Clutter Suppression with Doppler Frequency Shifted Least Mean Square Filtering in LEO Satellite-Based Passive Radar

Highlights

• Low-earth-orbit (LEO) internet satellite constellations can be utilized as illuminators of opportunity in passive radars. The fast-moving platform results in complicated clutter properties. Clutter property and suppression algorithms are discussed in this paper.

What are the main findings?

• The clutter model for a low-earth-orbit satellite (Starlink is an example)-based passive radar is deduced and the properties of the clutter are analyzed.
• A Doppler-frequency-shifted normalized least mean square filter with its fast implementation method is proposed to suppress the clutter.

What are the implications of the main findings?

• The clutter in a Starlink satellite-based passive radar shows shifted and broadened Doppler properties, but they are estimable.
• The proposed Doppler-frequency-shifted normalized least mean square filter and its block version can suppress the clutter with acceptable performance and high computational efficiency.

Abstract

With the rapid development of low-earth-orbit (LEO) internet satellite constellations, LEO satellites are becoming promising illuminators of opportunity for passive radar. However, the moving satellite platform results in a shifted Doppler frequency and increased Doppler spread of the clutter, leading to decreased clutter suppression performance. In this paper, the clutter model for a LEO satellite-based passive radar is analyzed. Based on the properties of the clutter, a Doppler-frequency-shifted normalized least mean square (LMS) filter is proposed to suppress the clutter. Furthermore, an efficient block adaptive method is introduced for fast implementation. Moreover, a Butterworth filter is designed to filter out the residual clutter. Simulations demonstrate the effectiveness of the proposed method.

1. Introduction

A passive radar is a type of bistatic remote sensing equipment that utilizes illuminators of opportunity, e.g., FM radio stations, digital video broadcast-terrestrial (DVB-T), cellular base stations (4G or 5G), and WiFi [1,2,3,4], to detect moving targets. These illuminators already exist in the surveillance area and are not part of the radar system, offering advantages such as covert operation, low cost, and reduced vulnerability. Terrestrial broadcast transmitters have been the most popular illuminators of opportunity for passive radars, with attention only relatively recently shifting to satellite illuminators [5]. Potential satellite illuminators include geostationary Earth orbit (GEO) satellites, such as global navigation satellite systems (GNSS) [6,7], digital video broadcast-satellite (DVB-S) [8,9], and Inmarsat [10], as well as low-Earth-orbit (LEO) satellites, such as Iridium. Compared to terrestrial illuminators, satellite-based illuminators of opportunity offer global coverage and persistent monitoring, particularly in remote areas lacking common terrestrial communication transmitters. However, the satellite-based passive radars face power budget limitations due to the far distance between the satellite transmitter and the radar receiver. This limitation is more severe when using geostationary orbit satellites as illuminators, necessitating long integration time and adequate signal processing gain in such passive radars. LEO satellites provide higher received power density on the ground compared to geostationary satellites. Consequently, passive radars using LEO satellite illuminators have attracted much attention in recent years. With the fast deployment of Internet satellite constellations, e.g., Starlink, these LEO constellations are becoming promising illuminators of opportunity because they provide a vast number of available satellites and can cover any location on Earth at any time [5].

During the early development of Internet satellite constellations, LEO satellites, including Starlink, were considered for positioning [11,12]. Since 2021, passive radar based on global Internet satellite constellations has become a new research focus. The feasibility of using LEO Internet constellations has been proven [13,14,15,16,17]. The power budgets of Starlink and OneWeb have been compared with those of GNSS. Signal-to-noise ratios (SNR) at both the reference antenna and target surveillance antenna were estimated using these satellites’ open access parameters, showing higher SNR compared to GNSS and enabling a longer target detection range. Range and Doppler migration have also been analyzed, demonstrating that Starlink-based satellites exhibit less severe range migration than GNSS [17]. These findings indicate that LEO constellations are becoming promising illuminators of opportunity for passive radars. A LEO satellite constellation-based passive radar architecture has been designed, utilizing a narrow-beam antenna to track and receive the reference signal from a LEO satellite [14]. Additionally, moving target detection schemes have been proposed [14,18]. Several experiments have also been carried out [19,20,21], analyzing properties of Starlink downlink signals, including the cross-ambiguity function (CAF), range resolution, and velocity resolution. Different surveillance antenna configurations were studied, taking system coverage into account [19]. State-of-the-art literature discusses the advantages of LEO satellite constellation-based passive radar, such as higher range and velocity resolution. LEO Internet satellites typically operate with wide signal bandwidth, e.g., 250 MHz for Starlink downlink signal-resulting in high range resolution and suitability for imaging applications. Since these constellations usually operate at Ku/Ka bands and a long integration time is required in satellite-based passive radars, high velocity resolution can be achieved in such systems.

Nevertheless, LEO satellite constellation illuminators of opportunity also introduce new challenges in passive radars. The rapidly moving LEO satellites produce a Doppler-shifted signal that can be problematic, requiring novel processing strategies to address this issue [5]. Range and Doppler migration caused by the high velocity of LEO satellites have been discussed, and range migration rectification methods have been designed [6,17,18]. Furthermore, the satellite’s motion complicates clutter distribution. Clutter may vary with the changes in the geometric relationship. Therefore, clutter characteristics and corresponding suppression methods require further investigation in LEO satellite constellation-based passive radars.

