Passive Radar-Based Parameter Estimation of Low Earth Orbit Debris Targets

Abstract

Major space agencies such as NASA and the ESA have long reported the growing dangers caused by resident space objects orbiting our planet. These objects continue to grow in number as satellites are imploded and space debris impacts each other, causing fragmentation. As a result, significant efforts by both the public and private sectors are geared towards enhancing space domain awareness capabilities to protect future satellites and astronauts from impact by these orbiting debris. Current approaches and standards implement very large radar arrays, telescopes, and laser ranging systems to detect and track such objects. These systems are very expensive, may take significant amounts of time to develop, and are still only sparingly able to efficiently track debris targets less than 10 cm in diameter. This work proposes a theoretical passive-radar-based method using illuminators of opportunity for detecting space debris while estimating motion direction and Doppler. We show that by using a signal processing chain based on the self-mixing technique and digital filters, Doppler information can be extracted and continuously tracked by a uniform linear receiver array. This can be achieved by a passive sensor system, which has the advantage of lower cost without the need to emit signals that constrain the spectrum sharing issues.

1. Introduction

Space situational awareness (SSA) was the traditional term that encompassed the knowledge of the dynamics of space debris objects, whether natural or man-made. SSA takes into consideration the past, present, and future characteristics of orbiting bodies, in addition to satellite capabilities and the space weather environment [1]. The space environment poses a constant and increasing threat to current and future satellites as the number of debris targets 1 cm or larger in diameter has surpassed 1.1 million, while current space surveillance networks actively track less than 37,000 targets [2]. As a result, substantial effort is required to increase SSA capabilities to achieve a more predictable space environment. The primary goal is collision avoidance, as resident space objects (RSOs) maintain speeds of about 8 km/s [3]. The kinetic energy contained in these low radar cross-section (RCS) missiles is enough to critically damage or decommission satellites and astronauts. Modern society depends on satellite capabilities for Global Positioning System (GPS) positioning, communications, and earth surveillance. The United States military has long recognized the importance of satellites and the overall space environment to the economic and cyber-safety of the country. As a result, the United States Space Command (USSPACECOM) and the US Space Force (USSF) were established in 2019 following the acknowledgment of space as a domain of warfare similar to air, land, and sea [4]. Since space was now seen as a domain of warfare, the SSA term was retired in favor of Space Domain Awareness (SDA). Both terms are still used interchangeably, but it is generally understood that surveillance techniques and technologies are significant to economic, cyber, and physical safety as a planet.

The importance of SSA is also recognized on a global scale. The Inter-Agency Space Debris Coordination Committee (IADC) was the name adopted for the unification of several space debris working groups in 1993 [5]. Currently, there are more than 13 agencies spread across 12 member nations [6] tasked with facilitating worldwide coordination with respect to space debris issues. A few years after the committee formation, the United Nations published the Technical Report on Space Debris [7], which highlighted debris measurement techniques, radar modalities, the space operational environment influence of the debris, environment modeling, and mitigation measures. The IADC has several working groups addressing these areas. The IADC Mitigation Guidelines [8] highlight the need for collision avoidance. The document highlights that public orbital states provided by the SSN are inadequate for maneuver determinations. The methods described in this paper are intended to support such efforts.

Modern efforts towards tracking space debris objects rely on billion-dollar radar arrays and optical telescopes. The United States Space Surveillance Network (SSN) is composed of over 30 ground-based systems spread across five continents, working in conjunction with six satellites [9]. Even with this extent of technological footprint, the catalog consists of less than 30,000 objects greater than 10 cm in diameter. The European counterpart to the SSN is the European Union Space Surveillance and Tracking (EU SST) framework established in 2014 [10] tasked with the objectives of collision avoidance, re-entry analysis, and fragmentation analysis. This network assimilates data across 51 sensors consisting of radars, such as the Bistatic Radar for LEO Survey (BIRALES) [11] and the Tracking and Imaging Radar (TIRA) [12], telescopes, such as the Joan Oró Telescope (TJO) [13], and light detection and ranging systems (LIDAR’s), such as the Matera Laser Ranging Observatory (MLRO) station [14]. The BIRALES system, in particular, employs a single 410–415 MHz transmitter and a 2800 m² receiver array [11]. Whereas this system has demonstrated space debris detection capability, the observation volume illuminated is still limited, and the RCS is dependent on backscatter in the Rayleigh region.

