A Cross-Track Interferometric Synthetic Aperture 3D Passive Positioning Algorithm

Abstract

High-precision, robust, and rapid three-dimensional (3D) passive positioning of the radiation source is critical for modern reconnaissance systems. While synthetic aperture technology has advanced 2D passive positioning performance, existing methods fail to achieve full 3D positioning with sufficient accuracy and computational efficiency. This is because of the inherent limitations of the single-station platform in resolving elevation-angle ambiguity. To address this gap, we propose a Cross-Track Interferometric Synthetic Aperture (CISA) 3D passive positioning algorithm. The algorithm innovatively realizes robust elevation-angle measurement by recursively deriving the long baseline unambiguous phase difference step-by-step from a virtual short baseline. The 3D positioning is achieved by combining passive synthetic aperture and interferometric angle measurement. Furthermore, we establish the incoherence model of synthetic aperture passive positioning for the first time and propose a compensation method based on static acquisition data to improve the practicability of CISA. Simulation and experimental results demonstrate that the proposed CISA algorithm achieves a positioning accuracy of 4.73‰R, improves computational efficiency by 1–2 orders of magnitude compared to conventional methods, and exhibits superior robustness to noise. The research can provide a reference for the method research and engineering realization of synthetic aperture 3D passive positioning.

1. Introduction

Single-station passive positioning is a widely concerned technology and has been applied in many fields due to its advantages in flexibility, system simplicity, and low cost [1,2,3,4]. It does not require a high-power transmitting device and is more conducive to achieving lightweight and miniaturization of the system compared to active systems. Unlike the distributed system, it does not require a data link or synchronization device. This eliminates the positioning accuracy loss caused by synchronization errors between nodes.

Single-station passive positioning is a technology that only uses a single platform to observe and locate the source target. It first obtains the observation data at different positions and then combines the differences in the information received at different positions to achieve passive positioning. Single-station passive positioning algorithms can be divided into two categories. One is the two-step positioning method based on parameter estimation [5,6,7]. For example, the classical BO (Bearing-Only Positioning) algorithm is achieved through angle measurement [6]. The advantages of a two-step positioning method are the simplicity of the algorithm and low computational complexity. However, it also has two main shortcomings. One is that the observation data from different angles are incoherent, and there are multiple initial errors in multiple observations, resulting in limited positioning accuracy. The other point is that in the two-step method, the positioning accuracy is highly sensitive to the geometric structure formed by multiple observation positions. And a larger observation angle is needed to achieve higher positioning accuracy. This results in the need for long-distance movement of the moving platform, slowing down the positioning speed.

The other type of single-station passive positioning algorithm is Direct Position Determination (DPD) [8,9,10], also known as the single-step positioning method. The core idea is similar to Matched Field Processing (MFP) in the field of Underwater Signal Processing [11,12]. It can be applied in both distributed systems and single-station systems. For instance, the DPD-MUSIC algorithm [13] and the DPD-Capon algorithm [9] are commonly used. DPD avoids information loss and has better robustness to noise. Many scholars have already carried out in-depth research on performance improvement methods [14,15], special signal direct positioning [16,17], array design [18,19], and other issues related to DPD. Regrettably, traditional DPD algorithms require a 3D grid search, which incurs a high computational resource overhead and leads to slow positioning.

Under the background, synthetic aperture passive positioning (SAPP) technology emerges as a computationally efficient single-station DPD method. Some scholars have first introduced the principle of synthetic aperture into the passive positioning of acoustic frequency band [20,21,22], that is, Passive Synthetic Aperture Sonar (PSAS) technology. The synthetic aperture passive positioning experiment proves that PSAS can effectively improve the detection ability of weak target signals.

In recent years, SAPP technology has been introduced into the microwave frequency band. It is mainly used for passive reconnaissance and positioning of radar or communication targets, as well as search and rescue applications. First, it overcomes the need for large observation angles required by the first kind of algorithms and is more conducive to the accurate positioning of the time-sensitive target and the occasional target. Additionally, it uniquely addresses three critical limitations of conventional DPD approaches: (1) Positioning accuracy: it uses the Doppler effect and pulse compression principle to make the positioning accuracy no longer limited by the grid accuracy. And it has a natural precision advantage in principle; (2) Computational complexity: it reduces the computational dimension of meshing from 3 to 1, which greatly optimizes the computational cost; (3) Noise robustness: the principle of phase-coherent reception and pulse compression makes the SAPP method more robust to noise. Some scholars have carried out research on the principle of the algorithm and carried out experimental verification based on the space-borne platform [2,23,24]. Subsequently, this technology has attracted many scholars to study system parameter optimization [25], pulse emitters positioning [26], multi-target positioning [27], wide-beam dispersion [28], frequency-hopping signal positioning [29], motion compensation [30], and other problems.

Although SAPP has received a lot of attention and research, the current research is almost carried out for 2D scenarios under the plane assumption, which is more applicable to the spaceborne platform. For low-altitude platforms such as vehicle-mounted and airborne platforms, the relative elevation of the target changes greatly, and the altitude information cannot be ignored. Therefore, it cannot achieve accurate 3D positioning for targets in highly undulating scenarios such as mountainous areas and cities. Typically, high-precision positioning of small targets is difficult with DEM (Digital Elevation Model) data and 2D positioning. This is due to the insufficient accuracy of current DEMs and that DEM is often not up to date. And for the communication hotspot target in the building, three-dimensional positioning is also needed to determine its floor information. Therefore, it is necessary to study the high-precision synthetic aperture 3D passive positioning technology to expanding the scenario adaptability of the method.

To address the challenge of 3D passive positioning for the radiation source target, we propose a Cross-track Interferometric Synthetic Aperture (CISA) 3D passive positioning algorithm. This algorithm makes full use of the information of the time, frequency, spatial, and phase degrees of freedom, and achieves a high precision passive positioning in the order of ‰R. It also reduces the operation time by two orders of magnitude compared to the DPD method, while achieving better noise robustness. Additionally, we introduce an innovative DOA (Direction Of Arrival) estimation method based on Step-by-step Recursion from a Second-Order Virtual Baseline (SRSVB). The accuracy of the proposed SRSVB method is improved by 57.1% (RMSE) compared with the MUSIC method, and the average resolving ambiguity correct rate is increased by 4.5%. Furthermore, we model the incoherence problem between the positioning system and the target and propose a compensation method based on static data acquisition.

This paper is organized as follows. In Section 2, the geometric model and the signal model are established, and the CISA 3D passive positioning algorithm proposed in this paper is presented. In addition, we model and analyze the influence of the incoherence between the target frequency source and the positioning system frequency source on the positioning result. We also provide a compensation method. In Section 3, we compare the proposed algorithm with three traditional 3D positioning algorithms. And we verify the effectiveness of the proposed algorithm, including positioning accuracy, robustness, and positioning speed, through simulation and experiment, respectively. The performance of the algorithm is discussed in Section 4. The conclusion is drawn in Section 5.