For passive radar systems, two antennas are typically employed: a surveillance antenna and a reference antenna. The surveillance antenna receives moving target echoes, while the reference antenna captures the direct signal from the illuminator of opportunity to obtain the original signal waveform, as these illuminators are uncooperative. Subsequent processing stages involve clutter suppression and coherent integration [22] to reduce interference and achieve processing gain, followed by target detection to identify moving targets [23]. Clutter suppression is essential because clutter can mask target echoes, and this masking effect critically impacts weak target detection.

Typical clutter suppression methods in passive radars include the CLEAN algorithm [24,25], the extensive cancelation algorithm (ECA) [26], and adaptive filtering. The CLEAN algorithm iteratively removes interference signals by estimating their range and Doppler parameters, but exhibits degraded performance against continuous scattering clutter [27]. The extensive cancelation algorithm (ECA) and ECA-B&S [26] suppress clutter by projecting received signals onto a subspace orthogonal to the clutter subspace, effectively handling continuous scattering clutter. However, ECA is computationally intensive and complex for practical implementation. The bandwidth of the Starlink downlink signalsandthe sampling rate is normally high. The massive matrix computation of ECA is a heavy burden for the system. Modified CLEAN algorithms have thus been developed to improve performance. The most efficient suppression methods belong to the adaptive filtering family, including least square (LS) filtering, least mean square (LMS) filtering, block least mean square (BLMS) filtering, and space-time adaptive processing (STAP). LS and LMS filtering adaptively adjust filter coefficients in the time domain by minimizing the error between interference signals and filter output under least square or least mean square criteria [28,29]. Since LS/LMS filters are iteratively calculated per time-domain sample, real-time efficiency is challenging. More efficient methods like block LMS (BLMS) filtering [30] balance effectiveness and efficiency by estimating filter coefficients in the frequency domain and processing data in blocks. However, standard LMS and BLMS methods cannot handle Doppler-shifted clutter effectively. STAP filtering is commonly used in airborne passive radar [31,32,33], leveraging joint spatial-temporal processing when clutter Doppler frequency couples with direction of arrival due to platform motion. For LEO satellite-based passive radars, high satellite velocities induce Doppler shifts and Doppler spread in clutter signals and might be more severe than in airborne radars since the satellite moves faster. Although STAP could be considered, its complexity impedes practical implementation. In addition, multi-channels are used to sample signals at the same time in STAP. Given that passive radar typically involves larger computational loads than active radar, efficient processing and accelerated implementations are essential [34].

This paper analyzes the clutter signal model and Doppler shift properties after coherent integration in a LEO satellite-based passive radar. Based on signal energy budgets and clutter distribution characteristics, we propose an efficient clutter suppression method named DFS-BLMS for this system. We estimate the clutter Doppler range and propose a Doppler-frequency-shifted filtering technique to enhance the performance of the BLMS method—an efficient approach for near-field clutter suppression. Furthermore, a Butterworth filter is employed to mitigate residual far-field clutter, enabling effective separation between targets and clutter interference. Simulations validate the performance of the proposed method.

2. Signal Model

Passive radar utilizes a surveillance antenna to receive target echoes and a reference antenna to receive the direct signal from the illuminator. Surveillance signals consist of target echoes, direct signals, clutter, and noise. The reference signal primarily contains direct signals and noise. In this section, we derive mathematical models for clutter and reference signals to analyze clutter distribution and properties after coherent integration.

2.1. Clutter Model

To derive the clutter model, we discretize the ground into grid cells, where each cell constitutes a clutter cell. The geometric relationship among the LEO satellite, passive radar receiver, and the ith clutter cell is shown in Figure 1. The satellite, receiver, and clutter cell collectively define a reference plane. $R_0$ denotes the baseline distance between the satellite and receiver at the initial reference time ($t=0$). $R_{r,i}$ denotes the fixed distance from the receiver to the clutter cell. $R_{t,i}$ denotes the distance from the clutter cell to the satellite at $t=0$. Let $v_{s,r}$ be the satellite speed in the reference plane. Then, at the time $t$, the satellite’s displacement from its initial position in the reference plane is $\Delta R_{s,r}(t)=$ $v_{s,r}t$. $v_{s,v}$ is the satellite speed in the direction perpendicular to the reference plane. The displacement in the perpendicular direction is $\Delta R_{s,v}(t)=v_{s,v}t$. Then the distance from the satellite to the receiver is $R_0(t)$. $R_{t,i}(t)$ is the distance from the clutter cell to the satellite. The echo from the ith clutter cell is

$$s_{\mathrm{clutter},\mathrm{i}}(t)=A_{c,i}{s}_T(t-{\tau }_{c,i}(t))\exp (j2\pi f_{\mathrm{c}}(t-{\tau }_{c,i}(t)))+n_1(t),$$

(1)

where $A_{c,i}$ is the amplitude of the signal. $f_{\mathrm{c}}$ is the carrier frequency of the LEO satellite signal. $n_1(t)$ is the receiver noise. $s_T(t)$ is the transmitted signal of the LEO satellite. ${\tau }_{c,i}(t)$ is the time delay of the received signal from the ith clutter cell, and

$${\tau }_{c,i}(t)=(R_r+R_{t,i}(t))/c.$$

(2)

where $c$ is the speed of light. According to the geometric relationship,

$$R_{t,i}(t)=\sqrt{{R_{t,i}}^2+v_{s,r}^2{t}^2-2R_t{v}_{s,r}t\cos ({\sigma }_i+{\displaystyle \frac{{\delta }_i}{2}})+{v_{s,v}}^2{t}^2}$$