Due to the increased demand for SDA technologies, the private sector has become involved. LeoLabs (Menlo Park, CA, USA), for example, has built several S-band phased array (electronically beamformed) radars in North and Central America, Europe, Australia, and New Zealand [15]. The Kiwi Space Radar is the New Zealand-based array that became operational in 2019 and was designed to detect targets as small as 2 cm in diameter.

Previous research efforts propose passive bistatic systems for imaging and tracking space debris [16]. In this study, the Murchison Widefield Array (MWA) in Western Australia can be additionally tasked to receive commercial FM reflections from the moon. The processing methods were shown to have high feasibility with some drawbacks due to the radar configuration and electromagnetic phenomena. The relatively large wavelength of FM signals incites Rayleigh scattering for smaller debris targets, which may result in significantly smaller RCS values. In addition, the debris motion results in RCS and radar geometry variation, causing decreased image quality. This results in reduced performance for targets less than 1 m in diameter.

Another novel approach involves the combination of active and passive radar modalities [17]. Logically, employing the benefits of both radar modalities would be ideal. However, this relies on the concept of sensor fusion, which introduces additional complexity to the signal processing, time synchronization, and sources of error. A point of concern noted in [17] was that the bistatic plots often have no correspondence with active plots for the same debris target.

It is obvious from the examples listed above that there are many techniques employed in pursuit of effective SDA. The common theme is to build or repurpose sensors across the globe to maximize orbital coverage. However, these designs come at significant time and capital costs. The arrays are large and require years to design and build. The problem of spectrum sharing must also be mentioned. Due to advancements in imaging, communications, navigation, and handheld devices, there is significant demand for operation over the same frequency bands. This has forced the need for radio spectrum frequency allocations [18]. The spectrum constraint has inspired research into adaptive frequency-hopping techniques [19,20,21,22].

This work proposes a forward scatter passive bistatic radar processing chain purposed towards enhancing SDA efforts through Doppler extraction. Most operational radars are active, i.e., they both transmit and receive signals. Passive radars are receive-only systems that exploit transmitters that are already available and in use for other applications, thereby allowing for several benefits [23]. The first benefit is time and cost. Only a receiver module or station must be designed and built. The next benefit is the lack of transmitted signals and the cost of generating these. This alleviates the issue of spectrum sharing, earning passive radar the title of ‘Green Radar’ [24] as the operation depends solely on illuminators of opportunity (IOOs), which are signals already present in the environment. An added advantage of green radar is its covert operation. Receive-only systems are generally impossible to detect and jam. The third benefit is the coverage. Since bistatic system performance is also dependent on the IOO [25], one can be chosen to maximize coverage. In this paper, it is proposed that the IOO be the L1 coarse/acquisition (C/A) GPS signal, which oscillates at 1575.42 MHz. This signal illuminates about 38% of the earth’s surface at any given instant [26], allowing receivers to be networked across continents. As a result of the GPS satellite altitude, the radar geometry is virtually constant, leading to consistencies in RCS values and overall system performance. Lastly, the bistatic configuration can allow for a larger target radar cross-section depending on the bistatic angle—typically $180°$. This configuration is known as forward scatter radar (FSR) [27].

This work explores Doppler extraction through self-mixing processing [28] and filtering of the received direct and target-reflected signals. The bistatic baseline is defined as the distance between the transmitter and receiver [23]. At the time of baseline crossing, the direct and reflected signals arrive at the receiver at virtually the same time. As a result, zero Doppler is detected. However, additional receivers at discrete separation distances from the initial receiver will extract discrete Doppler values, allowing for unique target identification. Previous FSRs implementing the self-mixing technique have been designed with a 4G Long Term Evolution (LTE) IOO [29] and an E-band frequency modulated continuous wave (FMCW) transmitter [30]. These cases, however, only looked at single targets. This paper will explore multiple targets in conjunction with multiple receiver elements.

This article is outlined as follows. A review of Mie scattering, bistatic radar geometry, and forward scatter RCS (FS-RCS) will be presented in Section 2. Next, the mathematical and time-frequency behavior of a baseline crossing event (BCE) will be explored in Section 3. Section 4 will describe the array geometry and self-mixing processing towards Doppler extraction. Section 5 will then present the simulation design parameters and a discussion of the results. Section 6 will describe the effects of noise and interference. Finally, the concluding remarks will be presented in Section 7.

2. Passive Bistatic Radar and Mie Scattering

Radars primarily fall under two configurations: monostatic and bistatic. In the monostatic configuration, the transmitter and receiver are either the same antenna or are co-located. Bistatic radars result from the separation of the transmitter ($T_x$) and receiver ($R_x$) modules by some significant distance, as seen in Figure 1.