2. Model and Methods

2.1. Signal Modeling

The platform moves uniformly in a straight line along the speed direction. The cross-track mounted linear array antenna receives the source signal from the right-side view. Taking the four-element antenna as an example, the geometric schematic is shown in Figure 1. Take the moment when the antenna starts to receive the signal as the azimuthal starting moment and take the vertical projection position of the platform center in the geoid at this moment as the origin of the coordinate system. The movement direction of the platform is considered the X-axis, the positive right-side view direction is the Y-axis, and the perpendicular to the upward direction of the geoid is the H-axis. The platform height is $h_{\mathrm{P}}$, and the speed is $V_{\mathrm{P}}$. The antenna is vertically installed on the platform, and the element installed at the center of the carrier is the primary element. The baseline lengths are $B_1$, $B_2$, and $B_3$. It is assumed that the target is stationary and located in the far-field area of the array antenna. The 3D position coordinates of the target area $T(x_{\mathrm{T}},y_{\mathrm{T}},h_{\mathrm{T}})$, with an incidence angle of ${\theta }_{\mathrm{T}}$.

In general, before passive positioning, pre-processing processes such as reconnaissance, signal recognition, and demodulation are needed. For narrowband modulated signals such as ASK, FSK, PSK, and QAM, the classical non-cooperative demodulation methods based on signal statistical characteristics are relatively mature, including signal spectrum, high-order cumulant, instantaneous characteristics, constellation diagram, and so on [31,32,33]. In recent years, the statistical feature recognition method based on the machine learning method has also developed rapidly and has made great progress [34]. After the recognition is completed, the carrier frequency signal with target position information can be obtained by parameter measurement and signal demodulation. There are also more mature solutions for parameter measurement and signal demodulation. In addition, for some wideband signals such as Chirp (Linear Frequency Modulation Signal), it is necessary to add a step of fast-time dechirp or pulse compression processing after signal identification and time-frequency parameter measurement. This way Chirp can be degraded to a single frequency signal. Therefore, most unencrypted deterministic steady-state signals can be demodulated to the single frequency signal after pre-processing. Taking the Chirp signal as an example, the target signal model is established as follows:

$$s\left(t\right)=A·\exp \left\{j\left(2\pi f_{c}t+{\phi }_0\right)\right\}·\exp \left\{j\pi K_{\mathrm{r}}{t}^2\right\}.$$

(1)

where $A$ is the amplitude, $f_c$ is the carrier frequency, ${\phi }_0$ is the initial phase, and $K_{\mathrm{r}}$ is the chirp rate. The fast-time and slow-time are denoted by $\tau$ and $\eta$, respectively. And their sum is time $t$.

Thus, the slant range of the $n$th array element is

$$r_n\left(\eta \right)=\sqrt{R_{\mathrm{T},n}^2+{\left(x_{\mathrm{T}}-V_{\mathrm{P}}\eta \right)}^2}=R_{\mathrm{T},n}\sqrt{1+{\left({\displaystyle \frac{x_{\mathrm{T}}-V_{\mathrm{P}}\eta }{R_{\mathrm{T},n}}}\right)}^2},$$

(2)

$$R_{\mathrm{T},n}=\sqrt{y_{\mathrm{T}}^2+{\left(h_{\mathrm{T}}-h_n\right)}^2}.$$

(3)

where, $h_n$ is the height of the $n$th array element, which is determined by the primary array element’s height $h_{\mathrm{P}}$ and the baseline of the antenna. When $x_{\mathrm{T}}=V_{\mathrm{P}}{\eta }_0$, the slant range from the target to the $n$th array element is minimized, and $r_n\left({\eta }_0\right)=R_{\mathrm{T},n}$, where ${\eta }_0$ is the zero Doppler moment and $R_{\mathrm{T},n}$ is the minimum slant range. The signal received by the $n$th element is

$$s_{\mathrm{r},n}\left(\tau ,\eta \right)=A·\exp \left\{j\left(2\pi f_c\left(\tau +\eta -{\displaystyle \frac{r_n\left(\eta \right)}{c}}\right)+{\phi }_0\right)\right\}·\exp \left\{j\pi K_{\mathrm{r}}{\left(\tau -{\displaystyle \frac{r_n\left(\eta \right)}{c}}\right)}^2\right\}.$$

(4)

where $c$ is the speed of light. According to the signal echo model, the carrier frequency ${\widehat{f}}_c$, time width, bandwidth and modulation frequency ${\widehat{K}}_{\mathrm{r}}$ of the fast-time signal can be estimated through time-frequency analysis. Set the frequency domain matching filter for fast-time:

$$H_{\mathrm{r}}\left(f_{\mathrm{r}}\right)=\exp \left\{j\pi {\displaystyle \frac{f_{\mathrm{r}}^2}{{\widehat{K}}_{\mathrm{r}}}}\right\}.$$

(5)

Perform pulse compression on fast-time signal, that is, perform the Fast Fourier Transform (FFT) for the signal along the fast-time dimension and transform to the frequency domain, multiply it with the frequency domain matching filter $H_{\mathrm{r}}\left(f_{\mathrm{r}}\right)$, and then perform the Inverse Fast Fourier Transform (IFFT) of the result, we get:

$$s_{\mathrm{r}\mathrm{c},n}\left(\tau ,\eta \right)={\mathrm{IFFT}}_{\mathrm{r}}\left[{\mathrm{FFT}}_{\mathrm{r}}\left[s_{\mathrm{r},n}\left(\tau ,\eta \right)\right]·H_{\mathrm{r}}\left(f_{\mathrm{r}}\right)\right].$$

(6)

Further, we get:

$$s_{\mathrm{r}\mathrm{c},n}\left(\tau ,\eta \right)=A·\exp \left\{j\left(2\pi f_c\left(\tau +\eta -{\displaystyle \frac{r_n\left(\eta \right)}{c}}\right)+{\phi }_0\right)\right\}·p_{\mathrm{r}}\left[\tau -{\displaystyle \frac{r_n\left(\eta \right)}{c}}\right].$$

(7)

where $p_{\mathrm{r}}\left[·\right]$ is the envelope of fast-time range compression, that is, the SINC function. And then change the fast-time signal to baseband. Here, after the range compression, the one-dimensional slow-time signal at the location of the fast-time peak point can be extracted:

$$s_{\mathrm{r}\mathrm{c},n}\left(\eta \right)\approx A·\exp \left\{j\left(2\pi f_c\left(\eta -{\displaystyle \frac{r_n\left(\eta \right)}{c}}\right)+{\phi }_0\right)\right\}.$$

(8)

When the target distance is much larger than the azimuthal position, the slant range can be approximated as a quadratic function of the slow-time by the McLaughlin expansion, that is, Equation (2) can be expressed as

$$r_n\left(\eta \right)\approx R_{\mathrm{T},n}\left(1+{\displaystyle \frac{{\left(x_{\mathrm{T}}-V_{\mathrm{P}}\eta \right)}^2}{2R_{\mathrm{T},n}^2}}\right).$$

(9)

Substitute Equation (9) into Equation (8), we get

$$s_{\mathrm{r}\mathrm{c},n}\left(\eta \right)\approx A·\exp \left\{j\left(2\pi f_c\eta +{\phi }_0\right)\right\}·\exp \left\{-j{\displaystyle \frac{{2\pi R}_{\mathrm{T},n}}{{\lambda }_c}}\right\}·\exp \left\{-j{\displaystyle \frac{\pi }{{\lambda }_c·R_{\mathrm{T},n}}}{\left(x_{\mathrm{T}}-V_{\mathrm{P}}\eta \right)}^2\right\}.$$