(3)

where ${\delta }_i$ is the angle between the satellite-to-clutter line and the baseline, called the irradiation angle. ${\sigma }_i$ is the angle between the satellite velocity vector and the bistatic bisector of the irradiation angle. ${\delta }_i$ and ${\sigma }_i$ are defined in the reference plane. Hence,

$${\tau }_{c,i}(t)=\left(R_{r,i}+\sqrt{{R_{t,i}}^2+v_{s,r}^2{t}^2-2R_{t,i}{v}_{s,r}t\cos ({\sigma }_i+{\displaystyle \frac{{\delta }_i}{2}})+{v_{s,v}}^2{t}^2}\right)/c$$

(4)

The clutter model is presented as

$$s_{\mathrm{clutter}}(t)={\displaystyle \sum _{i=1}^{N_c}{A}_{\mathrm{c},\mathrm{i}}{s}_T(t-{\tau }_{c,i}(t))\exp (j2\pi f_{\mathrm{c}}(t-{\tau }_{c,i}(t)))}+n_1(t),$$

(5)

where $N_c$ denotes the number of clutter cells. After down conversion of the signal, the clutter model is expressed as

$$s_{\mathrm{clutter}}(t)={\displaystyle \sum _{i=1}^{N_c}{A}_{\mathrm{c},\mathrm{i}}{s}_T(t-{\tau }_{c,i}(t))\exp (-j2\pi f_{\mathrm{c}}{\tau }_{c,i}(t))}+n_1(t).$$

(6)

From Equation (4), it can be inferred that the clutter phase is time-varying, with its variation dependent on ${\sigma }_i$ and ${\delta }_i$, which is determined by the geometric relationship of the clutter cell and the satellite.

2.2. Reference Signal

The reference signal is typically used for clutter suppression and coherent integration in passive radar. The radar coverage area can be divided into azimuth-range grids, as depicted in Figure 1. Each grid cell is considered a minimum scattering unit as the black dots in Figure 2.

For a LEO satellite-based system, the phase of the reference signal also varies with satellite motion. The reference signal is modeled as follows:

$$s_{ref}(t)=A_R{s}_T(t-{\tau }_R(t))\exp (j2\pi f_c(t-{\tau }_R(t)))+n_2(t),$$

(7)

where $A_R$ is the amplitude of the direct signal, and $n_2(t)$ denotes the receiver noise. ${\tau }_R(t)$ is the time delay of the received direct signal.

According to the geometry of a LEO satellite-based passive radar system depicted in Figure 1, the time delay of the direct signal is presented as

$${\tau }_R(t)=R_0(t)/c\hspace{0.33em}=\sqrt{R_0^2+v_{s,r}^2{t}^2-2R_0{v}_{s,r}t\cos (\theta )+v_{s,v}^2{t}^2}/c.$$

(8)

where $\theta ={\sigma }_i-{\delta }_i/2$. After down conversion, the reference signal is represented as

$$s_{ref}(t)=A_R{s}_T(t-{\tau }_R(t))\exp (-j2\pi f_c{\tau }_R(t))+n_2(t).$$

(9)

The reference signal’s phase also varies with LEO satellite motion.

2.3. Clutter Isorange Contours

Coherent integration is typically performed using the cross-ambiguity function between surveillance and reference signals. The bistatic range is defined as the difference between the baseline and the summation of the range from the satellite to $i$th clutter cell and the range from the clutter cell to the receiver. For each range cell, the clutter signal comprises echoes from all clutter cells sharing the same bistatic range.

Clutter isorange contours are used to analyze clutter properties within each range cell. The bistatic geometry is shown in Figure 3.

Let C represent the $i$th scatter cell, S denote the satellite, and R represent the receiver. O is the origin. $h_S$ is the orbital altitude and $h_R$ is receiver height. $r_S$ denotes the distance between the satellite and the clutter cell C. $r_R$ is the distance between the receiver and the clutter cell C. ${\theta }_R$ is the angle between OC and the x-axis. Thus, the position of clutter cell C is

$$(\sqrt{r_R^2-h_R^2}\cos ({\theta }_R),\sqrt{r_R^2-h_R^2}\sin ({\theta }_R),0).$$

(10)

Define the satellite position as $(L,0,h_S)$; therefore,

$$r_S=\sqrt{{(L-\sqrt{r_R^2-h_R^2}\cos ({\theta }_R))}^2+(r_R^2-h_R^2){\sin }^2({\theta }_R)+h_S^2}$$

(11)

where $L$ is the distance from the subsatellite point to the original point O. $r_L$ is the distance between the satellite and the receiver, and

$$r_L=\sqrt{L^2+{(h_S-h_R)}^2}$$

(12)

Let the bistatic range difference be

$$R_S=r_S+r_R-r_L.$$

(13)

According to Equations (11)–(13),

$$r_S=\sqrt{{(L-\sqrt{{(R_S+r_L-r_S)}^2-h_R^2}\cos ({\theta }_R))}^2+({(R_S+r_L-r_S)}^2-h_R^2){\sin }^2({\theta }_R)+h_S^2}$$

(14)

Therefore, we can figure out $r_S$ and $r_R$ by sweeping the parameter ${\theta }_R$ and obtain the clutter isorange contours for different $R_s$ values. Satellite constellation parameters for the Starlink satellite constellations are given in

Table 1. The Starlink satellite constellation parameters.

Parameters	Constellations
	Shell 1	Shell 2	Shell 3	Shell 4
Orbital planes	72	72	36	6/4
Satellite per plane	22	22	20	58/43
Altitude	550 km	540 km	570 km	560 km
Inclination	$53°$	$53.2°$	$70°$	$97.6°$

.