In Figure 1, $R_{Tx}$, $R_{Rx}$, and $L$ represent the transmitter-target, receiver-target, and bistatic baseline distances, respectively. The angle subtended at the target ($t_g$) by these distances is known as the bistatic angle $\beta$, while $T^{\theta }$ and $R^{\theta }$ are the transmitter and receiver look-angles with respect to a defined vertical direction or North.

2.1. Forward Scatter Radar

The forward scatter radar (FSR) configuration is achieved by modifying the bistatic geometry to where $\beta \approx 180°$. At this point, $T_x$ and $R_x$ are virtually diametrically opposed, and scattering based on Babinet’s principle becomes dominant. This principle states that the scattering characteristics from a target are equivalent to the radiation from a target silhouette-shaped aperture contained in an infinite perfect electrical conductor (PEC) plane [25]. Essentially, there is some signal blockage or electrical shadowing that disrupts the direct signal. This will be mathematically explored in Section 3.

As a result of this phenomenon, the FS-RCS and scattering beamwidth are simple expressions that depend only on the target dimensions and incident wavelength as described in Equations (1) and (2)

$${\sigma }_{FS-RCS}=4\pi {\left({\displaystyle \frac{A}{\lambda }}\right)}^2=4\pi {\displaystyle \frac{{\pi }^2{r}_{tg}^4}{{\lambda }^2}}$$

(1)

$${\sigma }_{bw}\approx {\displaystyle \frac{\lambda }{d}}$$

(2)

where $r_{tg}$ is the target radius, $A$ is the target silhouette area, $\lambda$ is the incident or operational wavelength, and $d$ is the largest dimension of the target, which is usually taken to be the diameter. From Equation (2), it is of note that the larger the wavelength is compared to the target, the larger the scattering beamwidth. This contributes to maximizing the coverage area of target reflections for space debris with diameters less than 10 cm.

2.2. Forward Scattering Phenomenology

FSR depends on the main lobe of scattering power propagating along the same direction as the incident wave, i.e., the forward direction. In general, significant FSR scattering behavior is observed from harmful space debris typically occupying the Mie and optical scattering regions depending on the size of the target radius in comparison to the incident wavelength [31]. Mie region scatter is defined as that which occurs over the range $0.1\lambda \le r_{tg}\le 2\lambda$, while scattering over the range $r_{tg}>2\lambda$ occupies the optical region. Therefore, if choosing the L1 C/A GPS signal to be our IOO with a wavelength of 19 cm, then FSR will be optimal for targets as small as 3.8 cm in diameter. This lower bound would significantly enhance global SDA efforts as the scattering energy is maximized towards the receiver. As a result, the signal-to-noise ratio (SNR) is improved.

The Mie region FS-RCS and scattering beamwidth provide additional benefits to target detectability and signal processing. Compared to the bistatic scatter cross sections (B-RCS), which occur at $\beta \approx 90°$, it was shown that the FS-RCS could be 4 to 22 dB larger depending on the complexity of the target shape [32]. This also holds true in comparison to the monostatic case for target diameters $\ge$ 1.5$\lambda$. Larger RCS values increase SNR and heighten correlation peaks. A Mie region pattern study depicts scattering curves with half-power beamwidths (HPBW) increasing inversely to relative target size [33], which further supports the approximation in Equation (2). In addition, it was also shown that FS-RCS was higher compared to the B-RCS for spheres of several sizes ranging from the Mie to the optical regime [33]. It is evident, therefore, that the FS-RCS phenomenon would be beneficial to the detection of the smaller targets that have proven difficult to track.

3. The Baseline Crossing Event

In FSR, the detection scheme is dependent on the modulation of signal amplitude that occurs as the target approaches, crosses, and departs the baseline. We will title this occurrence as a baseline crossing event (BCE). In the following sections, we explore the signal characteristics of a BCE in the time and frequency domains.

