac{d{R}_{R,i}\left(t\right)}{dt}}\right),$$

(6)

where $R_{T,i}\left(t\right)$ and $R_{R,i}\left(t\right)$ represent the slant range of the transmitting and receiving platform versus the $i$-th scattering unit, respectively. Doppler frequency can be approximately expressed as follows [16]:

$$f_{d0,i}={\displaystyle \frac{R_R^T{v}_R}{\lambda ‖R_R‖}}+{\displaystyle \frac{R_T^T{v}_T}{\lambda ‖R_T‖}}={\displaystyle \frac{‖v_T‖\cos {\psi }_{vT}+‖v_R‖\cos {\psi }_{vR}}{\lambda }},$$

(7)

where ${\psi }_{vT}$ and ${\psi }_{vR}$ represent the spatial cone angle of the transmitting and receiving platform velocity versus the $i$-th scattering unit, respectively. Because the spatial cone angle can be decomposed into the product of the cosine value of the azimuth angle and the cosine value of the downwards angle, the normalized Doppler frequency can be further derived as follows:

$$f_{d,i}=f_{dT,i}+f_{dR,i},$$

(8)

$$f_{dT,i}={\displaystyle \frac{‖v_T‖\cos {\alpha }_{vT}\sin {\beta }_T}{\lambda \cdot PRF}},$$

(9)

$$f_{dR,i}={\displaystyle \frac{‖v_R‖\cos {\alpha }_{vR}\sin {\beta }_R}{\lambda \cdot PRF}}$$

(10)

where $f_{dT,i}$ and $f_{dR,i}$ represent the normalized Doppler frequencies generated by the transmitting and receiving platform motion, respectively; ${\alpha }_{vT}$ and ${\alpha }_{vR}$ represent the azimuth angle of the transmitting and receiving platform velocity versus the $i$-th scattering unit, respectively; and ${\beta }_T$ and ${\beta }_R$ represent the downward angle of the transmitting and receiving platform versus the $i$-th scattering unit, respectively.

For the receiving uniform planar array, the columnar elements are synthesized in a column-oriented manner to form $N$ channels with channel spacing $d$. The spatial frequency of the $i$-th scattering unit is as follows [16]:

$$f_{s,i}={\displaystyle \frac{d}{\lambda }}\left({\displaystyle \frac{R_R^T{a}_{R0}}{‖R_R‖}}\right)={\displaystyle \frac{d}{\lambda }}\cos {\alpha }_{aR}\sin {\beta }_R,$$

(11)

where $a_{R0}$ is the unit vector of the receiving antenna orientation, and ${\alpha }_{aR}$ represents the azimuth angle of the receiving array orientation versus the $i$-th scattering unit.

The echo data is subsequently sampled across $N$ channels. After compensating for the quadratic and higher order introduced by the platform movement, the echo signal of the $l$-th range bin can be expressed as follows [1]:

$$x_l={\displaystyle \sum _{i=1}^{N_c}{a}_i{s}_i}={\displaystyle \sum _{i=1}^{N_c}{a}_i\left[e^{j2\pi f_{d,i}{\left[0:K-1\right]}^T}\otimes e^{j2\pi f_{s,i}{\left[0:N-1\right]}^T}\right]}$$

(12)

where $\otimes$ represents the Kronecker product; $N_c$ represents the number of scattering units on the $l$-th range bin; $a_i$ represents the clutter amplitude of the $i$-th scattering unit; and $s_i$ represents the space–time steering vector. The clutter covariance matrix can be expressed as follows [1]:

$$R_c={\displaystyle \sum _{i=1}^{N_c}{a}_i^2{s}_i{s}_i^H}.$$

(13)

The space–time steering vectors of different scattering units can form the following matrix:

$$S=\left[s_1,s_2,\cdots ,s_{N_c}\right].$$

(14)

where $S$ is an $NK\times N_c$ matrix, and the i-th column of the matrix is the space-time steering vector $s_i$.

In practice, the main lobe of the beam covers a limited area, and the received clutter signal is heavy but not reflective of the entire range bin. This is due to the modulation of the receiving and transmitting patterns, where the gain from the far side lobe is exceedingly low. The significant clutter primarily originates from the clutter signals of scattering units in the main and near side lobe areas. Suppose that $C_{BW,1}$ and $C_{BW,N_c}$ are the starting and ending scattering unit indexes in the main and near side lobe areas, respectively. The clutter covariance matrix and the space–time steering matrix, which account for the pattern modulation, can be expressed as follows:

$$R_{c,BW}={\displaystyle \sum _{i=C_{BW,1}}^{C_{BW,N_c}}{\alpha }_i^2{s}_i{s}_i^H}.$$

(15)

$$S_{BW}=\left[s_{C_{BW,1}},\cdots ,s_{C_{BW,N_c}}\right].$$

(16)

The space–time steering matrix and the clutter covariance matrix exhibits the following relationship [30]:

$$span\left(R_c\right)=span\left(S\right),$$

(17)

$$span\left(R_{c,BW}\right)=span\left(S_{BW}\right),$$

(18)

where $span\left(\cdot \right)$ represents the expanded space. Therefore, the ranks of $R_c$ and $S$ are equivalent, as are the ranks of $R_{c,BW}$ and $S_{BW}$.

3. Proposed Method

3.1. Challenge of Clutter Rank Estimation in Bistatic Radar

Bistatic radar configuration is influenced by the velocity and antenna vectors of two platforms. The Doppler frequency is determined by the velocity vectors of both the transmitting and receiving platforms, whereas the spatial frequency is determined solely by the receiving array and remains unaffected by the transmitting array. In this section, we consider arbitrary velocity vectors of the two platforms. For ease of representation, it is assumed that the receiving platform operates in side-looking mode, where the orientation of the receiving array aligns with the velocity direction (${\alpha }_{vR}={\alpha }_{aR}$). Consequently, the space-time steering vector of the $i$-th scattering unit on the range bin can be rewritten derived as follows:

$$\begin{array}{ll}{s}_i& =s_i^t\otimes s_i^s\hfill \\ & ={\left[\begin{array}{c}1\\ e^{j2\pi \left(f_{dT,i}+f_{dR,i}\right)}\\ ⋮\\ e^{j2\pi \left(K-1\right)\left(f_{dT,i}+f_{dR,i}\right)}\end{array}\right]}_{K\times 1}\otimes {\left[\begin{array}{c}1\\ e^{j2\pi f_{s,i}}\\ ⋮\\ e^{j2\pi f_{s,i}\left(N-1\right)}\end{array}\right]}_{N\times 1}\hfill \\ & ={\left[\begin{array}{c}1\\ e^{j2\pi f_{dT,i}}\\ ⋮\\ e^{j2\pi \left(K-1\right)f_{dT,i}}\end{array}\right]}_{K\times 1}\odot {\left[\begin{array}{c}1\\ e^{j2\pi \gamma f_{s,i}}\\ ⋮\\ e^{j2\pi \left(K-1\right)\gamma f_{s,i}}\end{array}\right]}_{K\times 1}\otimes {\left[\begin{array}{c}1\\ e^{j2\pi f_{s,i}}\\ ⋮\\ e^{j2\pi f_{s,i}\left(N-1\right)}\end{array}\right]}_{N\times 1}\hfill \end{array},$$

(19)

where $\odot$ represents the Hadamard product,

$$\gamma ={\displaystyle \frac{‖v_R‖}{d\cdot PRF}}.$$

(20)

The above equation can be further rewritten as follows:

$$\begin{array}{ll}{s}_i& =diag\left(\left[\begin{array}{c}1\\ e^{j2\pi f_{dT,i}}\\ ⋮\\ e^{j2\pi f_{dT,i}\left(K-1\right)}\end{array}\right]\otimes 1_N\right)\left({\left[\begin{array}{c}1\\ e^{j2\pi \gamma f_{s,i}}\\ ⋮\\ e^{j2\pi \left(K-1\right)\gamma f_{s,i}}\end{array}\right]}_{K\times 1}\otimes {\left[\begin{array}{c}1\\ e^{j2\pi f_{s,i}}\\ ⋮\\ e^{j2\pi f_{s,i}\left(N-1\right)}\end{array}\right]}_{N\times 1}\right)\hfill \\ & =\Gamma v_s\hfill \end{array},$$