Figure 4 shows the clutter isorange contours for $L=0.10 \mathrm{km},$$100 \mathrm{km}$ using the Shell 1 parameters of the first-phase Starlink constellation. Clutter cells sharing the same iso-range contour contribute to the same bistatic range cell in the cross-ambiguity function (CAF). As different clutter cells exhibit distinct Doppler frequency characteristics, analyzing the clutter Doppler frequency within each bistatic range cell is essential.

2.4. Analysis of Clutter Doppler Frequency

Coherent integration is performed as follows:

$$\chi (\tau ,f_d)={\displaystyle {\int }_{-\infty }^{\infty }{s}_{\mathrm{clutter}}(t)s_{ref}^{*}(t+\tau )e^{2\pi f_{d}t}dt}.$$

(15)

Let $\chi (\tau ,f_d)$ be a 2D matrix where time-delay $\tau$ corresponds to bistatic range ($\tau c$) and $f_d$ denotes Doppler frequency. The time delay ${\tau }_{clutte,i}(t)$ between the clutter and the reference signal is

$$\begin{array}{ll}{\tau }_{clutte,i}(t)& ={\tau }_{c,i}(t)-{\tau }_R(t)=\left(R_r+R_{t,i}(t)-R_0(t)\right)/c\\ & =\left(R_r+\sqrt{{R_{t,i}}^2+v_{s,r}^2{t}^2-2R_t{v}_{s,r}t\cos ({\sigma }_i+{\displaystyle \frac{{\delta }_i}{2}})+{v_{s,v}}^2{t}^2}-\sqrt{R_0^2+v_{s,r}^2{t}^2-2R_0{v}_{s,r}t\cos (\theta )+v_{s,v}^2{t}^2}\right)/c\end{array}$$

(16)

Perform the Taylor expansion of ${\tau }_{clutte,i}(t)$ at $t=0$ and ignore items higher than quadratic, yielding

$${\tau }_{clutte,i}(t)={\tau }_0+{\tau }_{1}t+{\tau }_2{t}^2$$

(17)

where

$${\tau }_0={\left.\left(R_r+R_{t,i}(t)-R_0(t)\right)/c\right|}_{t=0}$$

(18)

$${\tau }_1=d{\left.\left(\left(R_r+R_{t,i}(t)-R_0(t)\right)/c\right)/dt\right|}_{t=0}$$

(19)

$${\tau }_2=d^2{\left.\left(\left(R_r+R_{t,i}(t)-R_0(t)\right)/c\right)/d{t}^2\right|}_{t=0}$$

(20)

Then,

$${\tau }_0=\left(R_r+R_{t,i}-R_0\right)/c,$$

(21)

$${\tau }_1=\left(2v_{s,r}\sin {\sigma }_i\sin {\displaystyle \frac{{\delta }_i}{2}}\right)/c,$$

(22)

$${\tau }_2\approx -{\displaystyle \frac{v_{s,r}^2{\sin }^2\left({\sigma }_i-\frac{{\delta }_i}{2}\right)+v_{s,v}^2}{2c{R}_0}}\hspace{0.33em}+{\displaystyle \frac{v_{s,r}^2{\sin }^2\left({\sigma }_i-\frac{{\delta }_i}{2}\right)+{\left(v_{s,v}\right)}^2}{2c{R}_{t,i}}}.$$

(23)

Since the clutter Doppler frequency is

$$f_{d,clutter,i}(t)=f_{d,clutter,i}+\Delta f_{d,clutter,i}t=d\left({\tau }_{clutte,i}(t)c/\lambda \right)/dt,$$

(24)

where the Doppler frequency of the clutter after coherent processing is

$$f_{d,clutter,i}=\left(2v_{s,r}\sin {\sigma }_i\sin {\displaystyle \frac{{\delta }_i}{2}}\right)/\lambda ,$$

(25)

with the Doppler rate

$$\Delta f_{d,clutter,i}\approx \left(-{\displaystyle \frac{v_{s,r}^2{\sin }^2\left({\sigma }_i-\frac{{\delta }_i}{2}\right)+v_{s,v}^2}{R_0}}\hspace{0.33em}+{\displaystyle \frac{v_{s,r}^2{\sin }^2\left({\sigma }_i-\frac{{\delta }_i}{2}\right)+{\left(v_{s,v}\right)}^2}{R_{t,i}}}\right)/\lambda .$$

(26)

Equation (25) indicates that the clutter Doppler frequency is primarily determined by the satellite’s velocity and the geometric relationship among the clutter cell, receiver, and satellite. Consequently, the clutter distribution exhibits both temporal and spatial variations. The irradiation angle ${\delta }_i$ is typically very small since both the receiver and clutter cell are ground-based (separated by kilometers to tens of kilometers), while the satellite operates at several hundred kilometers altitude.

According to Equation (26), the Doppler rate may be moderate, given that $R_0$ and $R_{t,i}$ are hundreds of kilometers long; thus, the expansion may not be very severe in LEO satellite-based passive radar. However, the Doppler shift—which varies with spatial geometry—cannot be neglected.