3.1. BCE Signal Model

A baseline crossing signal model was presented in a previous characteristic study [34]:

$$s\left(t\right)=A_{BCE}\left(t\right)\cos \left(\varphi -{\displaystyle \frac{2\pi f_0}{c}}{R}_{\beta }\right)$$

(3)

For a received signal, $s\left(t\right)$, $A_{BCE}\left(t\right)$ is the scattering signal envelope, $\varphi$ represents the scattering phase, $c$ is the speed of light, $f_0$ is the operating frequency, and $R_{\beta }$ is the bistatic range. Generally, $R_{\beta }=R_{Tx}\left(t\right)+R_{Rx}\left(t\right)-L\left(t\right)$ [35]. The derivation of the scattering envelope, $A_{BCE}\left(t\right)$ is detailed in a Doppler phenomenology study [28]. Figure 2 resulted from minor changes to the distance measurements in [28], as the baseline midpoint is not considered in this study. The distances are determined by Equations (4) and (5):

$$T_{\alpha }^{\theta }\left(t\right)={\tan }^{-1}\left({\displaystyle \frac{x\left(t\right)}{L\left(t\right)-y\left(t\right)}}\right)$$

(4)

$$R_{\alpha }^{\theta }\left(t\right)={\tan }^{-1}\left({\displaystyle \frac{x\left(t\right)}{y\left(t\right)}}\right)$$

(5)

In these equations, $T_{\alpha }^{\theta }\left(t\right)$ and $R_{\alpha }^{\theta }\left(t\right)$ are the azimuthal look-angles from $T_x$ and $R_x$, respectively. $x\left(t\right)$ and $y\left(t\right)$ represent target position components in the bistatic plane, respectively. The bistatic plane is defined by the two-dimensional system in which $T_x$, $R_x$, and $L$ lie. Similar changes were adapted to the elevation look-angles.

Key simulation parameters were defined in [34], and adaptations were made to transmitter parameters to emulate the L1 C/A signal as defined in [36]. The ‘z’ variable shown in the legend represents the offset of the target’s center of body of mass and the bistatic baseline. The blue plot displays no offset, and the orange plot displays an offset of 20 m. In both cases, as the target approaches or departs from the baseline, there is a brief increase in signal power. At the crossing point, there is a prominent decrease in power. Therefore, the BCE is characterized by a ‘W’ or ‘U’ shaped waveform, which is easily seen on magnitude plots in low-noise environments or by correlation with an inverse Ricker wavelet [37].

3.2. BCE Doppler

Feature extraction often starts with Doppler analysis as it is inherently tied to the velocity of a target and/or signal source. Bistatic radar Doppler is described in Equation (6), given by

$$f_D={\displaystyle \frac{2v}{\lambda }}\cos \left(\delta \right)\cos \left({\displaystyle \frac{\beta }{2}}\right)$$

(6)

where $v$ is the target relative velocity, $\beta /2$ is the bistatic bisector angle (see Figure 1), and $\delta$ represents the angular difference between the target velocity vector and the bistatic bisector. As $\beta$ approaches $180°$, $\mathrm{c}\mathrm{o}\mathrm{s}(\beta /2)$ approaches $0$. This means that, at the crossing point, there is no path difference between the direct and reflected signals, i.e., the target lies along the baseline between $T_x$ and $R_x$ in the bistatic plane. This effect can be highlighted by taking the Short-Time Fourier Transform (STFT) of the BCE signal model, shown in Figure 3.

Figure 3 displays a similar magnitude modulation as Figure 2. It is shown that maximum power is reflected while the target is approaching and leaving the baseline with minimum power at the crossing point. Equation (5) is also confirmed, as there is no Doppler information at the crossing point. However, some unique frequency-spreading effects are noticed. The frequency harmonics appear to increase in power along with the signal magnitude. This effect follows a similar trend as the Doppler behavior due to a space debris target’s translational motion [38].

4. Self-Mixing Signal Processing

To obtain useful Doppler information for a particular target, an efficient signal processing scheme must be implemented along with multiple receivers in an array. Figure 4 displays a block diagram outlining the proposed processing steps.

The received signal must first pass through a low-noise amplifier (LNA) to increase the signal power. This step is especially necessary for GPS signals that are incident on the earth’s surface at −134 dBm [23]. Next, the square law detector (SLD) squares any input voltage. The low-pass filter (LPF) will then remove the center frequency. The high-pass filter (HPF) or direct current (DC) blocker will remove any DC frequencies. After the analog-digital converter (ADC) passes the signal to the host PC system, the Fast-Fourier Transform (FFT) can be performed to observe any Doppler shifts. The black-outlined boxes represent analog components, the red-outlined box represents the digital component, and the blue-outlined boxes represent processes that can be implemented either through analog or digital components. Such a system may be realized by a software-defined radio (SDR) in series with an LNA and SLD. The Ettus X310 (National Instruments, Austin, TX, USA), for example, has an onboard field programmable gate array (FPGA) [39] that may be tasked with signal acquisition to identify the space vehicle number(s) and satellite-induced Doppler shift(s). Afterward, the signals are downsampled and passed to the host PC for data binning and FFT processing.