(10)

where ${\lambda }_c$ is the signal wavelength. It is noteworthy that $\eta$ and $n$ form a two-dimensional data matrix. When $\eta ={\eta }_0$, the corresponding slow-time sliced signal is

$$s_{\mathrm{r}\mathrm{c},n}\left({\eta }_0\right)=A·\exp \left\{j{\phi }_0\right\}·\exp \left\{-j{\displaystyle \frac{{2\pi R}_{\mathrm{T},n}}{{\lambda }_c}}\right\}.$$

(11)

The above signal model shows that the signal modulation type with fast time is not critical for subsequent processing. In order to simplify the mathematical derivation of the proposed algorithm, the signal type is simplified to a single-frequency signal, that is, it is assumed that the signal has been demodulated into a single-frequency signal by pulse compression or other demodulation methods. Therefore, the single-frequency radiation source signal model is

$$s\left(t\right)=A·\exp \left\{j\left(2\pi f_{c}t+{\phi }_0\right)\right\}.$$

(12)

Similar to (4), the signal received by the $n$th element is

$$s_{\mathrm{r},n}\left(\tau ,\eta \right)=A·\exp \left\{j\left(2\pi f_c\left(\tau +\eta -{\displaystyle \frac{r_n\left(\eta \right)}{c}}\right)+{\phi }_0\right)\right\}.$$

(13)

After performing approximation (9), we get

$$\begin{gathered} s_{\mathrm{r},n}\left(\tau ,\eta \right)\approx A·\exp \left\{j\left(2\pi f_c\left(\tau +\eta \right)+{\phi }_0\right)\right\}·\exp \left\{-j{\displaystyle \frac{{2\pi R}_{\mathrm{T},n}}{{\lambda }_c}}\right\}· \\ \exp \left\{-j{\displaystyle \frac{\pi }{{\lambda }_c·R_{\mathrm{T},n}}}{\left(x_{\mathrm{T}}-V_{\mathrm{P}}\eta \right)}^2\right\}. \end{gathered}$$

(14)

The result of the interference caused by different array elements is as follows:

$$s_{\mathrm{r},n}\left(\tau ;{\eta }_0\right)·conj\left(s_{\mathrm{r},m}\left(\tau ;{\eta }_0\right)\right)=A^2·\exp \left\{-j{\displaystyle \frac{{2\pi (R}_{\mathrm{T},n}-R_{\mathrm{T},m})}{{\lambda }_c}}\right\}.$$

(15)

where $m,n\in \left[1,4\right]$ ($m>n$). When the far-field condition is satisfied, the electromagnetic waves arriving at different array elements are approximately plane waves, then,

$$R_{\mathrm{T},n}-R_{\mathrm{T},m}\approx -B·\cos {\theta }_{\mathrm{T}}.$$

(16)

where $B$ is the baseline length between the two array elements, and ${\theta }_{\mathrm{T}}$ refers to the incidence angle. Therefore, the phase difference of the signals received by the two array elements is

$$\varphi ={\displaystyle \frac{2\pi B}{{\lambda }_c}}\cos {\theta }_{\mathrm{T}}.$$

(17)

Thus, ${\theta }_{\mathrm{T}}$ can be estimated from $\varphi$. And by combining this with the estimated result of the minimum slant range ${\widehat{R}}_{\mathrm{T},n}$, the target’s ground range and elevation can be further calculated.

2.2. Cross-Track Interferometric Synthetic Aperture 3D Passive Positioning Algorithm

It is necessary to form observation capabilities in all three spatial dimensions. The resolution principle of synthetic aperture is similar to [23]. Specifically, it relies on the virtual aperture to achieve azimuth resolution and utilizes the law of the target slant range over time to achieve range resolution. However, different from [23], this letter utilizes the linear array antenna to construct non-uniform baselines in the elevation direction to obtain the resolution of the third dimension. And the interferometric angle measurement method will be improved to optimize the stability of resolving ambiguity and the phase error tolerance. The flowchart of the proposed CISA 3D passive positioning algorithm is shown in Figure 2.

Firstly, a fast-time Fast Fourier Transform (FFT) is applied to the signal of the primary element to estimate the carrier frequency. It should be noted that the derivation related to the signal of the primary element is no longer marked with the subscript “$n$”. Therefore, the result of the FFT is as follows:

$$\begin{array}{c}{s}_{\mathrm{r}}\left(f_c,\eta \right)=A·\delta (f-f_c)·\exp \left\{j\left(2\pi f_c\eta +{\phi }_0\right)\right\}·\exp \left\{-j{\displaystyle \frac{{2\pi R}_{\mathrm{T}}}{{\lambda }_c}}\right\}\\ ·\exp \left\{-j{\displaystyle \frac{\pi }{{\lambda }_c·R_{\mathrm{T}}}}{\left(x_{\mathrm{T}}-V_{\mathrm{P}}\eta \right)}^2\right\},\end{array}$$

(18)

where $\delta \left(f\right)$ is the impulse function, and it satisfies $\delta \left(f\right)=0$ ($when f\ne 0$), and ${\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}\delta \left(f\right)df=1$. According to the approximation of Equation (9), the effect of instantaneous Doppler is ignored in Equation (19). Reference [25] gives the constraint requirements of this approximation on synthetic aperture length $L_s$. Specifically, that is $L_s\le 4.32\sqrt{\delta R_T}$, where $\delta$ is the positioning error threshold, and $R_T$ is the minimum slant range.

Select the data at the frequency $f_c$, and change the slow-time signal to the baseband. The result is

$$s_{\mathrm{r}}\left(\eta ;f_c\right)=A·\exp \left\{j{\phi }_0\right\}·\exp \left\{-j{\displaystyle \frac{{2\pi R}_{\mathrm{T}}}{{\lambda }_c}}\right\}·\exp \left\{-j{\displaystyle \frac{\pi }{{\lambda }_c·R_{\mathrm{T}}}}{\left(x_{\mathrm{T}}-V_{\mathrm{P}}\eta \right)}^2\right\}.$$

(19)

Next, apply an FFT to the slow-time baseband signal, and the Doppler domain signal is derived using the Principle of Stationary Phase (POSP), that is,

$$S_{\mathrm{r}}\left(f_a\right)\approx A·\exp \left\{j{\phi }_0\right\}·\exp \left\{-j{\displaystyle \frac{{2\pi R}_{\mathrm{T}}}{{\lambda }_c}}\right\}·\exp \left\{j{\displaystyle \frac{\pi f_a^2}{K}}\right\}·\exp \left\{-j2\pi f_a·{\displaystyle \frac{x_{\mathrm{T}}}{V_{\mathrm{P}}}}\right\},$$

(20)

$$K={\displaystyle \frac{f_c{V_{\mathrm{P}}}^2}{c·R_{\mathrm{T}}}}.$$

(21)