(21)

where $diag\left(a\right)$ is the diagonal matrix whose $t$-th diagonal entry is the $t$-th entry of the vector $a$. Therefore, $\Gamma$ is a diagonal matrix that is solely determined by $f_{dT,i}$. The exponents of the elements in vector $v_s$ are arranged in ascending order, from smallest to largest, while eliminating any duplicates with identical values. This process yields the vector $z\in {ℂ}^{M\times 1}$ (M ≤ N) as follows:

$$z={\left[e^{j2\pi f_{s,i}{z}_1},e^{j2\pi f_{s,i}{z}_2},\cdots ,e^{j2\pi f_{s,i}{z}_M}\right]}^T,$$

(22)

where $z\in {ℂ}^{M\times 1}$ is the column vector, containing all values within $v_s$ and unrepeatable, and satisfies $z_{m+1}>z_m$. Therefore, the expression of $v_s$ can be transformed into the following form:

$$\begin{array}{ll}{v}_s& =Ez\hfill \\ & ={\left[\begin{array}{ccccc}1& 0& \cdots & & 0\\ 0& \cdots & 1& \cdots & 0\\ & & & \ddots & \\ 0& \cdots & & 0& 1\end{array}\right]}_{NK\times M}{\left[\begin{array}{c}{e}^{j2\pi f_{s,i}{z}_1}\\ e^{j2\pi f_{s,i}{z}_2}\\ ⋮\\ e^{j2\pi f_{s,i}{z}_M}\end{array}\right]}_{M\times 1}\hfill \end{array},$$

(23)

where $E\in {\mathbb{Z}}^{NK\times M}$ is the selection matrix, consisting of only 1 and 0. Equation (23) can be substituted into Equation (21), resulting in the following expression:

$$s_i\hspace{0.33em}=\Gamma Ez.$$

(24)

In monostatic radar, the transmitting Doppler frequency is equal to the receiving Doppler frequency. Equation (19) can be rewritten derived as follows:

$$S_i=\Gamma E{z}_{mo}=I_{NK}E\left[\begin{array}{c}{e}^{j2\pi f_{s,i}{z}_{mo,1}}\\ ⋮\\ e^{j2\pi f_{s,i}{z}_{mo,M}}\end{array}\right],$$

(25)

where $z_{mo}\in {ℂ}^{M\times 1}$ contains all values within ${\left[1,\cdots ,e^{j2\pi {\gamma }_{mo}{f}_{s,i}\left(K-1\right)}\right]}^T\otimes {\left[1,\cdots ,e^{j2\pi f_{s,i}\left(N-1\right)}\right]}^T$ and unrepeatable, ${\gamma }_{mo}=2\gamma$, and satisfies $z_{mo,m+1}>z_{mo,m}$. Each row’s phase can be expressed as follows:

$$s_i\left(n+kN+1\right)=f_{s,i}\cdot \left({\gamma }_{mo}k+n\right),$$

(26)

where $k\in \left[0:K-1\right]$, $n\in \left[0:N-1\right]$, and $s_i\left(n+kN+1\right)$ is the $\left(n+kN+1\right)$-th row’s phase of $s_i$. Therefore, the space–time steering vector $s_i$ can be regarded as a single-frequency signal, which is also time–frequency-limited derived as follows:

$$s_i\left(t,f_{s,i}\right)=\left\{\begin{array}{l}{e}^{j2\pi f_{s,i}t}\hspace{0.33em}\hspace{0.33em}0\le t\le T_m\\ 0\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}else\end{array}\right.,$$

(27)

when ${\gamma }_{mo}\left(K-1\right)<1$, the variation of $z_{mo,m}$ with $m$ is not continuous; therefore, $T_m$ should be calculated in segments. Conversely, when ${\gamma }_{mo}\left(K-1\right)\ge 1$, the variation of $z_{mo,m}$ with $m$ is continuous and $T_m$ can be determined by $z_{mo,M}$. The derived $T_m$ is summarized as follows:

$$T_m=\left\{\begin{array}{l}{\gamma }_{mo}\left(K-1\right)+N-1\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}{\gamma }_{mo}\left(K-1\right)\ge 1\\ {\gamma }_{mo}\left(K-1\right)N\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}{\gamma }_{mo}\left(K-1\right)<1\end{array}\right..$$

(28)

$f_{s,i}$ is limited within the scope $\min \left(f_{s,1},\cdots ,f_{s,N_c}\right)\le f_{s,i}\le \max \left(f_{s,1},\cdots ,f_{s,N_c}\right)$. The energy of this signal is predominantly confined to a specific time–frequency region. According to [36,37,38], such a signal can be represented by linear combinations of PSWF orthogonal bases. The number of PSWF orthogonal bases utilized for approximating the signal is referred to as the clutter rank, which can be expressed as follows:

$$N_r=⌈W_{fs}{T}_m+1⌉,$$

(29)

$$W_{fs}=\underset{i\in \left[1,N_c\right]}{\max }\left(f_{s,i}\right)-\underset{\in \left[1,N_c\right]}{\min }\left(f_{s,i}\right),$$

(30)

where $⌈x⌉$ represents the minimum integer no less than $x$, and $W_{fs}$ is the frequency bandwidth.

In bistatic radar, the transmitting Doppler frequency generally does not equal the receiving Doppler frequency. $\Gamma$ is a non-identity diagonal matrix, which signifies that the transmitting Doppler frequency induces varying degrees of modulation across different pulses and channels of the space–time steering vector. This is equivalent to increasing the broadening of $f_{dT,i}$ on the element of $Ez$, thereby causing changes in the frequency region. The phase of $s_i$ can be derived as follows:

$$\begin{array}{ll}{s}_i\left(n+kN+1\right)& =f_{s,i}\cdot \left(\gamma k+n\right)+f_{dT,i}\cdot k\hfill \\ & =\left\{\begin{array}{l}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}{f}_{s,i}n\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}k=0\\ \left(f_{s,i}+f_{dT,i}{\displaystyle \frac{k}{\gamma k+n}}\right)\cdot \left(\gamma k+n\right)\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}k>0\end{array}\right.\\ & =\left(f_{s,i}+\Delta f_i\left(n,k\right)\right)\cdot \left(\gamma k+n\right)\hfill \end{array}.$$

(31)

In this manner, the space–time coupling of bistatic radar results in the inability to express $s_i$ in the form of a single-frequency signal. The expression of $s_i$ is derived as follows:

$$s_i\left(t,f_{s,i},\Delta f_i\left(t\right)\right)=\left\{\begin{array}{l}{e}^{j2\pi \left(f_{s,i}+\Delta f_i\left(t\right)\right)t}\hspace{0.33em}\hspace{0.33em}0\le t\le T_m\\ 0\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}else\end{array}\right.,$$

(32)

where $f_{s,i}+\Delta f_i\left(t\right)$ is defined as the equivalent frequency value that exhibits temporal variations. The distribution of $f_{s,i}+\Delta f_i\left(t\right)$ is depicted in Figure 3. Due to the time-limited characteristic and the distribution of scattering units within the range bin, $s_i$ can still be considered time–frequency limited. Consequently, the key factor in determining the number of PSWF orthogonal bases required for signal approximation is effectively acquiring the equivalent frequency bandwidth in a bistatic configuration. The expression for the clutter rank in bistatic radar is as follows:

$$N_r=⌈W_{eq}{T}_m+1⌉,$$

(33)

where $T_m$ remains unchanged according to Equation (28), except for ${\gamma }_{mo}$ instead of $\gamma$. $W_{eq}$ represents the equivalent frequency bandwidth in bistatic configuration, which will be extensively discussed in the subsequent section.