4.1. Single Target Case

Starting with a single receive antenna and a single target, the received signal model can be described in Equation (7), given by

$$s_{Rx}\left(t\right)=A_{dir}\cos \left({\omega }_{0}t\right)+A_{ref}\sin \left(\left({\omega }_0+{\omega }_D\right)t\right)$$

(7)

where $A_{dir}$ is the direct signal amplitude, $A_{ref}$ is the signal amplitude reflected from the target, ${\omega }_0$ is the operational frequency, and ${\omega }_D$ is the Doppler shift due to the target’s motion. The SLD applies a transfer function responsible for squaring the input signal, resulting in Equation (8), given by

$$\begin{array}{l}{\left(s_{Rx}\left(t\right)\right)}^2=A_{dir}^2{\cos }^2\left({\omega }_{0}t\right)+2A_{dir}{A}_{ref}\cos \left({\omega }_{0}t\right)\sin \left(\left({\omega }_0+{\omega }_D\right)t\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+A_{ref}^2{\sin }^2\left(\left({\omega }_0+{\omega }_D\right)t\right)\end{array}$$

(8)

For simplification, the trigonometric power reduction and double-angle formulae can be applied to result in Equation (9).

$$\begin{array}{ll}{\left(s_{Rx}\left(t\right)\right)}^2& ={\displaystyle \frac{A_{dir}^2}{2}}+{\displaystyle \frac{A_{dir}^2}{2}}\cos \left(2{\omega }_{0}t\right)+{\displaystyle \frac{A_{ref}^2}{2}}-{\displaystyle \frac{A_{ref}^2}{2}}\cos \left(2{\omega }_{0}t+2{\omega }_{D}t\right)\\ & \hspace{1em}\hspace{1em} +2A_{dir}{A}_{ref}\cos \left({\omega }_{0}t\right)\sin \left({\omega }_{0}t\right)\cos \left({\omega }_{D}t\right)+A_{dir}{A}_{ref}\sin \left({\omega }_{D}t\right)\\ & \hspace{1em}\hspace{1em} +A_{dir}{A}_{ref}\left(\cos 2{\omega }_{0}t\right)\sin \left({\omega }_{D}t\right)\end{array}$$

(9)

In a practical application, Equation (9) may be scaled by some SLD factor [23]. For this work, we shall assume this value to be 1. Equation (10) contains the resulting expression after the signal is passed through the LPF:

$${\left(s_{Rx}\left(t\right)\right)}^2={\displaystyle \frac{A_{dir}^2}{2}}+{\displaystyle \frac{A_{ref}^2}{2}}+A_{dir}{A}_{ref}\sin \left({\omega }_{D}t\right)$$

(10)

We implemented an LPF at this stage as opposed to an HPF because it is expected that ${\omega }_0\gg {\omega }_D$. It has been shown that even with extremely high velocities, we can expect debris targets to produce Doppler shifts in the range of ±14 kHz [38]. Therefore, the LPF removes the terms containing ${\omega }_0$. The first two terms contain no frequency. These are the DC terms that can be removed either by an HPF or a DC-blocker. Equation (11) displays the resultant expression after the filters have been applied.

$$S\left(t\right)={\left(s_{Rx}\left(t\right)\right)}^2=A_{dir}{A}_{ref}\sin \left({\omega }_{D}t\right)$$

(11)

where $S\left(t\right)$ represents the Doppler expression after the processing stages have been completed. The Doppler frequency has been extracted and scaled by the product of the direct signal and reflected signal amplitudes.

4.2. Multiple Target Case

Including an additional target involves small modifications to Equation (7). Adding in terms for an additional magnitude and Doppler shift will result in Equation (12), given by

$$s_{Rx}\left(t\right)=A_{dir}\cos \left({\omega }_{0}t\right)+A_{ref,1}\sin \left(\left({\omega }_0+{\omega }_{D,1}\right)t\right)+A_{ref,2}\sin \left(\left({\omega }_0+{\omega }_{D,2}\right)t\right)$$

(12)

where $A_{ref,1}$, $A_{ref,2}$, ${\omega }_{D,1}$, and ${\omega }_{D,2}$ represent the signal amplitudes and respective Doppler shifts for targets 1 and 2. Following similar processing steps as in Section 4.1, the result is described in Equation (13), given by

$$S\left(t\right)=A_{ref,1}{A}_{ref,2}\cos \left(\left({\omega }_{D,1}-{\omega }_{D,2}\right)t\right)+A_{dir}\left[A_{ref,1}\sin \left({\omega }_{D,1}t\right)+A_{ref,2}\sin \left({\omega }_{D,2}t\right)\right]$$

(13)

Like the single target case, there are Doppler expressions scaled by the direct and reflected signal amplitudes. However, an additional sinusoid is also created with an oscillation frequency derived from the difference between the Doppler shifts of both targets. This signal is scaled by the magnitudes of the reflected signals. Therefore, it can be expected that this signal is extremely weak in comparison to the Doppler expressions and may be related to the distance between the targets [24]. However, since GPS signals contain Pseudorandom Noise (PRN) sequences that are modulated onto a continuous-wave carrier rather than a chirp signal [40], this may not be the case.