In the above equation, $f_a$ and $K$ are Doppler frequency and Chirp rate, respectively. Set up different matched filters based on different possible minimum slant ranges $R_{\mathrm{filter}}$. The frequency domain matched filter is

$$H\left(f_a;R_{\mathrm{filter}}\right)=\exp \left\{-j\pi {\displaystyle \frac{f_a^2}{K\left(R_{\mathrm{filter}}\right)}}\right\}.$$

(22)

The result of the matching filter is

$$s_{\mathrm{out}}\left(R_{\mathrm{filter}}\right)=IFFT\left[S_{\mathrm{r}}\left(f_a\right)·H\left(f_a;R_{\mathrm{filter}}\right)\right].$$

(23)

Then search for the matching filter result with the largest peak value, and we get an estimate of the minimum slant range

$${\widehat{R}}_{\mathrm{out}}=\underset{R_{\mathrm{filter}}}{\mathrm{arg\; max}}\left[\max \left(s_{\mathrm{out}}\left(R_{\mathrm{filter}}\right)\right)\right].$$

(24)

In addition, the matching peak of the optimal pulse compression results is located at $x_{\mathrm{T}}/V_{\mathrm{P}}$. And $V_{\mathrm{P}}$ is the speed of the radar, which can be obtained from the navigation system equipped with the passive radar system.

Next, an improved interferometric angle measurement method is designed to estimate the target’s incidence angle ${\theta }_{\mathrm{T}}$. Although there is much research on interference angle measurement methods [35,36], including the classical DOA least square estimation algorithm based on phase difference estimation (DOA-LS), it is difficult to avoid the tradeoff between accuracy and ambiguity solving stability. The most crucial parameter in the interferometric angle measurement method is the baseline, i.e., the distance between interferometric elements. The longer the baseline, the higher the accuracy theoretically.

However, a longer baseline also increases the likelihood of phase ambiguity. This is because interferometric angle measurement relies on observing the phase difference in the signals arriving at the array elements, and the periodicity of the phase often leads to differences between observed phase differences and actual phase differences by integer multiples of 2π, known as ambiguity number. A longer baseline leads to a larger ambiguity number, making the integer constraints for ambiguity solving more relaxed. This can result in an increased risk of ambiguity-solving failure in the presence of phase difference errors, meaning poor error tolerance, which is detrimental to the robustness of the system in practical applications.

Therefore, in order to accurately estimate the phase difference ambiguity number, the method of combining long and short baselines is often chosen. Generally, the long-short baseline angle measurement method needs to meet the requirement that the short baseline is less than half wavelength. However, for passive systems, especially high-frequency systems, not only the value of the half-wavelength condition is unknown, but also it is often so short that it is difficult to realize in the physical arrangement. When the condition of half-wavelength cannot be satisfied, it is necessary to carry out maximum likelihood estimation of ambiguity number through antenna design, so that the unique solution of the underdetermined equation can be obtained under constrained conditions when there is phase error. The baseline design affects both the error tolerance and the angle measurement accuracy, so a compromise optimization design is necessary.

Then, an interferometric angle measurement method based on a rotating phase plane and step-by-step extrapolation from virtual baselines is designed. Assume that the three baselines are $B_1$, $B_2$, and $B_3$ ($B_1$ < $B_2$ < $B_3$), and the corresponding phase difference measurements are ${\phi }_1$, ${\phi }_2$, and ${\phi }_3 ({\phi }_1,{\phi }_2,{\phi }_3\in \left(-\pi ,\left.\pi \right]\right.)$.

Design the baselines according to the following three criteria:

(1)

Second-order differencing of the baselines yields two virtual short baselines $B_{\mathrm{short},1}$ and $B_{\mathrm{short},2} (B_{\mathrm{short},1}<B_{\mathrm{short},2}<B_1)$;

(2)

The virtual baselines are prime multiples of the half-wavelength;

(3)

The prime numbers are as small as possible.

Then perform the same difference on ${\phi }_1$, ${\phi }_2$, and ${\phi }_3$ to obtain the phase differences ${\phi }_{\mathrm{short},1}$ and ${\phi }_{\mathrm{short},2}$, with the ambiguity numbers of $N_{\mathrm{short},1}$ and $N_{\mathrm{short},2}$. According to the geometric model, we have

$${\displaystyle \frac{B_{\mathrm{short},1}·cos{\theta }_{\mathrm{T}}}{{\lambda }_c}}·2\pi =2\pi ·N_{\mathrm{short},1}+{\phi }_{\mathrm{short},1}$$

(25)

$${\displaystyle \frac{B_{\mathrm{short},2}·cos{\theta }_{\mathrm{T}}}{{\lambda }_c}}·2\pi =2\pi ·N_{\mathrm{short},2}+{\phi }_{\mathrm{short},2}$$

(26)

As can be inferred, there is a linear relationship as follows:

$${\phi }_{\mathrm{short},2}={\displaystyle \frac{B_{\mathrm{short},2}}{B_{\mathrm{short},1}}}{\phi }_{\mathrm{short},1}+{\displaystyle \frac{B_{\mathrm{short},2}}{B_{\mathrm{short},1}}}2\pi ·N_{\mathrm{short},1}-2\pi {·N}_{\mathrm{short},2}$$

(27)

Within the angle measurement range of (0°, 180°), ${\phi }_{\mathrm{short},1}$ and ${\phi }_{\mathrm{short},2}$ form a series of parallel lines based on different combinations of ambiguity numbers, and the slope of the line is $B_{\mathrm{short},2}/B_{\mathrm{short},1}$, and the intercept of the line is determined by the ambiguity numbers. The essence of ambiguity solving is to determine which line it is from a series of parallel lines and then determine the corresponding ambiguity number combination. Taking $B_{\mathrm{short},1}=3\ast {\lambda }_c/2$ and $B_{\mathrm{short},2}=5\ast {\lambda }_c/2$ as an example, the phase plane is plotted as shown in the Figure 3.

If there is no phase difference error, the observed $({\phi }_{\mathrm{short},1},{\phi }_{\mathrm{short},2})$ will surely fall on a straight line in the phase plane. However, due to the noises and interferences, it may appear anywhere on the phase plane, and only the nearest line to the observation can be found to estimate the ambiguity number. Suppose $B_{\mathrm{short},2}/B_{\mathrm{short},1}=P/Q$, where $P$ and $Q$ are prime numbers. Rotate the phase plane to reduce the computation of ambiguity solving. Calculate the rotation angle, i.e., the angle between the line and the ${\phi }_{\mathrm{short},2}$ axis, that is,

$$\alpha =\arctan \left({\displaystyle \frac{Q}{P}}\right).$$

(28)

The distance between two adjacent lines is as follows:

$$V={\displaystyle \frac{2\pi }{\sqrt{P^2+Q^2}}}.$$

(29)

After rotation, the observed phase differences are

$$\left[\begin{array}{c}{\phi }_{\mathrm{short},1}^{\prime }\\ {\phi }_{\mathrm{short},2}^{\prime }\end{array}\right]=\left[\begin{array}{cc}{\displaystyle \frac{\cos \left(\alpha \right)}{V}}& -\sin {\displaystyle \frac{\left(\alpha \right)}{V}}\\ \sin \left(\alpha \right)& \cos \left(\alpha \right)\end{array}\right]\times \left[\begin{array}{c}{\phi }_{\mathrm{short},1}\\ {\phi }_{\mathrm{short},2}\end{array}\right].$$

(30)

The phase plane after rotation is shown in Figure 4. After rotation, each line is perpendicular to the horizontal axis and the horizontal coordinates are exactly at the position of the integer values. Therefore, in the presence of phase errors, the ambiguity number can be determined by rounding the phase difference after rotation, that is, $N_{\mathrm{short},1}=\left[{\phi }_{\mathrm{short},1}^{\prime }\right]$ ($[·]$ indicates the rounding operation). And $N_{\mathrm{short},2}$ can be calculated using Equation (27). Meanwhile, some lines correspond to two combinations of ambiguity numbers, so it is necessary to further combine the signs of the phase difference to uniquely determine the corresponding ambiguity numbers $N_{\mathrm{short},1}$ and $N_{\mathrm{short},2}$.