3.2. Calculation of W_eq

According to the expression of $\Delta f_i\left(n,k\right)$ in Equation (31), within the scope of $T_m$, the relationships described below hold true owing to $\gamma >0$ and $n,k\ge 0$:

$$0\le {\displaystyle \frac{k}{\gamma k+n}}\le {\displaystyle \frac{k}{\gamma k}}={\displaystyle \frac{1}{\gamma }}.$$

(34)

Therefore, the extreme value of $\Delta f_i$ is derived as follows:

$$0\le \Delta f_i\le \Delta f_{i,\max }={\displaystyle \frac{f_{dT,i}}{\gamma }}.$$

(35)

The frequency values $f_{si}+\Delta f_i$ are distributed within the range from $f_{si}$ to $f_{si}+\Delta f_{i,\max }$. According to Equation (31), the frequency values are specifically distributed in set ${\Omega }_i$, derived as follows:

$${\Omega }_i={\left[{\Omega }_{i,0},{\Omega }_{i,1},\cdots ,{\Omega }_{i,K-1}\right]}_{1\times NK},$$

(36)

where ${\Omega }_{i,k}\in {ℂ}^{1\times N}$. When $k=0$, based on Equation (31), the frequency values consistently correspond to $f_{s,i}$. Hence, ${\Omega }_{i,0}={\left[f_{s,i},\cdots ,f_{s,i}\right]}_{1\times N}$. When $k>0$, according to Equation (31), the frequency values $f_{s,i}+f_{dT,i}{\displaystyle \frac{k}{\gamma k+n}}$ exhibit variations dependent on both $k$ and n. Hence, ${\Omega }_{i,k}={\left[f_{s,i}+{\displaystyle \frac{f_{dT,i}}{\gamma }},f_{s,i}+{\displaystyle \frac{k{f}_{dT,i}}{1+k\gamma }},\cdots ,f_{s,i}+{\displaystyle \frac{k{f}_{dT,i}}{N-1+k\gamma }}\right]}_{1\times N}$. According to the aforementioned two cases, ${\Omega }_{i,k}$ can be expressed by the following equation:

$${\Omega }_{i,k}=\left\{\begin{array}{l}{\left[f_{s,i},f_{s,i},\cdots ,f_{s,i}\right]}_{1\times N}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}k=0\\ {\left[f_{s,i}+{\displaystyle \frac{f_{dT,i}}{\gamma }},f_{s,i}+{\displaystyle \frac{k{f}_{dT,i}}{1+k\gamma }},\cdots ,f_{s,i}+{\displaystyle \frac{k{f}_{dT,i}}{N-1+k\gamma }}\right]}_{1\times N}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}k>0\end{array}\right..$$

(37)

Figure 3 presents a schematic of the frequency values against the scattering units in a typical bistatic radar configuration. As previously mentioned, the frequency values in bistatic radar are not concentrated exclusively at $f_{si}$; instead, they are distributed within a range from $f_{si}$ to $f_{si}+\Delta f_{i,\max }$. $\Delta f_i$ is defined as the equivalent frequency broadening (EFB), which represents the discrepancy between the frequency value and its original reference frequency. The schematic of the EFB is further annotated in Figure 3.

3.2.1. Calculation Method A

$W_{eq}$ can be calculated through a direct approach, which involves determining the range of $f_{si}+\Delta f_{i,\max }$ or $f_{si}$ within the range bin by subtracting the minimum value from the maximum value. This represents the equivalent frequency range occupied by the red line or green line shown in Figure 3. $W_{eq}$ can be derived as follows:

$$W_{eq}=2\max \left\{R\left(f_{s,i}+\Delta f_{i,\max }\right),R\left(f_{s,i}\right)\right\},$$

(38)

where

$$R\left(f_{si}+\Delta f_{i,\max }\right)=\underset{i\in \left[1,N_c\right]}{\max }\left(f_{si}+\Delta f_{i,\max }\right)-\underset{i\in \left[1,N_c\right]}{\min }\left(f_{si}+\Delta f_{i,\max }\right).$$

(39)

$$R\left(f_{si}\right)=\underset{i\in \left[1,N_c\right]}{\max }\left(f_{si}\right)-\underset{i\in \left[1,N_c\right]}{\min }\left(f_{si}\right).$$

(40)

In reality, the frequency values of each scattering unit do not correspond to a single-frequency component. Each frequency value does not fully occupy the entire equivalent time $T_m$. This calculation method, which only considers the maximum and minimum frequency values within the range bin, can result in inaccuracies in estimating $W_{eq}$.

3.2.2. Calculation Method B

Clearly, a precise calculation of $W_{eq}$ requires considering the equivalent extended frequency distribution of each individual scattering unit. By integrating the equivalent extended frequency distribution, the expression of $W_{eq}$ can be derived when the range of extended frequency values surpasses that of $f_{s,i}$:

$$\begin{array}{ll}\hfill W_{eq}& ={\displaystyle {\int }_{0^{°}}^{{360}^{°}}\left(\frac{1}{T_m}{\displaystyle {\int }_0^{T_m}\left|f_{si}\right|+\left|\Delta f_i\left(t\right)\right|\hspace{0.33em}dt}\right)}\hspace{0.33em}d{\alpha }_{vT}\hfill \\ & ={\displaystyle {\int }_{0^{°}}^{{360}^{°}}\left|f_{si}\right|\hspace{0.33em}d{\alpha }_{vT}}+{\displaystyle {\int }_{0^{°}}^{{360}^{°}}\left(\frac{1}{T_m}{\displaystyle {\int }_0^{T_m}\left|\Delta f_i\left(t\right)\right|dt}\right)\hspace{0.33em}d{\alpha }_{vT}}\hfill \\ & =W_{fs}+\Delta W\hfill \end{array},$$

(41)

For the range of the extended values smaller than that of $f_{s,i}$, $W_{eq}$ should be represented as $W_{fs}-\Delta W$. Therefore, the expression of $W_{eq}$ is summarized as follows:

$$W_{eq}=\left\{\begin{array}{l}{W}_{fs}+\Delta W\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}R\left(f_{s,i}+\Delta f_{i,\max }\right)\ge R\left(f_{si}\right)\\ W_{fs}-\Delta W\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}R\left(f_{s,i}+\Delta f_{i,\max }\right)<R\left(f_{si}\right)\end{array}\right.,$$

(42)

where $W_{fs}$ is the frequency bandwidth of the single-frequency signal, which represents the frequency bandwidth of monostatic radar when ${\delta }_{t\alpha 0}\in \left\{0^{°},{180}^{°}\right\}$ and ${\delta }_{r\alpha }\in \left\{0^{°},{180}^{°}\right\}$. That means the symmetric scattering units are completely equivalent on the clutter rank. In other cases, $W_{fs}$ is twice the frequency bandwidth of monostatic radar.

$\Delta W$ is the variation in bandwidth attributed to the influence of $\Delta f_i$, which is denoted as follows:

$$\Delta W={\displaystyle {\int }_{0^{°}}^{{360}^{°}}\left(\frac{1}{T_m}{\displaystyle {\int }_0^{T_m}\left|\Delta f_i\left(t\right)\right|dt}\right)}\hspace{0.33em}d{\alpha }_{vT}={\displaystyle {\int }_{0^{°}}^{{360}^{°}}{F}_i}\hspace{0.33em}d{\alpha }_{vT}.$$

(43)

Firstly, the EFB integral average of the $i$-th scattering unit $F_i$ needs to be computed. The EFB values of the $i$-th scattering unit are distributed in the following set:

$${\Theta }_i=\left[{\displaystyle \frac{\left|f_{dT,i}\right|}{\gamma }},{\displaystyle \frac{\left|f_{dT,i}\right|}{1+\gamma }},\cdots ,{\displaystyle \frac{k\left|f_{dT,i}\right|}{n+k\gamma }},\cdots ,{\displaystyle \frac{\left(K-1\right)\left|f_{dT,i}\right|}{N-1+\left(K-1\right)\gamma }}\right].$$

(44)

Therefore, $F_i$ can be derived as follows:

$$\begin{array}{ll}{F}_i& ={\displaystyle \frac{1}{NK}}{\displaystyle \sum _{t=1}^{NK}{\Theta }_i\left(t\right)}\hfill \\ & ={\displaystyle \frac{{\displaystyle \sum _{k=1}^{K-1}{\displaystyle \sum _{n=0}^{N-1}\frac{k}{n+k\gamma }}}}{NK}}\cdot \left|f_{dT,i}\right|\hfill \\ & =\rho \cdot \left|f_{dT,i}\right|\hfill \end{array},$$

(45)

where under constant system parameters, $F_i$ is exclusively determined by the transmitting Doppler frequency $f_{dT,i}$.

