Ultra-Short Baseline Synthetic Aperture Passive Positioning Based on Interferometer Assistance

Abstract

The synthetic aperture passive positioning (SAPP) method has attracted the attention of researchers due to its high positioning resolution. However, there are still key technical issues regarding SAPP methods, such as residual frequency offset (RFO) coupling at Doppler frequency leading to decreased positioning accuracy, and non-periodic discontinuous signals emitted by unknown radiation sources (NRSs) causing positioning algorithm failure. Therefore, this paper proposes an ultra-short baseline SAPP method based on interferometer assistance. Firstly, conjugate multiplication is applied to the received signals of the interferometer’s dual antennas to obtain a single frequency received signal corresponding to the straight-line distance. Subsequently, the proposed step search (SS) algorithm based on cross-correlation analysis is used to obtain the receiving frequency of the single frequency signal, and the initial positioning distance is calculated using the corresponding mapping relationship based on this frequency. Finally, NRS positioning is completed in the two-dimensional coordinates of azimuth and range by combining with the signal arrival angle. The positioning results of this method are insensitive to RFO, and even if NRS emits non-periodic discontinuous signals, the proposed method can successfully locate them. In addition, the Cramer–Rao lower bound (CRLB) of the localization for this method is derived. The simulation and unmanned aerial vehicle (UAV) experimental results demonstrate the effectiveness and feasibility of this method.

1. Introduction

The precise positioning of unknown radiation sources (NRSs) is widely used in electronic information fields such as radio astronomy, navigation, and communication [1,2,3,4], and has significant research significance. Passive positioning (PP) has received widespread attention both domestically and internationally due to its long operating range, high concealment, and strong survivability [5,6,7].

The traditional single station PP technology determines the accurate location of NRSs by estimating the direction of arrival (DOA) [8,9], time of arrival (TOA) [10,11], and joint estimation of the received signals [12,13]. Although many high-resolution DOA estimation algorithms have been proposed, such as MUSIC, Capon, and other spectral estimation methods [14,15], the ability of DOA estimation is always limited by the aperture size of the receiving antenna array, and the number of NRSs also affects the performance of DOA estimation algorithms. The TOA estimation algorithm estimates the distance of an NRS by measuring the time it takes for the signal to arrive at the receiving end, which requires accurate threshold settings to ensure the accuracy of the estimation [16]. The joint estimation of the two is achieved by measuring the arrival angle and distance of the received signal for joint positioning. However, due to the inherent limitations of DOA, the accuracy of the estimation will decrease as the distance between the radiation source and the signal source increases [17].

In order to avoid positioning errors caused by simple estimation methods, a two-step approach is commonly used in subsequent research to achieve PL. This includes estimation of positioning parameters and determination of radiation source location, which is achieved by combining Doppler frequency with DOA estimation [18,19,20]. It uses mobile stations to receive signals from NRSs. As the platform is in a moving state, the DOA and Doppler of the received signals vary at different times, and the amount of these two changes is determined by the platform’s moving speed and the position of the NRS. Therefore, a two-step localization method using DOA and Doppler can be used to locate NRSs. However, the estimation error of DOA can also deteriorate the estimation of Doppler, resulting in a decrease in positioning accuracy [21].

The continuous deepening of research on synthetic aperture radar (SAR) [22,23] has brought new research ideas to PP [24], which uses the principle of synthetic aperture technology to locate NRSs. This method is also known as synthetic aperture passive positioning (SAPP) method [25]. Reference [26] first introduced the concept of long synthetic aperture into PP and proposed an SAPP model for locating radiation sources using range and azimuth dimensions. Specifically, mobile stations will continuously collect a certain amount of data during flight and process this data according to the SAR working mode. Perform fast Fourier transform (FFT) in the fast time domain to obtain the frequency of the radiation source signal, analyze the phase change process of the radiation source signal in the slow time domain, construct a Doppler domain matched filter using the stationary phase principle, and obtain the position information of the NRS in the range and azimuth coordinate system through focusing, with high positioning resolution. In SAPP, the beamwidth is much larger than that of SAR imaging, so the length of the synthetic aperture is an important parameter in SAPP, and its length, whether too long or too short, can affect the accuracy of positioning. In response to this, reference [27] analyzed the precise positioning expression based on the principle of SAPP, provided an estimation function for positioning error, and provided an approximate solution for engineering applications. In addition, it should be noted that SAPP is usually used for NRS positioning in the static state. If the NRS is in the moving state, it will lead to additional performance parameter characteristics for the synthetic aperture. Therefore, the NRS is also in the static state in the research object of this paper.

However, SAPP faces many technical challenges, mainly focused on the following two aspects. Firstly, the estimation of Doppler is inaccurate because the carrier frequency of the received radiation source signal cannot be accurately determined, resulting in uncertain residual frequency offset (RFO) in the down converted signal, which can cause Doppler offset and decrease the accuracy of azimuth distance [28]. The current literature mainly focuses on how to eliminate RFO to ensure positioning accuracy; for example, reference [29] determines RFO and Doppler by finding the maximum phase correlation value. However, the accuracy of estimation still has a significant impact on the positioning accuracy. Secondly, traditional SAPP uses FFT operation for two-dimensional matching in frequency domain or Doppler domain, which requires high requirements for NRS, that is, NRS needs to transmit continuous periodic signals. If NRS emits discontinuous and non-periodic signals, the slow time dimension in SAPP does not exist, which leads to the failure of Doppler domain matching filtering processing, resulting in localization failure and limited application scenarios. To address these two issues, reference [29] uses interferometers for assistance and dual antennas for localization. Firstly, the approximate azimuth distance is determined through least squares estimation, and then the map drift (MD) algorithm is used for calibration. This method utilizes the phase difference between the received signals of two antennas for processing, which can not only avoid the influence of RFO, but also achieve positioning for non-periodic and discontinuous signals. However, this method requires the signal receiving platform to operate in a forward and side view condition and the trajectory to pass through the leap point before it can be completed.