The clutter signal is the sum of echoes from all clutter cells; therefore, the clutter model can be rewritten as

$$s_{\mathrm{clutter}}(t)={\displaystyle \sum _{i=1}^{N_c}{A}_{\mathrm{c},\mathrm{i}}^{\prime }{s}_{ref}(t-{\tau }_{clutte,i}(t))\exp (-j2\pi f_{\mathrm{c}}{\tau }_{clutte,i}(t))}+n(t)$$

(27)

where $A_{\mathrm{c},\mathrm{i}}^{\prime }$ is the relative amplitude and $n(t)$ denotes noise signal. However, the Doppler frequency $f_{d,clutter,i}$ in each bistatic range cell $i$ results from the superposition of all clutter cells lying on the same isorange contour. Consequently, $f_{d,clutter,i}$ should be estimated by sweeping all possible angles in Equation (25). The Doppler frequency versus angle is calculated to analyze the clutter’s Doppler properties, as shown in Figure 5. The azimuth is the angle ${\theta }_R$ in Section 2.3.

In Figure 5a,c, the Doppler frequency from clutter cells on the same clutter isorange contour varies with azimuth angle, resulting in Doppler spreading after coherent integration. Clutter energy is distributed across multiple Doppler cells within each range cell of the CAF. Furthermore, Doppler shift magnitude increases with bistatic range. Fortunately, surveillance antennas at Ku/Ka-band (typical frequencies for LEO Internet satellite constellations) feature narrow beam widths. As shown in Figure 5b,d, when clutter azimuth is constrained to −30 to 30 degrees, Doppler variation is significantly reduced. However, residual Doppler variation remains non-negligible. Both Doppler shift and Doppler spreading must be compensated during clutter suppression processing.

3. Clutter Suppression and Target Detection

3.1. Traditional Clutter Suppression Methods

3.1.1. CLEAN Method

The CLEAN method [24,25] is commonly used in passive radars to suppress clutter. First, we need to find the maximum peak of the CAF and estimate the amplitude of the corresponding clutter signal. Then, the strongest clutter component is subtracted. The process is repeated iteratively until the residual energy stabilizes after each subtraction. Since the CAF must be recalculated and components removed iteratively, the CLEAN method is computationally expensive. The traditional CLEAN method assumes that the integration peak in the CAF is accurate; however, interference, noise, and discrete sampling may make the estimation inaccurate.

3.1.2. Normalized Least Mean Square Filtering and Block Least Mean Square Filtering

Adaptive filtering updates the filter parameters in real time based on previous filter parameters and outputs. Common methods include LMS, NLMS, and recursive LS (RLS), among others. NLMS offers good stability and is widely used in passive radar systems. The filter input is the reference signal, while the predicted error $\epsilon (t)$ represents the difference between the surveillance signal $s_{\mathrm{sur}}(t)$ and the filter output. The estimate of the expected signal $y(t)$ is

$$y(t)={\displaystyle \sum _{i=1}^M{w}_i}{s}_{\mathrm{ref}}(t-i+1)={\displaystyle \sum _{i=1}^M{w}_i}{A}_{\mathrm{R}}{s}_T(t-r_L/c-i+1)+n(t)$$

(28)

where $n(t)$ denotes noise. $w_i$ is the coefficient of the filter and is iteratively adjusted as

$$y(t)=w^T(t)x(t)$$

(29)

$$\epsilon (t)=s_{\mathrm{sur}}(t)-y(t)$$

(30)

$$w(t+1)=w(t)+u_{NLMS}{\displaystyle \frac{x(t)}{\left|\right|x(t)|{|}^2+e_{const}}}{\epsilon }^{*}(t)$$

(31)

where $w(t)={[w_1(t),w_2(t),...,w_M(t)]}^T$ is the filter coefficient vector at time $t$. The input of the filter is $x(t)={[s_{\mathrm{ref}}(t),\hspace{0.33em}{s}_{\mathrm{ref}}(t-1),\hspace{0.33em}{s}_{\mathrm{ref}}(t-M+1)]}^T$. $u_{NLMS}$ is the step size. $e_{const}$ is a small constant. $M$ is the filter order.

The NLMS process operates sample-by-sample in the time domain. Consequently, the block LMS (BLMS) algorithm, which is implemented in the frequency domain, was developed [30].

In BLMS filtering, input data is divided into blocks. The filter calculates the mean-square error gradient for each block, with the iterative process defined as follows:

$$w(k)=w(k-1)-{\mu }_B\mathrm{cov}(\epsilon (k-1),x(k-1))$$

(32)

$$\epsilon (k)=d(k)-\mathrm{cov}(x(k),w(k))$$

(33)

where $x(k)$ denotes the partitioned reference data block. $\epsilon (k)$ is the predicted error data block, and $d(k)$ is the surveillance data block. The convolution operation (denoted $\mathrm{cov}()$) implemented in the frequency domain as follows: First, transform $x(k)$ and the filter weights $w(k)$ to the frequency domain to obtain $\mathit{X}(k)$ and $\mathit{W}(k)$. Next, compute their element-wise product. Finally, transform the result back to the time domain.

The BLMS method divides data into blocks and leverages frequency-domain implementation, reducing computational complexity. However, both NLMS and BLMS suffer performance degradation when clutter is Doppler-shifted with Doppler spread, necessitating modified approaches.