After deriving the calculation formula for $F_i$ using Equation (45), the integral average of $F_i$ across the entire range bin is subsequently computed. Substituting Equation (43) into Equation (46) yields the following derived equation:

$$\begin{array}{ll}\Delta W& ={\displaystyle {\int }_{0^{°}}^{{360}^{°}}\rho \cdot \left|f_{dT,i}\right|\hspace{0.33em}}d{\alpha }_{vT}\hfill \\ & ={\displaystyle \frac{\rho \cdot ‖v_T‖}{\lambda \cdot PRF}}\cdot {\displaystyle {\int }_{0^{°}}^{{360}^{°}}\left|\cos {\psi }_{vT,i}\right|}\hspace{0.33em}d{\alpha }_{vT}\hfill \\ & ={\displaystyle \frac{\rho \cdot ‖v_T‖}{\lambda \cdot PRF}}\cdot {\displaystyle {\int }_{\underset{i\in \left[1,N_c\right]}{\min }\left({\psi }_{vT,i}\right)}^{\underset{i\in \left[1,N_c\right]}{\max }\left({\psi }_{vT,i}\right)}\left|\cos {\psi }_{vT,i}\right|}\hspace{0.33em}d{\psi }_{vT}\hfill \end{array}.$$

(46)

As Equation (46) represents an integral of the absolute value of $\cos {\psi }_{vT,i}$, the values for different scattering units may coincide. The detailed calculation of Equation (46) is presented in Appendix A, wherein the simplified expression for $\Delta W$ is derived as follows:

$$\Delta W=\left\{\begin{array}{l}{\displaystyle \frac{\rho \cdot ‖v_T‖}{\lambda \cdot PRF}}\cdot \left[\begin{array}{l}\sin \left(\underset{i\in {\mathbb{F}}_m}{\max }\left({\psi }_{vT,i}\right)\right)\\ -\sin \left(\underset{i\in {\mathbb{F}}_m}{\min }\left({\psi }_{vT,i}\right)\right)\end{array}\right]\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\begin{array}{c}\left({\delta }_{t\alpha 0}\in \left\{0^{°}\hspace{0.33em},{180}^{°}\right\}\right)\\ \wedge \left({\delta }_{r\alpha }\in \left\{0^{°}\hspace{0.33em},{180}^{°}\right\}\right)\end{array}\\ {\displaystyle \frac{\rho \cdot ‖v_T‖}{\lambda \cdot PRF}}\cdot {\displaystyle \sum _{m=1}^M\left[\begin{array}{l}\sin \left(\underset{i\in {\mathbb{F}}_m}{\max }\left({\psi }_{vT,i}\right)\right)\\ -\sin \left(\underset{i\in {\mathbb{F}}_m}{\min }\left({\psi }_{vT,i}\right)\right)\end{array}\right]}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}else\end{array}\right..$$

(47)

where $M$ is the count of monotone intervals as $\left|\cos {\psi }_{vT,i}\right|$ versus scattering units, and ${\mathbb{F}}_m\in \left[1,N_c\right]$ is the $m$-th monotone interval.

Once the expression of $\Delta W$ is derived from Equation (47), $W_{eq}$ can be obtained using Equation (41). Finally, the estimation of bistatic radar clutter rank can be obtained using the following equation:

$$N_r=\left\{\begin{array}{l}⌈W_{eq}\cdot \left[\gamma \left(K-1\right)+N-1\right]+1⌉\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\gamma \left(K-1\right)\ge 1\\ ⌈W_{eq}\cdot \left[\gamma \left(K-1\right)N\right]+1⌉\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\gamma \left(K-1\right)<1\end{array}\right..$$

(48)

Thus, the estimation of clutter rank under different bistatic configurations can be obtained based on the distribution of $f_{dT,i}$ and $f_{s,i}$. In summary, the flowchart of the clutter rank estimation method for bistatic radar is shown in Algorithm 1.

Algorithm 1. Proposed clutter rank estimation method

Input: $K$, $N$, $d$, $‖v_R‖$, PRF, ${\delta }_{t\alpha 0}$, ${\delta }_{r\alpha }$, $\left[f_{dT,i},\cdots ,f_{dT,N_c}\right]$, $\left[f_{s,1},\cdots ,f_{s,N_c}\right]$

Procedure:  (1)  Obtain $\gamma =‖v_R‖/\left(d\cdot PRF\right)$  (2)  For $i$ = 1 to $N_c$ do              Obtain $F_i=f_{dT,i}\cdot {\displaystyle \sum _{k=1}^{K-1}{\displaystyle \sum _{n=0}^{N-1}\frac{k}{n+k\gamma }}}/\left(NK\right)$        end for  (3)  Obtain $\left[F_1,\cdots ,F_{N_c}\right]$ after procedure (2)  (4)   Obtain $W_{fs}$  (5)  Obtain $\Delta W$ by Equation (47)  (6)  Obtain $W_{eq}$ by Equation (42)

Output: The estimation rank $N_r$ by Equation (48)

3.3. Extended Applicability Analysis of the Method

A.

Non-side-looking mode of receiving platform

When the receiving platform operates in non-side-looking mode, meaning the orientation of the receiving array does not align with the direction of velocity, the angle between the receiving array and the velocity direction is denoted as the yaw angle ${\Delta }_{aR}$.

The relation ${\alpha }_{vR}={\alpha }_{aR}+{\Delta }_{aR}$ implies that the normalized receiving Doppler frequency can be rewritten as

$$\begin{array}{ll}{f}_{dR,i}& ={\displaystyle \frac{‖v_R‖\cos \left({\alpha }_{aR}+{\Delta }_{aR}\right)\sin {\beta }_R}{\lambda \cdot PRF}}\\ & ={\displaystyle \frac{‖v_R‖\left(\cos {\alpha }_{aR}\cos {\Delta }_{aR}-\sin {\alpha }_{aR}\sin {\Delta }_{aR}\right)\sin {\beta }_R}{\lambda \cdot PRF}}\hfill \\ & ={\gamma }^{\Delta }{f}_{s,i}-f_{dR\Delta .i}\hfill \end{array},$$

(49)

where

$${\gamma }^{\Delta }={\displaystyle \frac{‖v_R‖\cdot \cos {\Delta }_{aR}}{d\cdot PRF}},$$

(50)

$$f_{dR\Delta ,i}={\displaystyle \frac{‖v_R‖\sin {\alpha }_{aR}\sin {\beta }_R\sin {\Delta }_{aR}}{\lambda \cdot PRF}}.$$

(51)

The presence of the yaw angle introduces nonlinearity in the relationship between the receiving Doppler frequency and spatial frequency. The receiving Doppler frequency can be expressed as the sum of a linear term related to the spatial frequency and a nonlinear term induced by the yaw angle. Each row’s phase of $s_i$ can be derived as follows:

$$\begin{array}{ll}{s}_i\left(n+kN+1\right)& =f_{s,i}\cdot \left({\gamma }^{\Delta }k+n\right)+\left(f_{dT,i}-f_{dR\Delta ,i}\right)\cdot k\hfill \\ & =\left\{\begin{array}{l}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}{f}_{s,i}n\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}k=0\\ \left[f_{s,i}+\left(f_{dT,i}-f_{dR\Delta ,i}\right){\displaystyle \frac{k}{{\gamma }^{\Delta }k+n}}\right]\cdot \left({\gamma }^{\Delta }k+n\right)\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}k\ne 0\end{array}\right.\hfill \\ & =\left(f_{s,i}+\Delta f_i^{\Delta }\left(n,k\right)\right)\cdot \left({\gamma }^{\Delta }k+n\right)\hfill \end{array}.$$

(52)