In response to the current technical difficulties in research, this paper proposes a more widely applicable SAPP method, namely the SAPP method based on interferometer-assisted ultra-short baseline. Specifically, we use the dual antennas of the interferometer to receive the radiation source signal. Due to the small distance between the dual antennas relative to the radiation source and the use of wide beam reception, it can be assumed that the difference in signal received by these two antennas is only reflected in the phase change corresponding to the receiving angle, while their carrier frequency, amplitude, etc., remain consistent. Therefore, the corresponding RFO, Doppler, etc., after down converting the two signals are also the same. We perform conjugate multiplication processing on two down converted baseband signals. Due to their strong correlation, RFO can be successfully eliminated, and a single frequency signal related to signal distance can be obtained. Subsequently, the proposed cross-correlation-based step search algorithm is used to estimate the frequency of the signal. Combine it with various known parameters such as arrival angle to solve for the coordinates of the position radiation source in azimuth distance, and complete passive positioning. The specific contributions of this paper are as follows:

A new SAPP method based on interferometer-assisted ultra-short baseline has been proposed, and the positioning results obtained by this method are insensitive to RFO faced in traditional SAPP. RFO in signal processing is eliminated by the signal correlation between two baselines.

The proposed method transforms the parameter estimation problem of traditional passive positioning based on two-dimensional matched filtering of azimuth distance into a one-dimensional single frequency signal frequency estimation problem, successfully solving the problem that the slow time dimension concept in traditional SAPP does not exist in the case of non-periodic and discontinuous signals sent by NRS, making the proposed method more widely applicable.

This paper analyzes the influence of signal-to-noise ratio and positioning distance on the positioning results, and provides the Cramer–Rao lower bound (CRLB) for positioning. The effectiveness of the proposed method has been evaluated through simulation calculations and unmanned aerial vehicle (UAV) experiments.

The organization of this paper is as follows: In Section 2, the SAPP method based on interferometer-assisted ultra-short baseline proposed in this paper and its implementation principle are presented. In Section 3, the frequency estimation model and SS algorithm of the method proposed in this paper are presented. Section 4 derives the positioning performance and error results of the method. Section 5 presents the experimental results of simulation calculations and measured data. Section 6 draws a conclusion.

2. Synthetic Aperture Passive Positioning Method

This paper proposes a passive positioning method using an interferometer-assisted ultra-short baseline synthetic aperture. The following will introduce the geometric structure and signal model, respectively.

2.1. Geometric Structure of SAPP

Figure 1 shows the geometric model of ultra-short baseline SAPP. The ultra-short baseline is located on the positive side of the UAV, arranged forward and backward (represented by RX1 and RX2), with a fixed interval of d. The UAV flies at a constant speed v parallel to the ground plane in a straight line and receives signals transmitted by the NRS (represented by P) during the flight. (To focus on the proposed algorithm, without loss of generality, we assume that there is only one RS transmitting the signal.) Establish a coordinate system with the initial position of RX1 as the origin, the direction of UAV flight as the X-axis (azimuth direction), and the direction perpendicular to UAV flight as the Y-axis (range direction). The coordinate of P is $\left(X,R\right)$, where X is the azimuth coordinate and R is the distance coordinate.

Use $R_1\left(t\right)$ and $R_2\left(t\right)$ to represent the instantaneous slant range of RX1 and RX2 during UAV flight when they receive NRS signals at time t. According to the geometric model shown in Figure 1 and the cosine theorem, it can be concluded that the instantaneous slant distance has the following relationship:

(1)$R_1\left(t\right)=\sqrt{R_{10}^2+{\left(vt\right)}^2-2R_{10}vtcos\theta }$

(2)$R_2\left(t\right)=\sqrt{R_{10}^2+{\left(d+vt\right)}^2-2R_{10}\left(d+vt\right)cos\theta }$

where $R_{10}$ is the distance between RX1 and P at the initial moment, and $\theta$ is the angle between the drone trajectory and $R_{10}$.

2.2. Signal Model

P may be a radar source or a communication source, which can operate in pulse mode or continuous signal mode. Assuming the signal expression emitted by NSR is the following:

(3)$s\left(t\right)=Aexp\left(j\varphi \left(t\right)\right)exp\left(j2\pi f_{c}t\right)$

where A is the amplitude of the signal, $\varphi \left(t\right)$ is the phase of the signal under a certain modulation method and $f_c$ is the carrier frequency. After spatial propagation, $s\left(t\right)$ arrived at UAV and was received by RX1 and RX2, represented by the following equation:

(4)$s_{R_1}\left(t\right)=A_{R_1}exp\left(j\varphi \left(t-{\tau }_1\right)\right)exp\left(j2\pi f_c\left(t-{\tau }_1\right)\right)$

(5)$s_{R_2}\left(t\right)=A_{R_2}exp\left(j\varphi \left(t-{\tau }_2\right)\right)exp\left(j2\pi f_c\left(t-{\tau }_2\right)\right)$

where $A_{R_1}$ and $A_{R_1}$ are the received signal amplitudes of RX1 and RX2, ${\tau }_1$ and ${\tau }_2$ are the time delays to RX1 and RX2, which are related to the propagation line distance. Firstly, by measuring the frequency, the carrier frequency of the received signal can be obtained. After down converting Equations (4) and (5), they can be rewritten as follows:

(6)$s_{R_1}\left(t\right)=A_{R_1}exp\left(j\varphi \left(t-{\tau }_1\right)\right)exp\left(j2\pi \Delta f_{c}t\right)exp\left(-j2\pi f_c{\displaystyle \frac{R_1\left(t\right)}{c}}\right)$

(7)$s_{R_2}\left(t\right)=A_{R_2}exp\left(j\varphi \left(t-{\tau }_2\right)\right)exp\left(j2\pi \Delta f_{c}t\right)exp\left(-j2\pi f_c{\displaystyle \frac{R_2\left(t\right)}{c}}\right)$

where $\Delta f_c=f_c-{\widehat{f}}_c$ is the RFO caused by frequency measurement error, ${\widehat{f}}_c$ is the measured carrier frequency. By performing Taylor series expansion on Equations (1) and (2) and substituting them into (6) and (7), ignoring higher-order terms, we can obtain the following:

(8)$\begin{array}{cc}\hfill s_{R_1}\left(t\right)& =A_{R_1}exp\left(j\varphi \left(t-{\tau }_1\right)\right)exp\left(j2\pi \Delta f_{c}t\right)\hfill \\ \hfill & \times exp\left(-j2\pi f_c{\displaystyle \frac{R_{10}-vtcos\theta +\frac{{\left(vt\right)}^2{sin}^2\theta }{2R_{10}}}{c}}\right)\hfill \end{array}$

(9)$\begin{array}{cc}\hfill s_{R_2}\left(t\right)& =A_{R_2}exp\left(j\varphi \left(t-{\tau }_2\right)\right)exp\left(j2\pi \Delta f_{c}t\right)\hfill \\ \hfill & \times exp\left(-j2\pi f_c{\displaystyle \frac{R_{10}-\left(d+vt\right)cos\theta +\frac{{\left(d+vt\right)}^2{sin}^2\theta }{2R_{10}}}{c}}\right)\hfill \end{array}$

By performing conjugate operation on Equation (8) and multiplying it with (9), we can obtain the following:

(10)$\begin{array}{cc}\hfill s_c\left(t\right)& ={s_{R_1}}^{*}\left(t\right)\times s_{R_2}\left(t\right)\hfill \\ \hfill & =A_{R_1}{A}_{R_2}exp\left(j\left(\varphi \left(t-{\tau }_2\right)-\varphi \left(t-{\tau }_1\right)\right)\right)\hfill \\ \hfill & \times exp\left(j2\pi f_c{\displaystyle \frac{dcos\theta -\frac{d^2{sin}^2\theta }{2R_{10}}-\frac{dvt{sin}^2\theta }{R_{10}}}{c}}\right)\hfill \end{array}$

It should be noted that the distance difference between the signals received by the receiving antenna RX1 and RX2 is $dcos\theta \left(t\right)$, where $\theta \left(t\right)$ is the angle between the instantaneous slant range of RX1 and NRS at time t and the direction of the flight of the UAV. Therefore, the difference between ${\tau }_2$ and ${\tau }_1$ can be expressed as ${\displaystyle \frac{dcos\theta \left(t\right)}{c}}$. Due to the fact that d is much smaller than c, the range of $cos\theta \left(t\right)$ is between −1 and 1. Therefore, it can be considered as ${\tau }_1\approx {\tau }_2$, which is $exp\left(j\left(\varphi \left(t-{\tau }_2\right)-\varphi \left(t-{\tau }_1\right)\right)\right)\approx 1$. So Equation (10) can be simplified as follows:

(11)$\begin{array}{cc}\hfill s_c\left(t\right)& \approx A_{R_1}{A}_{R_2}exp\left(j2\pi f_c{\displaystyle \frac{dcos\theta -\frac{d^2{sin}^2\theta }{2R_{10}}-\frac{dvt{sin}^2\theta }{R_{10}}}{c}}\right)\hfill \\ \hfill & =A_{R_1}{A}_{R_2}exp\left(j2\pi {\displaystyle \frac{-dv{f}_c{sin}^2\theta }{c{R}_{10}}}t\right)exp\left(j2\pi f_c{\displaystyle \frac{dcos\theta -\frac{d^2{sin}^2\theta }{2R_{10}}}{c}}\right)\hfill \end{array}$

Let

(12)$A_p=A_{R_1}{A}_{R_2}$

(13)$f_p={\displaystyle \frac{-dv{f}_c{sin}^2\theta }{c{R}_{10}}}$

(14)${\phi }_p=2\pi f_c{\displaystyle \frac{dcos\theta -\frac{d^2{sin}^2\theta }{2R_{10}}}{c}}$

Then, $s_c\left(t\right)=A_{p}exp\left(j2\pi f_{p}t\right)exp\left(j{\phi }_p\right)$ can be regarded as a single frequency signal with amplitude $A_p$, frequency $f_p$, and phase ${\phi }_p$. These signal parameters do not include $\Delta f_c$, indicating that RFO does not affect the results of $s_c\left(t\right)$. From Equations (13) and (14), we find that $R_{10}$ is included in the frequency and phase of $s_c\left(t\right)$, and only one of these two values needs to be obtained to complete the positioning of $R_{10}$.

Now, the ultra-short baseline positioning method proposed in this paper transforms the traditional SAPP two-dimensional matched filtering operation into an estimation problem of one-dimensional frequency or phase of the measured signal. As is well known, the measurement error of phase is very high relative to frequency (as derived from CRLB). Therefore, we use frequency estimation for more accurate NRS positioning.