In addition, it is generally believed that the tolerance of phase difference error is $V/2$, and when the error does not exceed this threshold, it will not lead to the failure of ambiguity solving. Therefore, when designing the baseline, the smaller $P$ and $Q$, the larger the tolerance for phase difference error.

Then, the unambiguous phase difference of the longest baseline is derived step by step. The process is as follows:

$${\varphi }_{\mathrm{short},2}=2\pi ·N_{\mathrm{short},2}+{\phi }_{\mathrm{short},2},$$

(31)

$${\varphi }_1=2\pi ·\left[{\displaystyle \frac{B_1{/B}_{\mathrm{short},2}·{\varphi }_{\mathrm{short},2}-{\phi }_1}{2\pi }}\right]+{\phi }_1,$$

(32)

$${\varphi }_2=2\pi ·\left[{\displaystyle \frac{B_2/B_1·{\varphi }_1-{\phi }_2}{2\pi }}\right]+{\phi }_2,$$

(33)

$${\varphi }_3=2\pi ·\left[{\displaystyle \frac{B_3/B_2·{\varphi }_2-{\phi }_3}{2\pi }}\right]+{\phi }_3.$$

(34)

where ${\varphi }_{\mathrm{short},2}$, ${\varphi }_1$, ${\varphi }_2$, and ${\varphi }_3$ are the unambiguous phase difference. Specifically, the rounding operation used in each recursion truncates the transmission of most phase difference error of the current baseline. This is one of the key factors contributing to the method’s high tolerance for error. And the unambiguous phase difference of the longest baseline is used to calculate the incidence angle. Substitute ${\varphi }_3$ into Equation (17), we get

$${\widehat{\theta }}_T=\arccos \left({\displaystyle \frac{{\lambda }_c·{\varphi }_3}{2\pi B_3}}\right).$$

(35)

Estimate the target ground range and elevation, that is,

$${\widehat{y}}_T={\widehat{R}}_T·\sin \left({\widehat{\theta }}_T\right),$$

(36)

$${\widehat{h}}_T=h_P-{\widehat{R}}_T·\cos \left({\widehat{\theta }}_T\right).$$

(37)

2.3. Effects of Incoherence on SAPP

The incoherence between systems is an unavoidable problem for passive positioning. The radiated source target and the passive system are non-cooperative, and their frequency sources are independent of each other. Therefore, it is necessary to consider the phase error introduced by the clock incoherence between the two, which will bring a non-negligible effect on the synthetic aperture passive positioning based on the phase history. To solve the problem, the phase noise of the frequency source needs to be modeled and compensated.

There has been some literature on the stability of a single frequency source [37,38,39,40], but there is no literature to analyze the influence of frequency source instability on synthetic aperture passive positioning. In the existing literature, it is generally accepted that frequency accuracy and frequency stability are used to evaluate the change of the output frequency of a frequency source with time. The mathematical description of the frequency source output signal is as follows:

$$V\left(t\right)=\left[V_0+\epsilon \left(t\right)\right]·sin\left[2\pi f_0(t+x\left(t\right))\right].$$

(38)

where, $V_0$ is the nominal amplitude, $\epsilon \left(t\right)$ is the amplitude fluctuation, and $f_0$ is the nominal frequency or long-term average frequency. x(t) is the instantaneous relative phase deviation, which is physically manifested as the time deviation, also known as the phase time. Common frequency sources are atomic clocks, crystal oscillators, and so on. The radar system generally uses the Oven Controlled Crystal Oscillator (OCXO). The following analysis takes the OCXO as an example. For common OCXO, $\left|\epsilon \left(t\right)\right|\ll V_0$ and it always holds. Among the many influencing factors of OCXO, temperature, and aging characteristics have the greatest influence on frequency stability. The influence of both on the instantaneous relative phase deviation x(t) can be described by the following mathematical expression:

$$x\left(t\right)=a+bt+{\displaystyle \frac{1}{2}}d·t^2+{\epsilon }_x\left(t\right).$$

(39)

where, $a$ is the initial relative phase deviation, $b$ is the initial frequency deviation, and $d$ is the linear frequency drift. Similar to the kinematic equation, $a,b,$ and $d$ are called clock difference, clock velocity, and clock drift (acceleration), respectively. Where the first three are deterministic errors, ${\epsilon }_x\left(t\right)$ establishes the random variation component of x(t). The random characteristics are mainly affected by various noises, there are five main cases: (1) phase-modulated white noise, (2) phase-modulated flicker noise, (3) frequency-modulated white noise, (4) Flicker noise, (5) frequency random walk noise. The random model of frequency source can be described by five independent energy spectrum noises, and the total noise is regarded as the linear superposition of five noises [39].

Since the physical process of frequency source noise is not very clear, and we are mainly concerned with phase errors not exceeding the second order, we consider fitting ${\epsilon }_x\left(t\right)$ with a second-order polynomial mode. And combining the deterministic phase error model, we can define the following simplified model of $x\left(t\right)$:

$$x\left(t\right)=A+Bt+{\displaystyle \frac{1}{2}}D·t^2+N\left(0,{\sigma }_x^2\right).$$

(40)

The physical meaning of $A,B$, and $D$ is the same as those of $a,b$, and $d$, but numerically the former is an observational fit of the latter. $N\left(0,{\sigma }_x^2\right)$ is Gaussian white noise with zero mean and variance ${\sigma }_x^2$.