3.2. Proposed Adaptive Filtering

3.2.1. DFS-NLMS Filtering

Considering Doppler-shifted clutter, the surveillance signal after down conversion is modeled as

$$\begin{array}{ll}{s}_{\mathrm{sur}}(t)& =s_{t\arg et}(t)+s_{\mathrm{clutter}}(t)+n_1(t)\\ & ={\displaystyle \sum _{j=1}^{N_t}{A}_j{s}_{ref}(t-{\tau }_j)\exp (j2\pi f_{\mathrm{d},\mathrm{T},j}t)}+{\displaystyle \sum _{i=1}^{N_c}{A}_{\mathrm{c},\mathrm{i}}^{\prime }{s}_{ref}(t-{\tau }_{clutte,i}(t))\exp (-j2\pi f_{d,clutter,i}(t)t)}+n_1(t)\end{array}$$

(34)

where $s_{t\arg et}(t)$ denotes the target signal. $N_t$ is the number of targets. $A_j$ is the amplitude of the $j$th target. $f_{\mathrm{d},T,j}$ is the Doppler frequency of the $j$th target. ${\tau }_j$ is the time delay of the $j$th target and $n_1(t)$ is the noise. Accounting for Doppler shifts, the filter output in Equation (28) can be rewritten as

$$\begin{array}{ll}y(t)& ={\displaystyle \sum _{i=1}^M{w}_i}\exp (j2\pi f_{\mathrm{d},i}t)s_{\mathrm{ref}}(t-i+1)\\ & ={\displaystyle \sum _{i=1}^M{w}_i}{A}_{\mathrm{R}}\left[s_{\mathrm{T}}(t-L_{\mathrm{R}}/C-i+1)\exp (j2\pi f_{\mathrm{d},i}t)\right]+n(t)\end{array}$$

(35)

where $M$ denotes the filter order. $f_{\mathrm{d},i}$ is the predefined Doppler shift used to modify the NLMS algorithm, giving the Doppler-Frequency-Shifted NLMS method (DFS-NLMS). The filter coefficient vector at time $t$ is denoted $w(t)={[w_1(t),w_2(t),...,w_M(t)]}^T$.

DFS-NLMS adjusts the coefficients $w_i$ to approximate $A_{\mathrm{c},\mathrm{i}}^{\prime }{s}_{ref}(t-{\tau }_{clutte,i}(t))$ in Equation (34). For the input signal $x(t)=$${[s_{\mathrm{ref}}(t),\hspace{0.33em}{s}_{\mathrm{ref}}(t-1),\hspace{0.33em}{s}_{\mathrm{ref}}(t-M+1)]}^T$ to the DFS-NLMS filter, the Doppler shift is estimated for each clutter range cell.

Thus, the iterative coefficient adjustment follows:

$$FD(t)=[\exp (j2\pi f_{d,1}t)\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\exp (j2\pi f_{d,1}(t-{\displaystyle \frac{1}{f_s}}))\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\exp (j2\pi f_{d,3}(t-{\displaystyle \frac{2}{f_s}}))\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\dots \hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\exp (j2\pi f_{d,M}(t-{\displaystyle \frac{(M-1)}{f_s}}))].$$

(36)

Therefore, the iterative filter adjustment is as follows:

$$\epsilon (t)=s_{\mathrm{sur}}(t)-y(t)$$

(37)

$$y(t)=w^T(t)\left[x(t)\circ FD(t)\right]$$

(38)

$$w(t+1)=w(t)+u_{FDNLMS}{\displaystyle \frac{x(t)\circ FD(t)}{\left|\right|x(t)\circ FD(t)|{|}^2+e_{const}}}{\epsilon }^{*}(t)$$

(39)

where $\epsilon (t)$ denotes the predicted error. $\circ$ represents the Hadamard product, and $u_{FDNLMS}$ is the step size. The DFS-NLMS filter structure is shown in Figure 6.

Two methods exist for estimating $FD(t)$: one detects the Doppler shift per range cell in the CAF, while the other estimates it using Equations (25) and (26), as illustrated in Figure 5. By incorporating Doppler compensation, DFS-NLMS achieves faster convergence around Doppler-shifted clutter. Section 4 evaluates the method’s performance, and a frequency-domain implementation is discussed to enhance computational efficiency.

3.2.2. DFS-BLMS Filtering

In this section, a Doppler-frequency-shifted BLMS (DFS-BLMS) method is proposed to enhance the computational efficiency of DFS-NLMS. The processing procedure is as follows:

Step 1: Construct the Doppler expanded reference signal matrix.

Let $s_{ref}=A_R{s}_T(n),0\le n\le N-1$ denote the reference signal.

$$s_{ref}=A_R{[s_T(0),s_T(1),s_T(2),\dots ,s_T(N-1)]}^T$$

(40)

where $N$ is the number of sampling points of the reference signal. The Doppler shift matrix for the $j$th range cell is

$$F_j=\left[\begin{array}{cccc}1& 0& \dots & 0\\ 0& \exp (j2\pi f_{\mathrm{d},j})& \dots & 0\\ ⋮& ⋮& \ddots & ⋮\\ 0& 0& \dots & \exp (j2\pi f_{\mathrm{d},j}(N-1))\end{array}\right]$$

(41)

where $f_{\mathrm{d},j}$ represents the Doppler shift in the $j$th range cell. The estimation of $f_{\mathrm{d},j}$ follows the same method as in DFS-NLMS. Therefore, the expanded reference signal is

$$S_{ref,fd}=\left[s_{ref}{F}_1;s_{ref}{F}_2;\dots ;s_{ref}{F}_j;\dots ;s_{ref}{F}_M\right]$$

(42)

Step 2: Partition signals into blocks.