3. Frequency Estimation Corresponding to Positioning

3.1. The Impact of NRS Location on Frequency Estimation

Here, we first analyze the distance values of the frequency $f_p$ to be measured. According to Equation (13), $f_p$ is a negative value, and the positive or negative frequency can be transformed into the positive or negative signal amplitude, without affecting the absolute value. Therefore, we calculated the range of $\left|f_p\right|$ using the parameters corresponding to Table 1 and obtained the results corresponding to Figure 2. It should be noted that 18 GHz and 35 GHz are the carrier frequencies of conventional ground-based radar signals.

From Figure 2, it can be seen that as the measurement distance $R_{10}$ increases, the corresponding $\left|f_p\right|$ decreases continuously. When the signal carrier frequency is 18 GHz and the distance is 50 km, the measured frequency is only 0.024 Hz. If conventional frequency measurement methods such as FFT are used for such a small frequency, at least one cycle of time needs to be accumulated for analysis, which is nearly 41.67 s or even longer. Such a long detection accumulation time is not conducive to real-time positioning of the target. Even at a distance of 10 km, the signal with a carrier frequency of 35 GHz needs to accumulate for 8.33 s. Therefore, conventional FFT frequency measurement method is not applicable. Short term (in other words, short period) frequency estimation algorithms are needed to complete the positioning requirements. In response to this, this paper proposes an improved auto regressive (AR) spectral estimation method for assisted localization, namely the step search (SS) algorithm based on cross-correlation analysis.

3.2. SS Algorithm Implementation

In the AR spectral estimation method, the processed digital input signal can be regarded as Equation (11) passing through a Gaussian white noise channel, which is manifested as follows:

(15)${\widehat{s}}_c\left(n\right)=s_c\left(n\right)+w\left(n\right)=A_{p}exp\left(j2\pi f_{p}n\right)exp\left(j{\varphi }_p\right)+w\left(n\right),n\in \left[0,N-1\right]$

where $w\left(n\right)$ is the Gaussian white noise signal, $N={\displaystyle \frac{T}{\Delta t}}$ is the accumulated number of signals, T is the accumulated time of signals, and $\Delta t$ is the sampling interval. At the same time, the signal can be seen as formed by $w\left(n\right)$ passing through a linear system with a response function of $h\left(n\right)$, specifically represented as follows:

(16)${\widehat{s}}_c\left(n\right)=w\left(n\right)\ast h\left(n\right)$

If $P_w\left(\omega \right)={{\sigma }_{\omega }}^2$ represents the power spectrum of $w\left(n\right)$, then the power spectrum of ${\widehat{s}}_c\left(n\right)$ can be written as follows:

(17)$P_{{\widehat{s}}_c}\left(\omega \right)=P_w\left(\omega \right)P_h\left(\omega \right)={{\sigma }_{\omega }}^2{\left|H\left(e^{j\omega }\right)\right|}^2={{\sigma }_{\omega }}^2{\displaystyle \frac{{\left|B\left(e^{j\omega }\right)\right|}^2}{{\left|A\left(e^{j\omega }\right)\right|}^2}}$

where $P_h\left(\omega \right)={\left|H\left(e^{j\omega }\right)\right|}^2$ is the power spectrum of the linear system, and the transfer function corresponding to $h\left(n\right)$ can be obtained by changing Z as follows:

(18)$H\left(z\right)={\displaystyle \frac{B\left(z\right)}{A\left(z\right)}}={\displaystyle \frac{{\displaystyle \sum _{k=0}^q}{b}_k{z}^{-k}}{{\displaystyle \sum _{k=0}^p}{a}_k{z}^{-k}}}$

In the AR model, $a_0=b_0=1$, $b_k=0$, $a_k$ are not all 0, where $1\le k\le p$. The system transfer function can be written as follows:

(19)$H_{AR}\left(z\right)={\displaystyle \frac{1}{A\left(z\right)}}={\displaystyle \frac{1}{1+{\displaystyle \sum _{k=1}^p}{a}_k{z}^{-k}}}$

where p represents the model order of AR, and the corresponding power spectrum of ${\widehat{s}}_c\left(n\right)$ changes from Equation (17) to the following:

(20)$P_{{\widehat{s}}_c}\left(\omega \right)={{\sigma }_{\omega }}^2{\displaystyle \frac{1}{{\left|1+{\displaystyle \sum _{k=1}^p}{a}_k{e}^{-j\omega k}\right|}^2}}$

At this point, only ${{\sigma }_{\omega }}^2$ and $a_k\left(1\le k\le p\right)$ need to be calculated to obtain the power spectrum estimation of the test signal, and thus obtain the frequency of the test signal. The commonly used solution method is the Yule–Walker method, and the autocorrelation function expression of ${\widehat{s}}_c\left(n\right)$ is

(21)$\begin{array}{cc}\hfill R_s\left(m\right)& =E\left[{\widehat{s}}_c\left(n\right){\widehat{s}}_c\left(n+m\right)\right]\hfill \\ \hfill & =E\left\{{\widehat{s}}_c\left(n\right)\left[-{\displaystyle \sum _{k=1}^p}{a}_k{\widehat{s}}_c\left(n+m-k\right)+w\left(n+m\right)\right]\right\}\hfill \\ \hfill & =-{\displaystyle \sum _{k=1}^p}{a}_k{R}_s\left(m-k\right)+E\left[{\widehat{s}}_c\left(n\right)w\left(n+m\right)\right]\hfill \end{array}$

where $-{\displaystyle \sum _{k=1}^p}{a}_k{\widehat{s}}_c\left(n+m-k\right)+w\left(n+m\right)$ is the difference equation of ${\widehat{s}}_c\left(n+m\right)$, and $E\left[{\widehat{s}}_c\left(n\right)w\left(n+m\right)\right]$ represents the cross-correlation between the test signal and white noise. Since white noise is uncorrelated at different times, according to Equation (16), it can be written as follows:

(22)$E\left[{\widehat{s}}_c\left(n\right)w\left(n+m\right)\right]=E\left[w\left(n\right)w\left(n\right)h\left(0\right)\right]=\left\{\begin{array}{cc}0& m\ge 1\\ {{\sigma }_{\omega }}^2& m=0\end{array}\right.$

Substituting Equation (22) into (21) yields the following:

(23)$R_s\left(m\right)=\left\{\begin{array}{cc}-{\displaystyle \sum _{k=1}^p}{a}_k{R}_s\left(m-k\right)& m\ge 1\\ -{\displaystyle \sum _{k=1}^p}{a}_k{R}_s\left(m-k\right)+{{\sigma }_{\omega }}^2& m=0\end{array}\right.$

After substituting $m=0,1,\dots ,p$ into Equation (23), the Yule–Walker equation of the AR model can be obtained, which is manifested as follows:

(24)$\left[\begin{array}{ccccc}{R}_s\left(0\right)& R_s\left(-1\right)& R_s\left(-2\right)& \cdots & R_s\left(-p\right)\\ R_s\left(1\right)& R_s\left(0\right)& R_s\left(-1\right)& \cdots & R_s\left(-\left(p-1\right)\right)\\ R_s\left(2\right)& R_s\left(1\right)& R_s\left(0\right)& \cdots & R_s\left(-\left(p-2\right)\right)\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ R_s\left(p\right)& R_s\left(p-1\right)& R_s\left(p-2\right)& \cdots & R_s\left(0\right)\end{array}\right]\left[\begin{array}{c}1\\ a_1\\ a_2\\ ⋮\\ a_p\end{array}\right]=\left[\begin{array}{c}{{\sigma }_{\omega }}^2\\ 0\\ 0\\ ⋮\\ 0\end{array}\right]$

This equation can easily solve for the value of $a_k$ as follows:

(25)$\left[\begin{array}{c}{a}_1\\ a_2\\ ⋮\\ a_p\end{array}\right]=-\left[\begin{array}{cccc}{R}_s\left(0\right)& R_s\left(-1\right)& \cdots & R_s\left(-\left(p-1\right)\right)\\ R_s\left(1\right)& R_s\left(0\right)& \cdots & R_s\left(-\left(p-2\right)\right)\\ ⋮& ⋮& \ddots & ⋮\\ R_s\left(p-1\right)& R_s\left(p-2\right)& \cdots & R_s\left(0\right)\end{array}\right]\left[\begin{array}{c}{R}_s\left(1\right)\\ R_s\left(2\right)\\ ⋮\\ R_s\left(p\right)\end{array}\right]$

Substitute the value of $a_k$ into Equation (24) and calculate the noise power ${{\sigma }_{\omega }}^2$ as follows:

(26)$\left[{{\sigma }_{\omega }}^2\right]=\left[\begin{array}{ccccc}{R}_s\left(0\right)& R_s\left(-1\right)& R_s\left(-2\right)& \cdots & R_s\left(-p\right)\end{array}\right]\left[\begin{array}{c}{a}_1\\ a_2\\ ⋮\\ a_p\end{array}\right]$

By substituting the obtained $a_k$ and ${{\sigma }_{\omega }}^2$ into Equation (20), the power spectrum of the test signal can be obtained, and the frequency of the test signal can be obtained by measuring the position of the peak.

Due to the unknown frequency of the signal and the influence of noise, fixed order AR models often cannot achieve the best fit to the input signal, resulting in significant spectral estimation errors. Traditional AR model order determination criteria such as final prediction error criterion, Akaike information criterion, discriminant autoregressive transfer function criterion, etc., are not very effective for short signal sequence lengths. We propose the SS algorithm for determining the order of autoregressive models for short sequences. This method solves the problem of model order ambiguity in the traditional method by thoroughly evaluating and detecting the cross-correlation features of short data sequences, significantly reducing spectral estimation errors and greatly improving the accuracy and reliability of the model.

The cross-correlation function is a commonly used tool in signal processing to measure the similarity between two signals. Assuming the maximum order of the SS algorithm is M-order, the frequency estimation value ${\widehat{f}}_{p,k}$ is obtained through the Yule–Walker method of the k-order AR model. So the reference signal can represent

(27)$x\left(n,{\widehat{f}}_{p,k}\right)=exp\left(j2\pi {\widehat{f}}_{p,k}n\right),1\le k<M$

The cross-correlation function between the input signal and the reference signal is the following:

(28)$R_{s,x}\left(m,{\widehat{f}}_p\right)=E\left[{\widehat{s}}_c\left(n\right)x\left(n+m,{\widehat{f}}_{p,k}\right)\right],-N<m<N$

The maximum modulus of the cross-correlation function is taken as the reference similarity between the frequency estimation value ${\widehat{f}}_{p,k}$ obtained by the k-order AR model and the true frequency $f_p$:

(29)$\theta \left({\widehat{f}}_p\right)=\underset{-N<m<N}{max}\left|R\left(m,{\widehat{f}}_p\right)\right|$

By repeatedly searching the M-order AR model in this way, all reference similarities $\theta \left({\widehat{f}}_p\right)$ corresponding to the M-order are obtained. The frequency estimation corresponding to the maximum reference similarity is the best frequency estimation value obtained:

(30)${\widehat{f}}_p=\underset{1\le k<M}{argmax}\left[\theta \left({\widehat{f}}_{p,k}\right)\right]$

When obtaining the estimated frequency value ${\widehat{f}}_p$, the initial distance required for localization can be solved by Equation (13) as follows:

(31)${\widehat{R}}_{10}=-{\displaystyle \frac{dv{sin}^2\theta }{\lambda {\widehat{f}}_p}}$

In the above equation, $\lambda ={\displaystyle \frac{c}{f_c}}$ is the wavelength of the NRS emission signal. Based on ${\widehat{R}}_{10}$ and $\theta$, P can be located in both azimuth and range directions in two dimensions, with coordinates of

(32)$\left[\begin{array}{c}\widehat{X}\\ \widehat{R}\end{array}\right]={\widehat{R}}_{10}\left[\begin{array}{c}cos\theta \\ sin\theta \end{array}\right]$

3.3. Positioning Method Processing Flow

The processing flow of the SAPP positioning method proposed in this paper is shown in Figure 3. It can be specifically divided into signal processing and frequency estimation parts.