Equations (14) and (15) include the received and output expressions in a scenario involving three moving targets.

$$\begin{array}{ll}{s}_{Rx}\left(t\right)& =A_{dir}\cos \left(\left({\omega }_0\right)t\right)+A_{ref,1}\sin \left(\left({\omega }_0+{\omega }_{D,1}\right)t\right)+A_{ref,2}\sin \left(\left({\omega }_0+{\omega }_{D,2}\right)t\right)\\ & \hspace{1em}\hspace{1em}+A_{ref,3}\sin \left(\left({\omega }_0+{\omega }_{D,3}\right)t\right)\end{array}$$

(14)

$$\begin{array}{ll}S\left(t\right)& =A_{dir}\left(A_{ref,1}\sin \left({\omega }_{D,1}t\right)+A_{ref,2}\sin \left({\omega }_{D,2}t\right)+A_{ref,3}\sin \left({\omega }_{D,3}t\right)\right)\\ & \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+A_{ref,1}{A}_{ref,2}\cos \left(\left({\omega }_{D,1}-{\omega }_{D,2}\right)t\right)+A_{ref,1}{A}_{ref,3}\cos \left(\left({\omega }_{D,1}-{\omega }_{D,3}\right)t\right)\\ & \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+A_{ref,2}{A}_{ref,3}\cos \left(\left({\omega }_{D,2}-{\omega }_{D,3}\right)t\right)\end{array}$$

(15)

Equation (15) shows three (3) Doppler signals scaled by the direct and respective reflected amplitudes. In addition, there are three pairs of Doppler-difference signals scaled by the respective reflected amplitudes. Observing Equations (11), (13) and (15), it is apparent that, for $N$ targets, the signal processing chain will produce an expression containing ($N+\sum _{k=N}^{1}k-1$) terms—of which, $N$ are Doppler sinusoids and ${}^{N}C_2$ are Doppler-difference sinusoids.

5. Simulations

The steps detailed in Section 4 represent the signal processing stages leading up to the FFT. Simulations involving the targets, equations, and filters can be designed in MATLAB Version R2024b. The self-mixing and FFT results will be discussed in this section.

5.1. Self-Mixing

Figure 5a displays the single-snapshot FFT results for a self-mixing scenario in which two signals were generated according to Equation (7) where $A_{dir}=1$, $A_{ref}=0.35$, ${\omega }_0=1.575$ GHz, and ${\omega }_D=1.5$ Hz. As expected, a single peak at the Doppler frequency is displayed. In this case, $A_{dir}{A}_{ref}=0.38$, which is only 8% off the expected value of 0.35.

Figure 5b displays the results of a similar scenario with an additional target added. In this case, $A_{ref,2}=0.25$, and ${\omega }_{D,2}=2.5$ Hz. As expected from Equation (13), three peaks resulted: two peaks corresponding to the Doppler signals and the remainder corresponding to the Doppler-difference signal.

5.2. Moving Target over a Forward Scatter Receiver Array

Doppler information is present, provided the target is not at the baseline crossing point. Therefore, Doppler shifts will be present in the signal as the target approaches and departs the baseline.

5.2.1. Single Receiver Geometry

Figure 6 displays the geometry involved for a single receiver-transmitter pair and an approaching target. This scenario is considered to be in a single bistatic plane. Therefore, the perpendicular distance to the target from the baseline is equal to the horizontal distance along the ground. The transmitter and receiver look-angles and bistatic angles can then be derived and are presented in Equations (16)–(18).

$$\theta ={\tan }^{-1}\left({\displaystyle \frac{h}{D}}\right)$$

(16)

$$\gamma ={\tan }^{-1}\left({\displaystyle \frac{D}{L-h}}\right)$$

(17)

$$\beta =180°-(\gamma +\left(90°-\theta \right))$$

(18)

In the above equations, $h$ is the target altitude, $D$ is the horizontal distance along the ground, $\theta$ is the receiver look-angle, and $\gamma$ is the transmitter look-angle.