When the yaw angle range is within $\left(-{90}^{°},{90}^{°}\right)$, ${\gamma }^{\Delta }$ remains positive, and Equation (34) remains valid. Therefore, $\Delta f_i^{\Delta }$ is distributed in the following range:

$$0\le \Delta f_i^{\Delta }\le \Delta f_{i,\max }^{\Delta },$$

(53)

$$\Delta f_{i,\max }^{\Delta }={\displaystyle \frac{d}{\lambda \cos {\Delta }_{aR}}}\left({\displaystyle \frac{‖v_T‖\cos {\psi }_{vT}}{‖v_R‖}}-\sin {\alpha }_{aR}\sin {\beta }_R\sin {\Delta }_{aR}\right).$$

(54)

The frequency values are distributed within the range from $f_{si}$ to $f_{si}+\Delta f_{i,\max }^{\Delta }$. Adopting the same concept as the $W_{eq}$ estimation approach employed in the side-looking mode, the estimation result of $W_{eq}^{\Delta }$ can be modified as

$$W_{eq}^{\Delta }=\left\{\begin{array}{l}{W}_{fs}+\Delta W^{\Delta }\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}R\left(f_{s,i}+\Delta f_{i,\max }^{\Delta }\right)\ge R\left(f_{si}\right)\\ W_{fs}-\Delta W^{\Delta }\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}R\left(f_{s,i}+\Delta f_{i,\max }^{\Delta }\right)<R\left(f_{si}\right)\end{array}\right..$$

(55)

where

$$\Delta W={\rho }^{\Delta }\cdot {\gamma }^{\Delta }\cdot {\displaystyle {\int }_{0^{°}}^{{360}^{°}}\left|\Delta f_{i,\max }^{\Delta }\right|\hspace{0.33em}d{\alpha }_{vT}}.$$

(56)

$${\rho }^{\Delta }={\displaystyle \frac{{\displaystyle \sum _{k=1}^{K-1}{\displaystyle \sum _{n=0}^{N-1}\frac{k}{n+k{\gamma }^{\Delta }}}}}{NK}}.$$

(57)

B.

Limited observation area

Considering the modulation of the transmitting and receiving patterns, the main lobe exhibits a limited range, while the energy in the side lobes remains significantly low. The schematic in Figure 4 illustrates the variation in the clutter-to-noise ratio (CNR) after beamforming within the coverage area under pattern modulation.

Figure 5 shows the relationship between clutter rank and the CNR of the far side lobe in a typical bistatic radar configuration. The clutter rank is determined by the number of large eigenvalues, where the ratio of the sum of these large eigenvalues to the total sum of all eigenvalues equals 99.99%. The reduction in the CNR of the far side lobe results in a diminished influence of the corresponding area on the clutter rank. When the CNR is less than 0 dB under pattern modulation, the clutter energy of the scattering unit diminishes to the noise energy level, rendering its influence on clutter rank negligible.

The limited observation area can be regarded as the area where the CNR after beamforming exceeds 0 dB, encompassing the pattern main lobe and near side lobes. $C_{BW,1}$ and $C_{BW,N_c}$ denote the starting and ending scattering unit indexes in these limited areas, respectively. The $i$-th scattering unit consists of $\left[C_{BW,1},C_{BW,N_c}\right]$, where $C_{BW,1},C_{BW,N_c}\in \left[1,\cdots ,N_c\right]$.

The frequency values corresponding to the scattering units in the limited observation area can be regarded as an intersection with those of the entire range bin. Figure 6 shows the schematic of the frequency value distribution within this limited observation area.

The equivalent frequency bandwidth of the limited observation area can be further derived as follows:

$$W_{eq,BW}=\left\{\begin{array}{l}{W}_{fs,BW}+\Delta W_{BW}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}{R}_{BW}\left(f_{s,i}+\Delta f_{i,\max }\right)\ge R_{BW}\left(f_{si}\right)\\ W_{fs,BW}-\Delta W_{BW}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}{R}_{BW}\left(f_{s,i}+\Delta f_{i,\max }\right)<R_{BW}\left(f_{si}\right)\end{array}\right.,$$

(58)

where

$$R_{BW}\left(f_{si}+\Delta f_{i,\max }\right)=\underset{i\in \left[C_{BW,1},C_{BW,N_c}\right]}{\max }\left(f_{si}+\Delta f_{i,\max }\right)-\underset{i\in \left[C_{BW,1},C_{BW,N_c}\right]}{\min }\left(f_{si}+\Delta f_{i,\max }\right),$$

(59)

$$R_{BW}\left(f_{si}\right)=\underset{i\in \left[C_{BW,1},C_{BW,N_c}\right]}{\max }\left(f_{si}\right)-\underset{i\in \left[C_{BW,1},C_{BW,N_c}\right]}{\min }\left(f_{si}\right).$$

(60)

When ${\delta }_{t\alpha 0}\in \left\{0^{°},{180}^{°}\right\}$ and ${\delta }_{r\alpha }\in \left\{0^{°},{180}^{°}\right\}$, the observation area is symmetrically distributed at the intersection of two platforms and the range bin, such as areas A and B in Figure 6. In this case, the calculation of $\Delta W_{BW}$ is derived as follows:

$$\Delta W_{BW}={\displaystyle \frac{\rho \cdot ‖v_T‖}{\lambda \cdot PRF}}\cdot \left[\sin \left(\underset{i\in \left[C_{BW,1},C_{BW,N_c}\right]}{\max }\left({\psi }_{vT,i}\right)\right)-\sin \left(\underset{i\in \left[C_{BW,1},C_{BW,N_c}\right]}{\min }\left({\psi }_{vT,i}\right)\right)\right].$$

(61)

In other cases:

$$\Delta W_{BW}={\displaystyle \frac{\rho \cdot ‖v_T‖}{\lambda \cdot PRF}}\cdot {\displaystyle \sum _{m=1}^M\left[\sin \left(\underset{i\in {\mathbb{F}}_{m,BW}}{\max }\left({\psi }_{vT,i}\right)\right)-\sin \left(\underset{i\in {\mathbb{F}}_{m,BW}}{\min }\left({\psi }_{vT,i}\right)\right)\right]},$$

(62)

where $M$ is the count of monotone intervals as $\left|\cos {\psi }_{vT,i}\right|$ versus scattering units, and ${\mathbb{F}}_{m,BW}\in \left[C_{BW,1},C_{BW,N_c}\right]$ is the $m$-th monotone interval.

However, when the range resolution of the observation area in a bistatic radar system is inferior, the spatial variability of bistatic range resolution may lead to the aggregation of various ranges within the same range bin. Consequently, the clutter signal in these areas is composed of various bistatic range sums of clutter. Figure 7 illustrates the distribution of $f_{si}+\Delta f_{i,\max }$ in this case.

It renders the following derived values of EFB:

$${\Theta }_i^L=\left\{\begin{array}{l}{\displaystyle \frac{\left|f_{dT,i}^1\right|}{\gamma }},{\displaystyle \frac{\left|f_{dT,i}^1\right|}{1+\gamma }},\cdots ,{\displaystyle \frac{k\left|f_{dT,i}^1\right|}{n+k\gamma }},\cdots ,{\displaystyle \frac{\left(K-1\right)\left|f_{dT,i}^1\right|}{N-1+\left(K-1\right)\gamma }},\cdots \\ {\displaystyle \frac{\left|f_{dT,i}^l\right|}{\gamma }},{\displaystyle \frac{\left|f_{dT,i}^l\right|}{1+\gamma }},\cdots ,{\displaystyle \frac{k\left|f_{dT,i}^l\right|}{n+k\gamma }},\cdots ,{\displaystyle \frac{\left(K-1\right)\left|f_{dT,i}^l\right|}{N-1+\left(K-1\right)\gamma }},\cdots \\ {\displaystyle \frac{\left|f_{dT,i}^L\right|}{\gamma }},{\displaystyle \frac{\left|f_{dT,i}^L\right|}{1+\gamma }},\cdots ,{\displaystyle \frac{k\left|f_{dT,i}^L\right|}{n+k\gamma }},\cdots ,{\displaystyle \frac{\left(K-1\right)\left|f_{dT,i}^L\right|}{N-1+\left(K-1\right)\gamma }}\end{array}\right\},$$

(63)

where $f_{dT,i}$ of the 1st, l-th, and L-th bistatic ranges within the area are denoted as $f_{dT,i}^1$, $f_{dT,i}^l$, and $f_{dT,i}^L$, respectively.