In the signal processing section, the two raw signals received by the interferometer are first analyzed and processed. After down conversion, the conjugate multiplication of the two signals is performed to eliminate the influence of RFO and obtain the corresponding single frequency signal, completing the first step. Subsequently, the proposed SS algorithm is used to estimate the frequency of a single frequency signal, thereby obtaining the signal frequency related to distance. Based on the mapping relationship, the distance estimation value of NRS is obtained, and the final positioning is achieved using the angle of arrival of the signal received by the interferometer. The main innovation of the method proposed in this paper is highlighted by the blue font and red box in the figure.

4. Performance Analysis

4.1. CRLB Derivation

According to the analysis in the third section, the proposed positioning method can be transformed into a frequency estimation problem for a single frequency signal. Therefore, the performance error of the method proposed in this paper can be analyzed from the perspective of frequency measurement of single frequency signals. The CRLB derivation for frequency estimation is

(33)$\mathrm{var}\left({\widehat{f}}_p\right)\ge CRL{B}_{f_p}={\displaystyle \frac{6{\sigma }^2}{{\left(2\pi \right)}^2{\left(A_p\right)}^{2}N\left(N^2-1\right)}}$

where ${\sigma }^2$ is the power of the noise signal. In terms of normalized power, the power of $s_c\left(n\right)$ can be expressed as ${\left(A_p\right)}^2$. In this case, the SNR can be defined as $SNR={\displaystyle \frac{{\left(A_p\right)}^2}{{\sigma }^2}}$. So Equation (33) can be expressed as

(34)$\mathrm{var}\left({\widehat{f}}_p\right)\ge {\displaystyle \frac{6}{{\left(2\pi \right)}^{2}N\left(N^2-1\right)SNR}}$

The CRLB of the distance measurement method proposed in this paper can be obtained from Equations (31) and (34) as follows:

(35)$\mathrm{var}\left({\widehat{R}}_{10}\right)={\left({\displaystyle \frac{d{R}_{10}}{d{f}_p}}\right)}^2\mathrm{var}\left({\widehat{f}}_p\right)=CRL{B}_{R_p}\ge {\displaystyle \frac{6{\lambda }^2{R}_{10}^4}{d^2{v}^2{sin}^4\theta {\left(2\pi \right)}^{2}N\left(N^2-1\right)SNR}}$

4.2. Distance Relative Error Analysis

CRLB represents dimensionless variance, so the standard deviation of the proposed distance measurement method can be obtained by taking the root of $\mathrm{var}\left({\widehat{R}}_{10}\right)$. In order to specify the accuracy error percentage of distance measurement, we use relative error (RE) $\delta$ instead of CRLB performance. The specific $\delta$ can be expressed as

(36)$\delta ={\displaystyle \frac{\sqrt{\mathrm{var}\left({\widehat{R}}_{10}\right)}}{R_{10}}}\ge {\displaystyle \frac{\lambda R_{10}}{dv{sin}^2\theta }}\sqrt{{\displaystyle \frac{6}{{\left(2\pi \right)}^{2}N\left(N^2-1\right)SNR}}}$

In order to obtain numerical calculation results, additional parameters shown in Table 2 were added on the basis of Table 1, namely the following: 1. Same distance, different SNR; 2. Under the same SNR and different distances, numerical calculations were performed to obtain the results corresponding to Figure 4 and Figure 5.

The curves of different colors in Figure 4 and Figure 5 represent the theoretical RE results of initial distance calculation corresponding to different cumulative times at different carrier frequencies. From Figure 4, it can be seen that as the SNR increases, the RE decreases continuously. The positioning RE of high carrier frequency NRS is better than that of low carrier frequency. As with the previous analysis results, as the accumulation time increases, the RE results of localization also continue to decline. This is because as the accumulation time increases, the information content carried by the signal becomes richer, and the useful information obtained through analysis becomes more accurate, resulting in a continuous decrease in RE. Due to the limitations of sampling interval and accumulation time, when $T=1$ s, only under high SNR conditions can the relative distance measurement error be lower than 0.01. When $T=4$ s is present, even if the carrier frequency is small, at 18 GHz, the SNR is still below −5 dB and the RE is still below 0.01.

In Figure 5, it can be seen that as the positioning distance increases, the measured RE continues to increase. This is because as the distance increases, the measurement frequency converted gradually decreases, resulting in a continuous increase in the required accumulation time. By comparing the red plus line and the blue meter line, it can be seen that with the same carrier frequency of 35 GHz, even at a short distance and SNR of 0 dB, the RE cannot be reduced below 0.01 within 1 s of accumulation time. In the case of a cumulative time of 10 s, even if the distance increases to 50 km, the RE cannot exceed 0.01. Therefore, with a fixed initial distance, radiation source carrier frequency, and transmission power, the signal accumulation time can be increased to ensure that the method has better positioning performance.