5.2.2. Linear Receiver Array Geometry

Figure 7 depicts the basic geometry of a uniform linear array setup for FSR. Each receiver module is unique, but we observe the L1 C/A signal from only a single transmitter. Since the GPS satellites are at a sufficiently high altitude (20,200 km [23]), the incident signal at each receiver module arrives at approximately the same phase. It was shown in [41] that by applying a first-order Taylor series expansion to the Pythagorean theorem, we have $h_2\approx h_1+{\displaystyle \frac{1}{2}}\left({\displaystyle \frac{d^2}{h_1}}\right)\approx h_1$ where $h_1$ is the vertical distance of the $T_x$ above $R_{x1}$, $d$ is the distance to $R_{x2}$, and $h_2$ is the hypotenuse distance to $T_x$.

The major benefit of implementing an array is persistent Doppler information. When the target reaches the crossing point of the first receiver, $R_{x1}$, the shift measurement at that time instance is 0 Hz. However, the other receivers will register a non-zero shift as $\beta$ is close to but not equal to $180°$. Therefore, Doppler can be extracted and tracked for the entire duration of the target path visible to the array.

The secondary benefit is some processing gain. An example scenario could be two receivers observing a single moving target. For a single snapshot, if the received signals were added and then filtered, the signal would be of the form in Equation (19).

$$y\left(t\right)=A_{ref,1}{A}_{ref,2}\cos \left(\left({\omega }_{D,1}-{\omega }_{D,2}\right)t\right)+2A_{dir}\left[A_{ref,1}\sin \left({\omega }_{D,1}t\right)+A_{sc2}\sin \left({\omega }_{sc2}t\right)\right]$$

(19)

This result is an improvement on Equation (13) as the magnitudes of the Doppler signals have doubled. Therefore, it can be expected that an $N$-element array will provide an $N$-fold increase in Doppler signal power. This behavior aligns with the $N$-fold improvement in SNR due to coherent integration [42].

5.2.3. FFT Results

The simulation parameters are displayed in

Table 1. Simulation Parameters.

Parameter	Value
$\mathrm{Velocity}, \nu$	$7.5$ km/s
$\mathrm{Altitude}, h$	400 km
$\mathrm{Baseline}, L$	20,200 km
$\mathrm{Incident} \mathrm{Wavelength}, \lambda$	0.1905 m
$\mathrm{Element} \mathrm{spacing}, d$	$\lambda /2$
$\mathrm{Horizontal} \mathrm{Distance}, D$	$\pm {10}^3\cdot \lambda /2$

. The FFT results are displayed in Figure 8.

Figure 8a highlights a linear trend of Doppler increasing as the target is further away from the baseline. At the crossing point, there is no Doppler, thus indicating a successful target detection. The graphs show absolute Doppler values. According to Equation (5), $\delta$ is positive as the target approaches the baseline and negative as the target departs the baseline. Therefore, asymmetry as the Doppler shifts from positive to negative is expected.

Figure 8b displays a similar trend as Figure 8a but over all four elements in the receiver array. The plot is zoomed in for clarity. In this simulation, the target is traveling leftward. Therefore, it arrives at $R_{x1}$ first and departs $R_{x4}$ last. This is obvious from the Doppler behavior as $R_{x1}$ sees a zero Doppler instance first. $\lambda /2$ m later, $R_{x2}$ sees the zero Doppler. This information can provide an initial estimate direction value for the target translational motion with respect to the receiver array.

5.2.4. Discrete Doppler Relationship

During the time instance while the Doppler at $R_{x1}$, $f_{D,R_{x1}}=0$ Hz, the Doppler at $R_{x2}$, $f_{D,R_{x2}}=\Delta f$. Furthermore, as highlighted by the black circles in Figure 8b, the Doppler values at the remaining receivers are $f_{D,R_{x3}}=2\Delta f$ and $f_{D,R_{x3}}=3\Delta f$. Therefore, a pattern of discrete Doppler shifts is recorded at subsequent receiver elements. This pattern can be summarized by Equation (20) when the reference receiver records zero Doppler shift. The shifts at the adjacent receivers are given by

$$f_{D,R_{xn}}=\left(n-1\right)\cdot \Delta f=\left(n-1\right)\cdot f_{D,R_{x2}}$$

(20)

Knowledge of the discrete Doppler behavior may assist data preparation and analysis processes as data periods with no BCEs may be removed.