Divide the expanded reference signal into blocks:

$$S_{\mathrm{ref},fd,l}=\left[s_{ref,l}{F}_{1,l};s_{ref,l}{F}_{2,l};\dots ;s_{ref,l}{F}_{M,l}\right]$$

(43)

$$\hspace{0.33em}{s}_{ref,l}=s_{\mathrm{ref},l}(n+(l-1)\times N_{\mathrm{FD}-\mathrm{BLMS}}),\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}l=1,2,\dots ,M_{\mathrm{FD}-\mathrm{BLMS}}$$

(44)

$$F_{j,l}=\left[\begin{array}{cccc}\exp (j2\pi f_{\mathrm{d},j}((l-1)N_{\mathrm{FD}-\mathrm{BLMS}}))& 0& \dots & 0\\ 0& \exp (j2\pi f_{\mathrm{d},j}((l-1)N_{\mathrm{FD}-\mathrm{BLMS}}+1))& \dots & 0\\ ⋮& ⋮& \ddots & ⋮\\ 0& 0& \dots & \exp (j2\pi f_{\mathrm{d},j}((l-1)N_{\mathrm{FD}-\mathrm{BLMS}}+N_{\mathrm{FD}-\mathrm{BLMS}}-1))\end{array}\right].$$

(45)

Partition the surveillance signal $s_{sur}=s_{\mathrm{sur}}(n),0\le n\le N$ into blocks:

$$s_{\mathrm{sur},l}(n)=s_{\mathrm{sur},l}(n+(l-1)N_{\mathrm{FD}-\mathrm{BLMS}})\hspace{0.33em}$$

(46)

where $\hspace{0.33em}l=1,2,...,M_{\mathrm{FD}-\mathrm{BLMS}}$.$M_{\mathrm{FD}-\mathrm{BLMS}}$ is the number of blocks. $N_{\mathrm{FD}-\mathrm{BLMS}}$ is the block length.

Step 3: Adaptive coefficient estimation.

Define the DFS-BLMS coefficients as $w=[w_1,w_2,\dots ,w_j,\dots ,w_L].$ Hence,

$$w_j(l)=w_j(l-1)-{\mu }_j\mathrm{cov}(\epsilon (l-1),s_{ref,l}{F}_{j,l}),\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}l=1,\dots ,L_{\mathrm{r}}$$

(47)

$$\epsilon (l)=y(l)-{\displaystyle \sum _{j=1}^L\mathrm{cov}(s_{ref,l}{F}_{j,l},w_i(l))}$$

(48)

where $L_{\mathrm{r}}$ is the iteration count. Typically, $L_{\mathrm{r}}=M_{\mathrm{FD}-\mathrm{BLMS}}$. The prediction error vector at the $j$th iteration is $\epsilon (l)$ and $y(k)$ is the filter output vector. The step size ${\mu }_j$ controls adaptation. $\mathrm{cov}()$ is the convolution process, which is implemented in the frequency domain, like standard BLMS.

The proposed DFS-BLMS achieves faster convergence near Doppler-shifted clutter and may reduce computation versus DFS-NLMS. Performance is analyzed in Section 4.

3.3. Identification and Removal of Clutter Residue

During clutter suppression, the primary objective is to reduce the weak targets’ masking effect. Thus, filtering may be applied only to near-range cells in passive radar systems. This approach typically preserves target detection capability in terrestrial passive radar (using fixed illuminators), as clutter concentrates near zero Doppler and is easily identifiable.

However, LEO satellite-based passive radar experiences Doppler-shifted clutter, making clutter residue separation from moving targets more challenging. This section designs a digital filter to suppress residual clutter.

The filter is designed using Doppler shift and Doppler spread parameters derived in Equations (25) and (26) for each range cell. A Butterworth filter is employed:

$${\left|H(j\omega )\right|}^2={\displaystyle \frac{1}{1+{(\omega /{\omega }_c)}^{2n}}}$$

(49)

where ${\omega }_c$ is the normalized cutoff frequency. $\omega$ is the normalized angular frequency. $n$ is the filter order.

$${\omega }_c=[f_{d,clutter,\min },f_{d,clutter,\max }]/f_{d,\max }$$

(50)

where $f_{d,clutter,\min }$ and $f_{d,clutter,\max }$ denote the minimum and maximum Doppler frequencies of clutter for each range cell, while $f_{d,\max }$ represents the maximum clutter Doppler frequency in the CAF.

3.4. Moving Target Detection

Moving target detection is performed using constant false alarm rate (CFAR) processing after the identification and removal of clutter residue. Let $Z$ denote the background level estimate

$$Z={\displaystyle \frac{1}{N_{\mathrm{CFAR},\mathrm{r}}{N}_{CFAR,fd}}}{\displaystyle \sum _{j=1}^{N_{CFAR,r}}{\displaystyle \sum _{i=1}^{N_{\mathrm{CFAR},\mathrm{fd}}}{\chi }_{i,j}}}>T_{CFAR}$$

where $N_{\mathrm{CFAR},\mathrm{r}}$ and $N_{\mathrm{CFAR},\mathrm{fd}}$ represent the number of reference cells along range and Doppler dimensions, respectively, in the CAF. ${\chi }_{i,j}$ denotes the CAF value at the Doppler cell and $j$th range cells. $T_{CFAR}$ is the detection threshold.

4. Simulation Results

4.1. Clutter Simulation

This section presents simulation analyses of clutter characteristics in LEO satellite-based passive radar systems and evaluates the performance of proposed clutter suppression methods. The LEO satellite constellation parameters are listed in Table 1, while simulated signal parameters appear in

Table 2. The simulation parameters.