The variation in range resolution along the range bin is observed to extend the EFB range, which differs from merely considering the clutter rank based on a single bistatic range. $F_i$ is modified as follows:

$$F_i={\displaystyle \frac{\left({\displaystyle \sum _{k=1}^{K-1}{\displaystyle \sum _{n=0}^{N-1}\frac{k}{n+k\gamma }}}\right)}{NK}}\cdot {\displaystyle \frac{{\displaystyle \underset{l=1}{\overset{L}{\int }}\left|f_{dT,i}^l\right|}}{L}}={\displaystyle \frac{\rho }{L}}\cdot {\displaystyle \underset{l=1}{\overset{L}{\int }}\left|f_{dT,i}^l\right|}$$

(64)

Therefore, $\Delta W_{BW}$ is the average of the $\Delta W_{BW}^l$ of various bistatic ranges, which can be modified as follows:

$$\Delta W_{BW}={\displaystyle \frac{1}{L}}{\displaystyle \sum _{l=1}^L\Delta W_{BW}^l}$$

(65)

According to Equation (58), under the assumption of a fixed observation area and constant bistatic system parameters, $W_{fs,BW}$ remains constant. The closer $\Delta W_{BW}$ of the observation area is to zero, the smaller the clutter rank of the observation area. Therefore, the observation area with a lager clutter Doppler bandwidth exhibits a correspondingly larger clutter rank. With an increase in Doppler bandwidth, a greater number of Doppler units are encompassed within the observation area due to the constant Doppler resolution $PRF/K$, thereby resulting in enhanced azimuth resolution across the area. This elucidates the variation rule of clutter rank, indicating a positive correlation between the clutter rank of the observation area and the Doppler bandwidth, and a negative correlation with azimuth resolution.

$$N_r\propto B_d\propto {\displaystyle \frac{1}{{\rho }_a}},$$

(66)

where $B_d$ and ${\rho }_a$ are the Doppler bandwidth and azimuth resolution of the observation area, respectively.

4. Numerical Examples

In the subsequent experiments, the scattering units on the range bin are divided using a super-resolution method to derive the theoretical covariance matrix $R_{cn}$. The eigenvalue decomposition of $R_{cn}$ is performed, and the eigenvalues are sorted in descending order to obtain $p$.

$$R_{cn}\stackrel{eig}{=}U\left[\begin{array}{ccc}{\lambda }_1& & \\ & \ddots & \\ & & {\lambda }_{NK}\end{array}\right]U^H,$$

(67)

$$p=sort\left(\left[{\lambda }_1,\cdots ,{\lambda }_{NK}\right]\right).$$

(68)

where ${\lambda }_i$ is the $i$-th eigenvalue of the decomposition; $U$ is the matrix that consists of eigenvectors; and $p$ is the eigenvalue vector arranged in descending order, from the largest to the smallest.

In order to directly reflect the accuracy of the estimated clutter rank, the percentage of the sum of the first $N_r$ large eigenvalues and the sum of all eigenvalues is defined as $\eta$ [40]. $\eta$ represents the occupancy in the total energy, which can be represented as follows:

$$\eta =\left[{\displaystyle \sum _{r=1}^{N_r}p\left(r\right)}/{\displaystyle \sum _{r=1}^{NK}p\left(r\right)}\right]\times 100\%,$$

(69)

where $p\left(r\right)$ represents the $r$-th eigenvalue. The closer the value of $\eta$ is to 99.99%, the more accurate the estimation result.

Since the RVCS is insufficient for visually comparing the relationship between the velocity direction of two platforms and the range bin, the $x_{vR}$ and $z_{vR}$ axes of RVCS are rotated ${\delta }_{r\alpha }$ counterclockwise around the $y_{vR}$ axis to obtain the relative coordinate (RC) system $oxyz$ as shown in Figure 8. In the RC system, the $x$ axis coincides with the direction of the projection of baseline in the $x_{vR}o{z}_{vR}$ plane. Thus, ${\beta }_{rt}$ is the only angle needed to describe the position of the transmitting platform. In RVCS, we can obtain the following: ${\delta }_{t\alpha }={\delta }_{r\alpha }-{\delta }_{t\alpha 0}$.

4.1. Experiment 1: Clutter Rank Estimation for Airborne Bistatic Radar

The existing methods for estimating clutter rank in bistatic radar are specifically applicable to airborne bistatic systems where two aircraft are flying at the same altitude and maintaining a steady horizontal flight, implying ${\beta }_{rt}=0^{°}$ and ${\delta }_{t\beta }=0^{°}$. The proposed method is compared with the existing method [32] in this experiment. Furthermore, we also conducted simulations to estimate the clutter rank corresponding to $W_{eq}$ calculation method A (described in Section 3.2).

In this experiment, ${\delta }_{r\alpha }=0^{°}$ and $-{30}^{°}$ are held constant while the clutter rank and $\eta$ are determined via both the proposed method and the method presented in [32] for varying values of ${\delta }_{t\alpha }$. The relationship between the transmitting and receiving aircraft velocities in the $xoz$ plane is shown in Figure 9. The experimental parameters can be found in

Table 1. Simulation parameters of Experiment 1.

(a) Configuration Parameters
Parameter	Value	Unit
Bistatic range sum	60	km
PRF	4000	Hz
Altitude of transmitting aircraft	10	km
Altitude of receiving aircraft	10	km
Velocity of transmitting aircraft	200	m/s
Velocity of receiving aircraft	200	m/s
The distance between the sub-aircraft points of two aircrafts	15	km
(b) System Parameters
Parameter	Value	Unit
Signal wavelength	0.2	m
Channel spacing	0.1	m
Number of channels	6	-
Number of pulses	128	-
CNR	50	dB

.

Figure 10 and Figure 11 illustrate the estimated clutter rank and $\eta$ of various configurations at ${\delta }_{r\alpha }=0^{°}$ and $-{30}^{°}$, respectively. The consistency between the clutter rank estimated by the proposed method and the method presented in [32] with the actual clutter rank is prominently observed under different configurations, both achieving a remarkable $\eta$ exceeding 99.85%. However, the clutter rank estimation results obtained by employing $W_{eq}$ in calculation method B tend to overestimate, resulting in $\eta$ exceeding 100%. Consequently, the subsequent simulation will not include the analysis of proposed method A.

In Figure 10, when ${\delta }_{t\alpha }={\delta }_{r\alpha }=0^{°}$, the clutter rank suddenly decreases due to the symmetric distribution of frequency values between the UHR and LHR’s scattering units. Figure 12 further simulates the eigenvalue distribution curves of the clutter covariance matrix with ${\delta }_{r\alpha }=0^{°}$, ${\delta }_{t\alpha }=0$–${10}^{°}$. As ${\delta }_{t\alpha }$ approaches $0^{°}$, a portion of the clutter component approaches the noise level, reflecting a decrease in distinguishability.

As shown in Figure 11, due to ${\delta }_{r\alpha }=-{30}^{°}$, a shift is observed in the location where the minimum clutter rank occurs. The extended frequency values distribution of UHR and LHR symmetric position scattering units in the configuration exhibits disparities, thereby eliminating the sudden decrease in clutter rank. This experiment verifies the consistency of the clutter rank estimation results between the proposed method and the method presented in [32] for specific bistatic radar configurations.

For more complicated airborne bistatic configurations, such as two platforms at varying altitudes, the method described in [32] is not suitable. This is because the signal model of the method proposed in [32] assumes the two platforms have the same altitude H. Due to this limitation, H in more complicated airborne configurations, such as two platforms at varying altitudes, cannot be set, ultimately causing the failure of method. Therefore, the proposed method is employed to conduct experiments on airborne bistatic radar at varying altitudes. In the simulation, the altitude and velocity of transmitting aircraft is set at 50 km and 400 m/s, while the bistatic range sum is set at 100 km. Other parameters are the same as in Table 1. The selection of such parameters can result in significant differences in altitude and velocity between the transmitting and receiving aircraft, thereby validating the applicability of the proposed method in more complex airborne bistatic configurations. ${\delta }_{r\alpha }=-{30}^{°}$ is held constant, while the clutter rank and $\eta$ are determined via the proposed method for varying values of ${\delta }_{t\alpha }$.