It is worth noting that since the angle measurement of elevation is achieved by multi-channel signal interference, the incoherent effects between the transmitter and the receiver cancel each other out. Therefore, incoherence only affects the two-dimensional synthetic aperture positioning. Next, based on the above instantaneous relative phase deviation model of frequency source, a transfer model from incoherent phase error to synthetic aperture 2D passive positioning error is established. Similar to (13), when there is an incoherent error, the signal received by the primary element is:

$$s_{\mathrm{incoh}}\left(\tau ,\eta \right)=A·e^{j\left(2\pi f_c\left(\tau +\eta +x\left(t\right)\right)+{\phi }_0\right)}·e^{-j{\displaystyle \frac{2\pi r\left(\eta \right)}{{\lambda }_c}}}.$$

(41)

Due to the fast-time $\tau$ is commonly $\mu s$ magnitude, during which the secondary phase ${\displaystyle \frac{1}{2}}D{\tau }^2$ caused by linear deviation changed little. Therefore, the following approximation is considered to be valid:

$$x\left(t\right)=x\left(\tau ,\eta \right)\approx A+B\left(\tau +\eta \right)+{\displaystyle \frac{1}{2}}D·{\eta }^2+N\left(0,{\sigma }_x^2\right).$$

(42)

Substituting Equation (42) into Equation (41), the result of using the fast-time signal to estimate the carrier frequency is ${\widehat{f}}_c=(1+B)f_c$, that is,

$$\begin{gathered} s_{\mathrm{incoh}}\left({\widehat{f}}_c,\eta \right)=A·\delta \left(f-\left(1+B\right)f_c\right){·e}^{j2\pi f_c\left(\eta +A+B\eta +{\displaystyle \frac{1}{2}}D·{\eta }^2+N\left(0,{\sigma }_x^2\right)\right)}{·e}^{j{\phi }_0} \\ ·e^{-j{\displaystyle \frac{{2\pi R}_{\mathrm{T}}}{{\lambda }_c}}}·e^{-j{\displaystyle \frac{\pi }{{\lambda }_c·R_{\mathrm{T}}}}{\left(x_{\mathrm{T}}-V_{\mathrm{P}}\eta \right)}^2}. \end{gathered}$$

(43)

According to ${\widehat{f}}_c$, change the slow-time signal to baseband. The result is

$$\begin{gathered} s_{\mathrm{incoh}}\left(\eta ;f_c\right)\approx A·e^{-{\displaystyle \frac{j\pi }{{\lambda }_c·R_{\mathrm{T}}}}\left[{\left(V_{\mathrm{P}}^2-c·R_{\mathrm{T}}·D\right)\left(\eta -{\displaystyle \frac{V_{\mathrm{P}}{x}_{\mathrm{T}}}{V_{\mathrm{P}}^2-c·R_{\mathrm{T}}·D}}\right)}^2+\left(x_{\mathrm{T}}^2-2c{R}_{\mathrm{T}}·A-{\displaystyle \frac{{\left(V_{\mathrm{P}}{x}_{\mathrm{T}}\right)}^2}{V_{\mathrm{P}}^2-c·R_{\mathrm{T}}D}}-2c{R}_{\mathrm{T}}·N\left(0,{\sigma }_x^2\right)\right)\right]} \\ {·e}^{j\left({\phi }_0-{\displaystyle \frac{{2\pi R}_{\mathrm{T}}}{{\lambda }_c}}\right)}. \end{gathered}$$

(44)

By analyzing the above slow-time signal model under the incoherent error, it can be concluded that the impact of the incoherent error on the signal includes the following aspects: (1) because of the influence of frequency deviation B, carrier frequency estimation is not accurate, from ${\widehat{f}}_c=f_c$ into ${\widehat{f}}_c=\left(1+B\right)f_c$; (2) affected by linear frequency drift D, azimuth chirp rate changes from $K={\displaystyle \frac{{V_{\mathrm{P}}}^2}{{\lambda }_c·R_{\mathrm{T}}}}$ to $K_{\mathrm{incoh}}={\displaystyle \frac{V_{\mathrm{P}}^2-c·R_{\mathrm{T}}·D}{{\lambda }_c·R_{\mathrm{T}}}}$; (3) affected by the frequency deviation B and linear frequency drift D, the peak position of pulse compression result changes from ${\displaystyle \frac{x_{\mathrm{T}}}{V_{\mathrm{P}}}}$ to ${\displaystyle \frac{V_{\mathrm{P}}{x}_{\mathrm{T}}}{V_{\mathrm{P}}^2-c·R_{\mathrm{T}}·D}}$; (4) the initial relative phase deviation A and linear frequency drift D, together form a constant phase ${\displaystyle \frac{-\pi }{{\lambda }_c·R_{\mathrm{T}}}}\left(x_{\mathrm{T}}^2-2c{R}_{\mathrm{T}}·A-{\displaystyle \frac{{\left(V_{\mathrm{P}}{x}_{\mathrm{T}}\right)}^2}{V_{\mathrm{P}}^2-c·R_{\mathrm{T}}D}}\right)$, which has no influence on the synthetic aperture history; (5) the phase also is affected by the random white noise ${\displaystyle \frac{2\pi ·c{R}_{\mathrm{T}}·N\left(0,{\sigma }_x^2\right)}{{\lambda }_c·R_{\mathrm{T}}}}$. Because the characteristics of the white noise are random and uniform, it is believed that the effect of this term is small after pulse compression.

According to Equation (24), the results of the optimal search for pulse compression can be obtained as follows:

$${\widehat{x}}_{\mathrm{T}}=V_{\mathrm{P}}·{\displaystyle \frac{V_{\mathrm{P}}{x}_{\mathrm{T}}}{V_{\mathrm{P}}^2-c·R_{\mathrm{T}}·D}},$$

(45)

$${\widehat{R}}_{\mathrm{T}}={\displaystyle \frac{{V_{\mathrm{P}}}^2{R}_{\mathrm{T}}(1+B)}{\left(V_{\mathrm{P}}^2-c·R_{\mathrm{T}}·D\right)}}.$$

(46)

Based on the above analytic results of the passive positioning, the effects of incoherence can be quantitatively and numerically analyzed. The results show that the farther the target is, the more serious the incoherent effect is. And the slower the speed of the passive positioning platform, the more serious the incoherent impact. Especially for small and slow platforms, incoherence can cause positioning accuracy to deteriorate by 2–3 orders of magnitude. Therefore, the incoherent effects must be compensated.

We propose a simple and effective compensation method from the system point of view: before positioning, the system collects the target signal without any movement, performs polynomial fitting to the phase change curve with time, and estimates the key instability parameters $B$ and $D$. The phase compensation function is generated according to $B$ and $D$ to compensate for the subsequent synthetic aperture positioning data.

3. Results

In this section, numerical simulation data and vehicle experimental data are used to verify the effectiveness and performance of the proposed CISA 3D passive positioning algorithm. In order to verify the adaptability of the algorithm to both airborne and vehicle-mounted scenes, the simulation takes the airborne parameters as an example to verify, and the experiment takes the vehicle-mounted scene as an example to verify. In the simulation part, a one-dimensional precision simulation of the proposed SRSVB method is carried out first. Then, taking the airborne remote passive positioning scenario as an example, the positioning accuracy, speed, and robustness of the proposed CISA 3D passive positioning algorithm are verified. In the experimental part, constrained by the experiment conditions, the 3D positioning verification experiment is carried out with the close-range vehicle environment as the experiment scenario. The effect of non-coherent compensation is also verified.

3.1. Simulation Results

In the simulation part, the effectiveness of the proposed SRSVB interferometric angle measurement method is first verified by simulation. And the proposed SRSVB method is compared with the classical DOA-LS algorithm and the advanced MUSIC algorithm in terms of accuracy, error tolerance, and computational complexity. Next, the effectiveness and performance advantages of the proposed CISA 3D passive positioning algorithm are verified by comparing the BO, DPD-MUSIC, and DPD-Capon algorithms with numerical simulation data. The main simulation parameters are shown in

Table 1. Simulation Parameters.