Parameters
Carrier frequency	11.575 GHz
Bandwidth	250 MHz
Sampling rate	640 MHz
Noise power	−119.9978 dBw
Signal power of target 1	−177.6624 dBw
Velocity of target 1	300 m/s
Signal power of target 2	−172.8309 dBw
Velocity of target 2	800 m/s
Clutter power	−104.0425 dBw
Baseline length	550.16 km
Integration time	0.1 s

.

Section 2.4 discusses clutter characteristics. Doppler shift and Doppler spread correlate with clutter azimuth. Since most LEO internet satellites operate in Ku/Ka bands, their antenna beam width is typically narrow. Consequently, clutter signals are constrained to azimuth angles between −30 and 30 degrees. Figure 5b,d illustrate the Doppler frequency shift, while Figure 7 presents simulated clutter distributions. In this scenario, the clutter broaden is not very severe owing to the narrow beam of the Ku band antenna.

4.2. Clutter Suppression Results

The proposed DFS-NLMS and DFS-BLMS are applied to suppress clutter simulated in Section 4.1. Figure 8 presents DFS-NLMS results.

The step size of DFS-NLMS is chosen as 0.1, achieving a clutter suppression ratio (CSR) of 14.7003 dB. Most clutter has been removed, revealing previously masked weak target peaks in Figure 8b that were obscured by clutter side lobes, verifying DFS-NLMS effectiveness. In the figures, the Z-axis indicates the SNR of the target peak. Targets above 13 dB can be detected with a detection probability of 90% and a false alarm probability of 10⁻⁶. The target peaks are masked by the noise before cancelation. Figure 9 shows DFS-BLMS results with a block size of 2000 and step size $1.2\times {10}^{12}$ yielding a CSR of 14.4358 dB. The performance reduction compared to DFS-NLMS results from block processing effects.

However, both DFS-NLMS and DFS-BLMS demonstrate superior performance compared to the NLMS, BLMS, and CLEAN methods. Figure 10 compares the mean CAF values per range cell after clutter suppression, while

Table 3. CSR comparison using different suppression methods.

Algorithm	CSR (dB)
CLEAN	12.2171 dB
NLMS	14.0956 dB
DFS-NLMS	14.7003 dB
BLMS	13.8976 dB
DFS-BLMS	14.4358 dB

presents CSR.

The CLEAN method required 8000 iterations. NLMS and DFS-NLMS used filter order 2000; BLMS and DFS-BLMS employed block length 2000. The step sizes of NLMS filtering and BLMS filtering are 1.2 and $1.6\times {10}^{13}$, respectively. DFS-NLMS and DFS-BLMS outperform conventional methods by compensating for clutter Doppler shift. Although DFS-NLMS achieves the highest CSR (Table 3), its sample-by-sample processing incurs high computational complexity. Conversely, DFS-BLMS maintains competitive suppression while reducing processing time versus DFS-NLMS (

Table 4. Processing time comparison using different methods without parallel processing.

Algorithm	Processing Time (s)
CLEAN	55,765
NLMS	4304.1
DFS-NLMS	8510.1
BLMS	13.8976
DFS-BLMS	48.3641

). The signal length is $64\times {10}^6$ points. The test is taken using a 3.10 GHz CPU (single-threaded).

While BLMS is the computationally fastest method, DFS-BLMS provides optimal balance, retaining suppression performance. Thus, DFS-BLMS is recommended for LEO satellite-based passive radar. Adaptive filtering methods are iterative algorithms. BLMS and DFS-BLMS update M points for each iteration, while NLMS and DFS-NLMS update one point for each iteration. Therefore, BLMS and DFS-BLMS need fewer iterations than NLMS and DFS-NLMS. In Figure 11a, the mean squared error (MSE) is compared between NLMS and DFS-NLMS. Since the Starlink signal is noise-like for each burst, it is difficult to see the average variation. The average value of 200,000 points is displayed. Meanwhile, the MSEs of BLMS and DFS-BLMS are presented in Figure 11b. The MSE for each iteration is calculated as the average value of the M points (M = 2000 in this experiment). Then the average value of 100 points is displayed. It can be figured out that the DFS-NLMS and DFS-BLMS converge a little faster than NLMS and BLMS.

4.3. Target Detection Results

After near-range clutter suppression, target detection is performed. A moving target (210 m/s) is simulated with signal power ≈ −172.8311 dBw at range cell 30,437. Figure 12 shows the estimated Doppler shift.

The CAF in range cell 30,437 (Figure 13a) shows that the clutter peaks are difficult to separate from moving targets. A Butterworth filter, tuned to the estimated clutter Doppler shift, is therefore applied to suppress residual clutter. In Figure 13, residual clutter is identified and suppressed using the estimated clutter distribution. This enhances target detectability as shown in Figure 13b.

5. Conclusions

This paper analyzes the Doppler characteristics of clutter in LEO satellite-based passive radar using illuminators of opportunity. Clutter Doppler shift and spread can be estimated with satellite-ground geometry. Leveraging these estimates, we design DFS-NLMS and DFS-BLMS filters to suppress strong near-range clutter, mitigating target masking effects. Additionally, a Butterworth filter suppresses residual clutter to enhance moving target detection. Simulations demonstrate that DFS-NLMS achieves superior CSR versus CLEAN, NLMS, and BLMS. DFS-BLMS maintains competitive suppression while reducing computational load. The −30 to 30 degree clutter azimuth range accounts for high antenna directivity, with both methods performing effectively within these constraints. In LEO systems, sparse short-burst downlink signals may degrade suppression performance. Future work will address clutter suppression for such signal conditions.