Figure 13 illustrates the estimated clutter rank and $\eta$ of various configurations at ${\delta }_{r\alpha }=-{30}^{°}$. It can be seen that the proposed method demonstrates accurate estimation results under different aircraft altitudes, with η exceeding 99.8% (the minimum value of η is 99.82%). Due to the limitation of other methods requiring both platforms to have the same altitude, the proposed method is employed to simulate scenarios where other methods are inapplicable, thereby validating the applicability of the proposed method in more complicated airborne bistatic configurations.

4.2. Experiment 2: Clutter Rank Estimation for Spaceborne Bistatic Radar

For spaceborne bistatic radar, the curvature of the earth cannot be disregarded due to the satellite’s high altitude. Consequently, it is imperative to acknowledge that the ground can no longer be perceived as a flat surface, but rather as a curved one [39]. The method proposed in [32] is based on a planar ground surface, necessitating that the velocities of both aircraft lie within the same plane. Consequently, this method lacks applicability to spaceborne bistatic radar systems. This experiment examines several typical spaceborne bistatic radar configurations, including low earth orbit (LEO)-LEO, middle earth orbit (MEO)-LEO, and geosynchronous earth orbit (GEO)-LEO, to validate the applicability of the proposed method across general bistatic radar configurations.

In spaceborne radar detection, the echo signal is typically received by a subarray consisting of multiple array elements, where the receiving channel spacing exceeds half of the wavelength. To adhere to the Nyquist sampling theorem, both the Doppler frequency and the spatial frequency of the entire range bin must be unambiguous. However, the PRF and channel spacing utilized in actual engineering do not satisfy the requirements for unambiguity. Therefore, to verify the accuracy of the clutter rank estimation, $PRF$ is set to $2\left(‖v_T‖+‖v_T‖\right)/\lambda$, and $d$ is set to $\lambda /2$. The experimental parameters of different configurations are shown in

Table 2. Simulation parameters of Experiment 2.

Parameter		Value		Unit
Bistatic range sum	LEO-LEO	2500		km
	MEO-LEO	6200		km
	GEO-LEO	19,000		km
PRF	LEO-LEO	147,074		Hz
	MEO-LEO	126,200		Hz
	GEO-LEO	115,950		Hz
		Transmitter	Receiver
Orbit altitude		1000/8000/ 35,786	1000	km
Orbit inclination		0/45/135	0/−30/−30	°
Right ascension of ascending node		15	0	°
Argument of perigee angle		0	0	°
Eccentricity		0	0	°
True near location angle		0	0	°

. Other parameters are the same as in Table 1. The relationship between the transmitting and receiving satellite velocities in the $xoz$ plane is shown in Figure 14. Furthermore, the velocity directions of the transmitting and receiving platforms are parallel to the ground in these configuration parameters.

As shown in Figure 15, we simulate the distribution curves of the extended frequency values against the scattering units in a LEO-LEO bistatic configuration. The red and green lines represent the distribution curve of $f_{s,i}+\Delta f_{i,\max }$ and $f_{s,i}$, respectively, with other frequency values distributed between these two lines. When ${\delta }_{t\alpha }={\delta }_{r\alpha }=0^{°}$, the distribution curves of UHR and LHR exhibit symmetry; however, under different conditions, the distribution curves of UHR and LHR diverge significantly.

The distribution curves of the extended frequency values against the scattering units in MEO-LEO and GEO-LEO bistatic configurations are shown in Figure 16. Due to space constraints, only the configurations of ${\delta }_{t\alpha }={\delta }_{r\alpha }=0^{°}$ are included. The EFB of these configurations is significantly smaller than that of the LEO-LEO bistatic configuration, resulting in a considerably narrower curve width in comparison.

Figure 17a–c display the eigenvalue distribution curves of the clutter covariance matrix in various LEO-LEO, MEO-LEO, and GEO-LEO bistatic configurations. The dashed lines in the figures represent the estimated results of clutter rank using the proposed method. This method accurately estimates the boundary points of large and small eigenvalues under different configurations.

Table 3. Statistical results and proportions of the clutter rank estimation under different spaceborne bistatic radar configurations.

	LEO-LEO		MEO-LEO		GEO-LEO
	Nr	η	Nr	η	Nr	η
${\delta }_{t\alpha }={\delta }_{r\alpha }=0^{°}$	105	99.58%	79	99.86%	67	99.76%
${\delta }_{t\alpha }={45}^{°},{\delta }_{r\alpha }=-{30}^{°}$	172	99.64%	140	99.95%	128	99.99%
${\delta }_{t\alpha }={135}^{°},{\delta }_{r\alpha }=-{30}^{°}$	79	99.96%	108	99.99%	125	99.99%

provides statistics on the proportion of eigenvalues corresponding to the estimated clutter rank, with the proportion in different cases exceeding 99.5%. These simulation results confirm the accuracy of the proposed method across various spaceborne bistatic radar configurations. It can be seen from the statistical results that a relatively lower value of $\eta$ was obtained in LEO-LEO bistatic configurations compared to MEO-LEO and GEO-LEO bistatic configurations. This is due to the fact that, in accordance with Equation (47), $\Delta W$ induced by the transmitting Doppler frequency exhibits a direct proportionality to the velocity of the transmitting platform. The velocity of the satellite decreases as its orbital altitude increases, resulting in a reduced velocity for GEO and MEO relative to LEO. Consequently, the smaller $\Delta W$ of the LEO transmitting satellite results in a more concentrated frequency distribution and makes it closer to that of a monostatic radar, thereby leading to a larger $\eta$.

4.3. Experiment 3: Clutter Rank Estimation of Non-Side-Looking Mode

In previous experiments, we employed the side-looking mode; however, in this experiment, we validated the clutter rank estimation using the non-side-looking mode. According to Equation (53), the proposed method is valid for the non-side-looking mode with a yaw angle ${\Delta }_{r\alpha }$ range of $\left(-{90}^{°},{90}^{°}\right)$. Therefore, it is imperative to limit the yaw angle ${\Delta }_{r\alpha }$ to within $\left(-{90}^{°},{90}^{°}\right)$. We utilized the same LEO-LEO bistatic configuration and system parameters as in Experiment 2, while adjusting the yaw angle ${\Delta }_{r\alpha }$ to −85°, −60°, −45°, −30°, −10°, 10°, 30°, 45°, 60°, and 85°.

Figure 18a–c present the eigenvalue distribution curves of the clutter covariance matrix with various yaw angles ${\Delta }_{r\alpha }$ in LEO-LEO bistatic configurations of ${\delta }_{t\alpha }={\delta }_{r\alpha }=0^{°}$, ${\delta }_{t\alpha }={45}^{°},{\delta }_{r\alpha }=-{30}^{°}$, and ${\delta }_{t\alpha }={135}^{°},{\delta }_{r\alpha }=-{30}^{°}$, respectively. The proposed method effectively estimates the boundary points of large and small eigenvalues under varying yaw angles ${\Delta }_{r\alpha }$, demonstrating its accuracy across different conditions.

The statistics of the estimated clutter rank and the proportion of eigenvalues corresponding to ${\Delta }_{r\alpha }$ are presented in Figure 19a and Figure 19b, respectively. The proportion of different cases exceeded 99.2%. The impact of $\Delta W$ on $\eta$ remained consistent with experiment 2. The accuracy of clutter rank estimation increased as $\Delta W$ decreased. When ${\delta }_{t\alpha }={\delta }_{r\alpha }=0^{°}$, the clutter rank increased as ${\Delta }_{r\alpha }$ approached zero. However, when ${\Delta }_{r\alpha }$ equalled zero precisely, the clutter rank experienced a sharp decline. When ${\delta }_{t\alpha }={45}^{°}, {\delta }_{r\alpha }=-{30}^{°}$, under positive ${\Delta }_{r\alpha }$, the clutter rank exhibited an increasing trend as it approached zero. Conversely, for negative yaw angles, larger deviations from zero resulted in elevated clutter rank. When ${\delta }_{t\alpha }={135}^{°}, {\delta }_{r\alpha }=-{30}^{°}$, the clutter rank tended to increase as the deviation of ${\Delta }_{r\alpha }$ from 0° increased.