4.3. RE Analysis Under RFO Conditions

Although the influence of RFO does not exist for distance positioning RE, the fundamental reason for RFO is inaccurate carrier frequency measurement of NRS. We use $\xi ={\displaystyle \frac{\Delta f_c}{f_c}}$ to represent the RE of the carrier frequency. According to Equation (36), for $\delta$, $\xi$ affects $\lambda$. In this regard, we will add error $\xi$ to $\lambda$ and then re-substitute it into Equation (36) for numerical calculation, providing the distance RE in the presence of RFO. The specific results are shown in Figure 6 and Figure 7.

At this time, the carrier frequency of NRS is set to 35 GHz, and the cumulative time is 10 s. Other conditions should be consistent with the parameter settings in Section 4.2. Set $\xi$ to five values: 0 (without RFO), 0.0001, 0.001, 0.05, and 0.1. Figure 6 shows the results with RFO influence in the range of 10 km, and Figure 7 shows the RFO influence results with SNR of 0 dB. Comparing the curves in the graph, it can be seen that when the value of $\xi$ is very small, it has almost no effect on $\delta$. Even when $\xi$ increased to 0.1, the performance degradation of $\delta$ did not exceed 0.001. This indicates that the theory $\delta$ of the method proposed in this paper is insensitive to RFO.

5. Results

This section conducted experiments on simulated and collected data to verify the advantages of the proposed algorithm and the effectiveness of practical operations.

5.1. Simulation Results

To verify the accuracy of the method proposed in this paper, we conducted simulation experiments using linear frequency modulation signal as the transmission signal of NRS. Place the NRS shown in Table 3 on a two-dimensional plane, and the parameters of the receiver are given in Table 4.

We conducted simulation experiments on NRS transmission of periodic continuous signal and non-periodic discontinuous signal under SNR of 0 dB. The $s_c\left(t\right)$ result corresponding to Equation (11) and the SS algorithm fitting result are shown in Figure 8 and Figure 9, respectively. The abscissa in the figures represent the signal accumulation time, and the ordinate represents the real part signal value of the single frequency complex signal obtained after the processing method.

Equation (27) indicates that the performance of the SS algorithm proposed in this paper is related to the order p of the Yule–Walker matrix used in its calculation. To demonstrate the impact of p on the localization performance of the proposed method, we set p to vary from 8 to 128 with a step size of 8. At each p, we conducted 500 Monte Carlo simulation experiments with an SNR of 0 dB and obtained the results shown in Figure 10.

To demonstrate the superiority of the SS algorithm proposed in this paper over traditional AR spectral estimation method and the insensitivity of the localization method to RFO, two sets of comparative experiments were conducted here. Firstly, with a fixed distance of 10 km, the SNR was set to vary from −5 to 5 dB in increments of 1 dB. Subsequently, set the SNR to 0 dB and vary the distance direction from 5 to 50 km in increments of 5 km. Set four values of 0, 0.0001, 0.001, and 0.1, respectively. The relative measurement error was obtained by conducting 500 Monte Carlo simulations at each SNR and distance, and the results are shown in Figure 11 and Figure 12.

In order to highlight the excellent performance of the method proposed in this paper, we compared it with the SAMDP algorithm proposed in reference [29], and the specific parameter settings were the same as those in Table 3 and Table 4. Figure 13 shows the theoretical CRLB results of the relative positioning error between the SAMDP algorithm and the algorithm proposed in this paper under different SNR conditions. The blue plus line represents the result of the SAMDP algorithm, and the red triangle represents the result of the method proposed in this paper.

Meanwhile, we conducted scene simulation localization experiments on the radiation source using the proposed algorithm and SAMDP algorithm to compare their performance. The SNR from −5 to 5 dB in increments of 1 dB. At each SNR, 500 Monte Carlo simulations were conducted to obtain the relative measurement error, and the results are shown in Figure 14. The blue asterisk line in the figure represents the coarse distance estimation error result obtained by using the preliminary LS estimation in the SAMDP algorithm, the red plus line represents the final SAMDP error result, and the yellow triangle line represents the relative error result of the method proposed in this paper.

5.2. Actual Measurement Data Results

We also use the obtained data to validate the effectiveness of the proposed algorithm. The UAV implementation system is shown in Figure 15. We use the first and fifth antennas for signal reception. The parameters of the UAV experimental platform are given in Table 5.

In the experiment, the average speed of the UAV was 2.09 m/s. At the same time, various factors can lead to undesirable flight processes, such as deviation in heading angle causing changes in trajectory. Therefore, we use the average or minimum quadratic fitting method to compensate for relative errors caused by motion errors. The measurement and fitting results of the UAV flight trajectory, as well as the real position and estimated position results of the positioning target are shown in Figure 16. Figure 17 shows the result of signal processing. Figure 16a shows the true trajectory position of the UAV receiving NRS signals, as well as the corresponding fitted trajectory results. It can be seen that the UAV is in a relatively stable straight flight path. At the same time, the signals emitted by NRS are non-periodic and discontinuous. Figure 16b shows the true location of NRS and the estimated location using the method proposed in this paper. It can be seen that the error between the two is very small, with only a few meters of error.

6. Discussion

6.1. Simulation-Based Comparative Analysis

From the results shown in Figure 8 and Figure 9, it can be seen that regardless of whether the NRS sends a periodic continuous signal or a non-periodic discontinuous signal, the method proposed in this paper can perform well in processing and fitting. According to the set NRS position calculation, the actual frequency that needs to be measured is −0.23333 Hz. The frequency obtained by receiving, processing, and estimating periodic continuous signal is −0.23326 Hz, while the frequency obtained by processing non-periodic discontinuous signal is −0.23321 Hz, with a difference of only 0.00005 Hz between the two. This indicates that the SAPP method proposed in this paper is universally applicable in various scenarios. For the convenience of calculation and analysis, periodic continuous signal are used for analysis in subsequent simulation calculations and processing.