Parameter	Value
Carrier frequency	10.5 GHz
Moving speed	60 m/s
Platform height	2 km
Sampling frequency	2.4 GHz
Synthetic aperture length	800 m
Observation intersection angle	0.38°
Antenna baseline	[0.17 m, 0.37 m, 0.58 m]
Target position	(380 m, 120 km, 100 m)

.

According to the designed baseline and Equation (29), the phase error tolerance of the DOA-LS algorithm is calculated to be 6.63$°$, while that of the proposed SRSVB method is 54$°$, which is an 8-fold improvement. The error tolerance of the MUSIC algorithm is related to the signal length and SNR and is less related to the baseline design scheme. Therefore, it is difficult to calculate the theoretical value of MUSIC’s tolerance to phase difference error.

Figure 5 shows the correct rate results of 200 groups of Monte Carlo resolving ambiguity simulation experiments. Each group of Monte Carlo simulations traversed a range of −89° to 89°, in steps of 1 degree. Therefore, the ordinate represents the average correct rate of resolving ambiguity at all angles. The phase errors set in the simulation conform to the Gaussian distribution of 30° standard deviation. The results show that the average resolving ambiguity correct rate of the traditional DOA-LS method is 48.2686%, the MUSIC method is 68.8966%, and the proposed SRSVB method is 72%. The average resolving ambiguity correct rate of the proposed algorithm is similar to MUSIC, but slightly higher than it, while SRSVB improves 49.17% compared with DOA-LS.

Figure 6 shows the variation curve of the average angle measurement results and accuracy with the angle. And 200 simulations were performed for each angle. The upper subgraph is the curve of average angle measurement results. The results show that the error of DOA-LS at large angles increases abruptly, while MUSIC and the proposed method are more stable. The proposed method has some slight precision advantages over MUSIC. The following subgraph shows the RMS (Root-Mean-Square) accuracy curve, and the results show that the average accuracy of the proposed SRSVB method is better than that of DOA-LS and MUSIC. Through calculation, the average RMS accuracy of DOA-LS under different angles is 31.0144°, that of MUSIC is 14.2430°, and the proposed SRSVB is 6.1096°. The SRSVB is improved by 57.1% compared with MUSIC.

Assume that the number of antenna elements is $K$ and the number of sampling points is $N$. The algorithm complexity of DOA-LS depends on the ratio of wavelength to the baseline. If the complexity of loss function calculation is $O\left(M\right)$, the baseline used for searching is $d$, the radar wavelength is $\lambda$, and the digital phase discrimination is realized by FFT. So the algorithm complexity of DOA-LS is $O\left(KN·{\log }_2\left(N\right)+M·\left(2\ast round\left(d/\lambda \right)+1\right)\right)$. Under the same parameters, $O\left(K^{2}N\right)$ is required to calculate the covariance matrix in MUSIC, and the complexity of feature decomposition of the covariance matrix is $O\left(K^3\right)$. When $K$ is large, the computational complexity will be greatly increased. The complexity of calculating the spatial spectrum at $P$ angle points is $O\left(P{K}^2\right)$, which may become the dominant term when $P$ is large. Therefore, the total algorithm complexity of MUSIC is $O(K^{2}N+K^3+P{K}^2)$. Suppose the complexity of calculating the distance from a point to a line is $O\left(M\right)$, and the number of lines used for searching in Figure 3 is $q$. Then the complexity of calculating the distance between the point and the line is $O(KN·{\log }_2\left(N\right)+qM)$, and the complexity of the step-by-step recursion itself is the constant $O\left(1\right)$. The total algorithm complexity of SRSVB is $O(K{N·\log }_{2}N+qM+1)$.

To visually compare the values,

Table 2. Single Monte Carlo simulation runtime for different DOA estimation methods.

Algorithm	Run Time/s
DOA-LS	0.0048
MUSIC	0.0345
Proposed SRSVB Algorithm	0.0050

shows the runtime required to complete a single simulation based on MATLAB R2021 b. The results show that the computational complexity of the proposed SRSVB is equivalent to that of DOA-LS, and it is one order of magnitude faster than MUSIC.

The results of Figure 5 and Figure 6 and Table 2 verify that the proposed SRSVB method not only greatly improves the angle measurement accuracy but also improves the phase error tolerance compared with other existing interference angle measurement methods. In addition, SRSVB has the advantage of small computation and is more suitable for engineering practice. Using the SRSVB method as the third-dimension incidence angle estimation method for CISA 3D passive positioning is beneficial to realize the 3D passive positioning with high precision and high robustness. The performance of 3D passive positioning is verified by the simulation below.

Figure 7 shows the 3D positioning results obtained from 200 Monte Carlo simulations with an SNR (Signal-to-Noise Ratio) of 10 dB. The positioning accuracy is 0.32 ‰R (RMSE), and the positioning error is 38.45 m.

Figure 8 shows the variation curves of 3D passive positioning accuracy with SNR obtained using the four algorithms respectively. The simulation scene and the longest baseline are the same, the azimuth baseline is 800 m, and the elevation baseline is 0.58 m. The result shows that even under the condition of very poor SNR, the positioning accuracy of the proposed algorithm will not deteriorate rapidly and can still maintain a high level, which indicates that the algorithm has good robustness to noise.

In addition,

Table 3. Single Monte Carlo simulation runtime for different 3D positioning methods.

Algorithm	Run Time/s
BO	1.50
DPD-MUSIC	115.62
DPD-Capon	109.76
Proposed Algorithm	2.99

gives the runtime required to perform a single Monte Carlo simulation of the four algorithms based on the MATLAB R2021 b platform. From the simulation results, it can be seen that the proposed algorithm performs well on positioning accuracy and SNR robustness. In addition, compared with the two DPD algorithms, the proposed algorithm achieves an improvement of two orders of magnitude in positioning speed.

Next, the robustness to frequency measurement accuracy will be analyzed by simulation. The frequency measurement error mainly affects the optimal search results of the matching filter. It can be deduced that the minimum slant range error is caused by the frequency measurement error $∆f_c$ is $∆{\widehat{R}}_{\mathrm{T}}=∆f_c/f_c·R_T$, which leads to the deterioration of the positioning accuracy by $(∆f_c/f_c·1000)‰\mathrm{R}$. Currently, the common frequency measurement accuracy is in the order of 1 × 10⁶~1 × 10⁷ Hz [41,42], and the effect is very small and can be tolerated for passive systems in the X-band or even higher frequency bands. Figure 9 shows the simulation curve of positioning accuracy under different frequency measurement accuracies, and the results also verify the above analysis. The results show that the positioning accuracy of the proposed CISA 3D passive positioning algorithm is affected by frequency measurement errors, but the influence is not sensitive to the distance of the target. The relative positioning accuracy of targets at different distances has a similar curve trend with frequency measurement accuracy, and the magnitude is the same. We conclude that the proposed algorithm has great robustness to the frequency measurement accuracy.

3.2. Experiment Results

In the experimental verification part, the vehicle is selected as the moving platform, which carries a four-antenna cross-track interferometric synthetic aperture passive receiver system and POS 610 Integrated Navigation System (INS) to complete the principle verification experiment. The system comprises four components: a four-element line array antenna, RF channels, a signal processing unit, and INS.