4.4. Experiment 4: Clutter Rank Estimation in a Limited Area

In this experiment, we further validated the accuracy of the clutter rank estimation method within a limited observation area. The LEO-LEO bistatic configuration was consistently selected for simulation, with all parameters, except for orbit inclination, remaining as presented in Table 2.

Initially, we simulated the clutter rank with various ${\delta }_{t\alpha }$ and ${\delta }_{r\alpha }$ when the velocity directions of the transmitting and receiving platforms are parallel to the ground. The estimated clutter rank and proportions under different transmitting and receiving velocity directions are shown in

Table 4. Proportions of the clutter rank estimation under different ${\delta }_{t\alpha }$ when the velocity directions of the transmitting and receiving platforms are parallel to the ground.

δtα	0°	45°	90°	135°	180°	−45°	−90°	−135°
${\delta }_{r\alpha }=0^{°}$	99.83%	99.50%	99.89%	99.93%	99.59%	99.76%	99.77%	99.82%
${\delta }_{r\alpha }={45}^{°}$	99.78%	99.75%	99.80%	99.54%	99.93%	99.86%	99.62%	99.69%
${\delta }_{r\alpha }={90}^{°}$	99.33%	99.76%	99.78%	99.33%	99.60%	99.65%	99.63%	99.40%
${\delta }_{r\alpha }={135}^{°}$	99.84%	99.82%	99.76%	99.28%	99.52%	99.71%	99.92%	99.46%

and Figure 20. As shown in Table 4, the proportion of clutter rank estimation results in different transmitting and receiving velocity directions can exceed 99.3%. Therefore, the proposed clutter rank estimation method continued to yield effective results within a constrained area.

To verify the above clutter rank variation rule with azimuth resolution, we selected ${\delta }_{r\alpha }=0^{°}, {\delta }_{t\alpha }={135}^{°}$ and ${\delta }_{t\alpha }={\delta }_{r\alpha }=0^{°}$ bistatic configurations to divide the LRH into Doppler units according to Doppler resolution. As shown in Figure 21, the interval between two triangle marks on the range bin is a Doppler unit. Compared with Figure 21a,b, the overall azimuth resolution of ${\delta }_{t\alpha }={\delta }_{r\alpha }=0^{°}$ is significantly better than that of ${\delta }_{r\alpha }=0^{°}, {\delta }_{t\alpha }={135}^{°}$ on the same range bin. We selected four typical observation areas according to the azimuth resolution distribution of ${\delta }_{r\alpha }=0^{°}, {\delta }_{t\alpha }={135}^{°}$, which are denoted as observation areas 1 to 4. Observation area 1 exhibited the best azimuth resolution, while observation area 2 demonstrated the worst.

The distribution of the eigenvalue and the statistical results in observation 1 to 4 are shown in Figure 22 and

Table 5. Statistical results and proportions of the clutter rank estimation in observation area 1 to 4.

Observation Area		1	2	3	4
${\delta }_{r\alpha }=0^{°}, {\delta }_{r\alpha }={135}^{°}$	$N_r$	44	6	26	11
	$\eta \left(\%\right)$	99.99	99.93	99.93	99.68
${\delta }_{r\alpha }={\delta }_{t\alpha }=0^{°}$	$N_r$	25	143	140	59
	$\eta \left(\%\right)$	99.20	99.83	99.79	99.80

, respectively. The simulation results show that the clutter rank of the four observation areas of ${\delta }_{r\alpha }=0^{°}, {\delta }_{t\alpha }={135}^{°}$ and ${\delta }_{t\alpha }={\delta }_{r\alpha }=0^{°}$ are 1st > 3rd > 4th > 2nd and 2nd > 3rd > 4th > 1st, respectively. The clutter rank of observation area 2 of ${\delta }_{t\alpha }={\delta }_{r\alpha }=0^{°}$ reaches 143, which is significantly larger than the max rank of observation area 1 of ${\delta }_{r\alpha }=0^{°}, {\delta }_{t\alpha }={135}^{°}$. The simulation results of clutter rank correlate with the distribution of azimuth resolution, thus verifying the correctness of the clutter rank variation rule. This rule is supported both theoretically and experimentally and offers valuable design guidance for optimizing bistatic radar configurations.

Lastly, we simulated the clutter rank with various ${\delta }_{t\beta }$ when ${\delta }_{t\alpha }$ and ${\delta }_{r\alpha }$ remain constant. The simulation employed ${\delta }_{r\alpha }=0^{°}$, with ${\delta }_{t\alpha }$ values of 0°, ±15°, ±30°, ±45°, ±60°, ±75° and ±90°. Figure 23 illustrates the variation in clutter rank with respect to various ${\delta }_{t\beta }$ values. The closer ${\delta }_{t\alpha }$ is to ±90°, the change in clutter rank with respect to ${\delta }_{t\beta }$ reduces. The proportion of clutter rank estimation results in various ${\delta }_{t\beta }$ can exceed 99.5%, as illustrated in Figure 24. The proposed method exhibits robustness across diverse conditions, as evidenced by the experimental results.

Finally, the proposed method was subjected to error analysis based on the extensive experimental results mentioned above. It can be observed from the experimental results that the minimum value of η reaches 99.20%, while the maximum value attains 99.99% of its theoretical counterpart. In general, the mean value serves as a measure of the central tendency in a dataset, while the root-mean-square error (RMSE) quantifies the discrepancy between estimated and theoretical values [41]. The equations representing the mean value and the RMSE of η are as follows [41]:

$$\overline{\eta }={\displaystyle \frac{1}{M}}{\displaystyle \sum _{\mu =1}^M{\eta }_{\mu }},$$

(70)

$$RMSE=\sqrt{{\displaystyle \frac{1}{M}}{\displaystyle \sum _{\mu =1}^M\left[{\left({\eta }_{\mu }-{\eta }_{true}\right)}^2\right]}},$$

(71)

where ${\eta }_{\mu }$ is the µ-th experimental result; M is the total number of the experimental results; and ${\eta }_{true}=99.99\%$ is the theoretical value. For the total 474 experimental results, the mean value and RSME of η were calculated by Equations (70) and (71), respectively. The corresponding statistical results are presented in

Table 6. Statistical results of the mean value and RMSE of η.

	The Mean Value	RMSE
Experiment 1	99.94%	0.07%
Experiment 2	99.85%	0.20%
Experiment 3	99.83%	0.24%
Experiment 4	99.82%	0.23%
Total Experiments	99.84%	0.20%

.

From the error analysis results, it evident that for airborne bistatic radar (Experiment 1), the mean value of η exceeds 99.94%, with an RMSE of merely 0.07%. Although the accuracy of spaceborne bistatic radar estimation results (Experiments 2 to 4) is not as high as that of airborne bistatic radar, the mean value of η still surpasses 99.82%, and the RMSE remains below 0.24%. This further substantiates the robustness of the proposed method across different bistatic configurations.

5. Conclusions

This study presented a novel method for estimating clutter rank in bistatic radar, a critical aspect of radar signal processing, with significant implications for enhancing the accuracy and reliability of remote sensing data. The challenge of accurately estimating the clutter rank in bistatic radar was first addressed. It is particularly challenging to precisely calculate the time-bandwidth product when the space-time steering vector is represented using PSWF. A method to calculate the equivalent time-bandwidth product based on the space-time distribution characteristics of bistatic radar was then introduced. This method facilitates the estimation of clutter rank for either the entire range bin or a limited observation area in any bistatic radar configuration, whether it is side-looking or operates in partial non-side-looking mode. The percentage of the sum of the large eigenvalues with the first estimated clutter rank and the sum of all eigenvalues surpassed 99% in all experiments. This approach can provide essential design indicators for reduced dimension and reduced rank STAP in bistatic radar. Additionally, it clarified the influencing factors and variation rules of clutter rank, enabling rapid evaluation of clutter rank and supporting optimization efforts in radar system design. The implications of our findings can provide a valuable tool for engineers and scientists working in the field of remote sensing. In our future work, we will further exploit the estimated clutter rank results to investigate reduced rank STAP algorithms for bistatic radar.