6. Effects of Noise and Interference

In practical applications, the target environment and system components will degrade the processing performance by adding noise. Simultaneously, it is expected that transmissions from other GPS systems may interfere with desired signals and reduce SNR. In this section, we address concerns related to interference, thermal noise, and measurement errors.

6.1. Interference

As previously stated, the L1 C/A GPS frequency is 1575.42 MHz. It is assumed that the receiver system will be stationary. Therefore, the signal will undergo some Doppler shift due to the motion of the satellite. Typically, we can expect this value to be ≈±5 kHz [43]. Additional Doppler shift due to the orbital motion of the space debris can be ≈±14 kHz, depending on the altitude. Therefore, received signals can be bandpass filtered over a passband defined by 1575.42 $\pm$ 0.019 MHz.

The C/A modulation is a pseudorandom noise (PRN) code that uniquely identifies an L1 signal transmission. Through the signal acquisition process [43], the C/A codes can be replicated and convolved with the received signals to eliminate extraneous transmissions from other GPS satellites. A similar process is applied to other GPS-based systems, providing a range of IOOs from which to select.

Target Masking

It is possible that two targets may orbit with small separation distances between them. In a situation where one target is slightly higher in altitude than the other, the BCE at a single receiver may not be distinguishable from the BCE of one target. The resulting magnitude reduction and zero Doppler readings will look identical to that of Figure 2 and Figure 8a. Therefore, one target essentially masks the other since only one target is detected instead of two. This necessitates a receiver array, as when two targets are crossing the baseline over one receiver, the other receivers would still detect unique Doppler shifts. This suggests that a multiple-receiver setup would achieve higher detection rates. However, a formal analysis will be conducted in a later study.

6.2. Thermal Noise

In the operating environment, noise sources can include cosmic/galactic noise and solar noise [44]. However, since the L1 C/A center frequency is >1 GHz, there are negligible effects from galactic noise. In addition, solar noise is only significant when the receiver is pointed towards the sun. Therefore, in this case, the only noise to note is that derived from the internal components of the receiver system. i.e., thermal noise. While many techniques exist to reduce noise, such as averaging, windowing, and matched filtering [44], we will demonstrate the self-mixing performance gains by coherently summing the received inputs from array elements.

It was shown in Section 5.2.2 that the phase difference of the direct signal between the ULA elements is effectively zero. Therefore, if the signals are coherently summed, the amplitude of the direct signal is increased by the number of elements being summed. This, in turn, increases the amplitude of the filtered signal as there is dependence on both the reflected and direct signal amplitudes, as seen in Equation (19). Figure 9 displays the FFT performance in a severely noisy environment when receiver inputs from elements are added.

In Figure 9, a single target reflection is received at the first receiver ($R_{x1}$) with a Doppler value of ${\omega }_{D,1}$ = 1.5 Hz. The next two elements also receive some of the signal energy from the same target at slightly different Doppler values (due to bistatic geometry), which we have arbitrarily equated to 2.5 and 3.5 Hz. At the first receiver, the SNR present is −12.7 dB. The target Doppler is not discernable. However, after coherently summing signals from the subsequent elements, we achieve an SNR increase relative to the factor increase of $A_{dir}$, and the magnitude of the 1.5 Hz component is much greater than the noise level.

6.3. Frequency Estimation Errors

Frequency shift estimation errors are introduced by the receiver components and processing stages. In the case of analog filter components, multiple identical low-pass or high-pass filters may be installed in series to achieve more robust filtering of unwanted frequency content. As it pertains to signal processing, the FFT size should be sufficiently large to allow for frequency resolutions on the order of 10⁻² Hz to account for a continuous spectrum of speeds at which RSOs may travel. Too large of an FFT size may result in frequency ambiguation as the signal energy is spread across several FFT bins.

7. Conclusions

This work explored fundamental principles concerning the baseline crossing phenomenon for a forward scatter radar configuration. It was highlighted that a unique time–frequency behavior occurs, from which deductions about the target location in relation to a specific receiver can be made. Zero Doppler is accompanied by a magnitude reduction at the baseline crossing point. Maximum signal reflection magnitude occurs just before and after a baseline crossing event. Maximum Doppler is displayed at the visible points furthest from the receiver. In addition, the self-mixing signal processing technique was investigated and shown to be an efficient method for extracting Doppler information when utilized in conjunction with analog or digital filters and the FFT. Doppler extraction is maximized when using a linear array. When one element records zero Doppler, the other elements will record discrete Doppler differences. Future efforts will be towards target localization and harnessing the Doppler and direction information to build space debris angular tracks.