The experiment is carried out in Pinggu District, Beijing, near the Juhe bridge. The experiment scene is shown in Figure 10, the vehicle moves on a bridge, and the target radiation source is set up on another bridge 600 m away. The R&S^®SMA100B signal generator serves as the radiation source. The position of the target is observed by the differential GPS device, and the INS records the motion trajectory of the platform in real time. Before each set of experiments, the passive positioning system first statically samples the radiation source signal and estimates the frequency drift parameters, which is used to compensate for the positioning data sampled later.

Under the near ideal trajectory, that is, the uniform linear motion trajectory, the radiation source signal is sampled by a passive radar system. The signal first enters the four-element array antenna, then is downconverted to an intermediate frequency, followed by sampling by an ADC (Analog to Digital Converter). At the same time, the trajectory is recorded by the POS 610 INS. The time of the radar system is synchronized with the INS. Then CISA 3D passive positioning algorithm is used to process the data. The experimental parameters are shown in

Table 4. Experiment Parameters.

Parameter	Value
Carrier frequency	11.5 GHz
Intermediate frequency	1.8 GHz
Platform height (elevation)	56.23 m
Moving speed	10 m/s
Synthetic aperture length	56.55 m
PRT	1 ms
Sampling window length	5 µs
Sampling frequency	2.4 GHz
Antennae baseline	[0.17 m, 0,37 m, 0.58 m]
Target position	[40.00 m, 605.887 m, 58.039 m]

. And the observed intersection angle is about 5.36°.

Figure 11 shows the time-domain and frequency-domain forms of the signals sampled by the four antennas. And Figure 12 shows the slow-time phase history and Doppler spectrum of the primary element signal. The slow-time phase history shows an obvious pattern of quadratic curves.

Figure 13 shows the slow-time phase history obtained from static sampling of passive radar system and the result of fitting it by quadratic polynomial. According to the fitting results, the frequency deviation $B=-2815/(2\pi ·11.5\times {10}^9)=-3.896\times {10}^{-8}$, which represents the relative frequency accuracy between the target and the passive radar. Similarly, the linear frequency shift $D=-1.253/(\pi ·11.5\times {10}^9)=-2.72\times {10}^{-11}$. The instruction manual for the R&S^®SMA100B signal generator used indicates that the frequency accuracy of the instrument is better than $1\times {10}^{-8}$ and the frequency stability is better than $5\times {10}^{-10}$. That validates the estimated results in Figure 13. The phase compensation curve is constructed according to $B$ and $D$, and the slow-time domain signal is compensated.

Figure 14 shows the search results of the optimal slow-time matching filter. The result shows a peak of pulse compression, and the position of the peak represents the azimuth position of the target. And the corresponding matching filter indicates the slant range of the target.

The positioning results are shown in Figure 15, and the 3D positioning accuracy is 4.73 ‰R. After measurement, the time-domain SNR is about 0 dB, and the robustness of the proposed algorithm under low SNR conditions is also verified. Then, using the experimental motion trajectory data, setting the same scene parameters and system parameters as the experiments, simulation data are generated for semi-physical experiments. And the positioning accuracy is 3.6 ‰R, which is close to the vehicle experiment accuracy. The two accuracy results verify each other.

In order to verify the necessity of static sampling compensation, the 3D positioning accuracy of compensated and uncompensated is compared. The results show that the positioning accuracy without compensation is 77.16 ‰R, and the positioning accuracy after compensation is 4.73 ‰R. This shows that the effect of incoherence on synthetic aperture is significant, and compensation is necessary. In addition, in order to evaluate the incoherent level influence of the experimental system on the positioning accuracy, a set of two-dimensional positioning accuracy simulations were carried out according to Equations (45) and (46). The result is shown in Figure 16. The contour map of positioning accuracy is also drawn in the figure, and the experimental uncompensated positioning accuracy 77.16 ‰R is similar to the theoretical influence 52.36 ‰R.

Then, the experiment data are processed using three classical single-station passive positioning algorithms respectively, and the positioning results are shown in

Table 5. Positioning Results Based on Different Algorithms.

Algorithm	Positioning Result	Positioning Error	Positioning Accuracy
BO	[40.01 m, 2.75 m, 57.57 m]	603.14 m	995.46 ‰R
DPD-MUSIC	[55.80 m, 550.75 m, 74.00 m]	59.53 m	98.25 ‰R
DPD-Capon	[5.00 m, 619.00 m, 81.20 m]	43.97 m	72.57 ‰R
Proposed Algorithm	[39.97 m, 603.29 m, 56.82 m]	2.87 m	4.73 ‰R

. The comparison results show that the proposed algorithm has higher positioning accuracy in the same scenario, which is one order of magnitude higher than traditional DPD algorithms.

4. Discussion

The advantages of the proposed CISA 3D passive positioning algorithm are that, firstly, the synthetic aperture process makes full use of the phase history information from coherent observations, improving the positioning accuracy and noise robustness. Secondly, in the elevation direction, the rotation of the phase plane reduces the computational complexity of ambiguity solving. The interferometric angle measurement method, which uses virtual short baselines to calculate step by step, solves the bottleneck of the antenna in processing short baselines while balancing the advantages of unambiguity in short baselines and high accuracy in long baselines. Finally, compared with traditional DPD algorithms that require a three-dimensional grid search, the proposed algorithm replaces the grid search of elevation with array estimation and utilizes the coupling relationship between the minimum slant range and azimuth position to compress the joint search process of two-dimensional positions into a one-dimensional search, greatly improving the dimensionality disaster of computational complexity and enhancing the positioning speed of the algorithm.

Through the simulation and experiment, it is verified that the proposed algorithm has higher precision, lower computational complexity and better noise robustness than traditional algorithms such as BO and MUSIC-DPD. In the current experiment scenario, the positioning accuracy of 4.73 ‰R is obtained. However, it is worth noting that the value of positioning accuracy is related to the stability characteristics of the target radiation source, the target distance, the noise environment, and the polynomial fitting estimation accuracy. Therefore, the proposed algorithm is competent in positioning the radar radiant source with good stability crystal oscillator.

5. Conclusions

In the field of single-station 3D passive positioning, the traditional two-step method has poor positioning accuracy and robustness. Although the accuracy of the DPD method is improved, the positioning speed is slow because it relies on 3D grid search with a huge computational load. To solve this problem, this paper proposes a coherent positioning algorithm, that is, a Cross-Track Interferometric Synthetic Aperture 3D passive positioning algorithm, which can balance the positioning accuracy, noise robustness, and positioning speed, and gives detailed processing steps and flow diagram. In addition, the problem of incoherence between the receiver and transmitter system is modeled and analyzed, and the compensation method is given. Simulation results and vehicle experiment results verify the effectiveness of the proposed 3D positioning algorithm and its performance advantages compared with other passive positioning algorithms. However, it should be noted that the basic models in this paper are derived based on the assumption that the number of targets is sparse and only focuses on the case of a single target, without the algorithm universality analysis for multiple unknown signals. These are the problems and challenges that the proposed algorithms need to face in practical applications, and the research on these contents will be part of future work.