From Figure 10, it can be seen that the positioning performance continues to improve with the increase of p. But when p increases to 64, the performance improvement is minimal and the relative error curve converges. Meanwhile, an increase in p leads to an increase in the size of the Yule–Walker matrix, resulting in a more complex calculation process. Therefore, for the convenience of calculation, the value of p is used as 64 in subsequent simulation experiments.

From Figure 11, it can be seen that as the SNR increases, the positioning errors of both methods continue to decrease. However, the RE of traditional AR estimation method consistently remains above 0.01, while the method proposed in this paper can be lower than 0.01 when the SNR is −3 dB. Figure 12 shows that as the distance increases, the estimation error continues to increase, but the performance of the method proposed in this paper has always been better than traditional AR method, with high positioning accuracy. Comparing the cases with RFO in the two figures, it can be found that even with the largest RFO, its positioning RE is still close to the case without RFO. The reason why the performance with RFO is better than without RFO is due to the insufficient number of Monte Carlo simulations and the influence of random noise.

We found that there is still a certain gap between the results obtained by any estimation algorithm and the theoretical minimum value. This is because the method proposed in this paper is an improvement based on the AR spectral estimation method, and its estimation performance still has certain shortcomings. At the same time, the setting of p is fixed at 64, which is not the best performance setting point. If higher positioning accuracy is required, the value of p can be increased to make the results closer to theory.

From Figure 13, it can be seen that as the SNR increases, the RE lower bounds of both algorithms continue to decrease. However, the RE lower bound of the method proposed in this paper has always been lower than that of the SAMDP method, indicating that the theoretical performance of the method proposed in this paper is superior to that of SAMDP.

From Figure 14, it can be seen that the error of the preliminary LS estimation of SAMDP algorithm is very large, and when the SNR is below −2 dB, the calculated relative error even exceeds 1. Even with an SNR of 5 dB, the relative error is not less than 0.1. Although the relative error result has been greatly improved after the second complete SAMDP, it still maintains a high relative error value. The relative error of the method proposed in this paper still maintains a positioning error accuracy of less than 0.01 even at −5 dB, demonstrating high positioning performance.

The reason why the measurement results of SAMDP differ significantly from those given in Reference [29] is that the radiation source used in this paper emits a linear frequency modulation signal, while the single frequency signal source used in reference [29]. This will result in the $\psi$ corresponding to Equation (3) in reference [29] being a quadratic time-varying phase, not a constant, and the residual phase $\varphi$ in Equation (5) after down conversion is also a quadratic phase related to $\psi$. This leads to errors in calculating the phase difference of the received signal, resulting in a rapid decline in estimation performance. The method proposed in this paper utilizes conjugate multiplication to ignore the phase error caused by the baseband linear frequency modulation signal, transforming the positioning requirement into a frequency estimation problem, thereby greatly reducing estimation errors and improving positioning accuracy.

6.2. Experimental Performance Analysis

The result corresponding to Equation (24) obtained by processing the measured data is represented by blue dots in Figure 16. It can be seen that within 150 s of UAV receiving, the result of received signal processing is not a complete continuous single frequency signal, but discontinuous. The frequency value obtained by estimating the data using the proposed SS algorithm is −0.0315 Hz, and the corresponding red curve in the Figure 17 is fitted. It can be seen that the fitting effect is good, and the large fluctuations in blue dots may be due to unstable drone flight and environmental noise. When using the method proposed in this paper for localization, the estimated values of azimuth and range distance are 188.2943 m and 1334.7125 m, respectively. The true azimuth and range distance of the target are 188.4859 m and 1336.0702 m, respectively. At this time, the estimation errors of azimuth and range distance are 0.1916 m and 1.3577 m, respectively. The total distance positioning error is 1.3711 m, and the relative positioning error is 0.001016.

7. Conclusions

In response to the technical difficulties of traditional SAPP algorithms, this paper proposes a SAPP method based on interferometer-assisted ultra-short baseline. This method transforms the traditional SAPP’s two-dimensional matched filtering parameter estimation problem in azimuth distance into a one-dimensional frequency estimation problem. The obtained positioning results are insensitive to RFO, and even if the relative offset of RFO is 0.1, the corresponding RE will not deteriorate by more than 0.001. At the same time, it solves the problem of the traditional SAPP method not having a slow time dimension when NRS emission is non-periodic and discontinuous, and increases the application scenarios of SAPP. The implementation principle of this method was introduced through a reasonable mathematical model analysis, and CRLB was derived to describe the positioning performance of this method. The simulation calculation shows the feasibility of the proposed method. The UAV experimental results show that the method proposed in this paper has a high positioning accuracy with an error of 0.001016 in the NRS positioning of non-periodic and discontinuous signals, demonstrating the effectiveness of the proposed method. The reason why the simulation results did not meet the theoretical performance is due to the limitations of the frequency estimation method. In future research, we will consider using frequency estimation methods with better performance to achieve localization.

Author Contributions

Conceptualization, G.L. and Q.Z.; Methodology, G.L., Q.Z., Z.H. and B.M.; Software, G.L., J.X., J.Z. and H.G.; Validation, G.L., Q.Z., J.X., J.Z. and Z.H.; Formal analysis, B.M.; Investigation, Q.Z. and J.X.; Resources, H.G.; Data curation, G.L., Q.Z., J.X., J.Z., Z.H. and B.M.; Writing—original draft, Q.Z.; Writing—review & editing, G.L., Q.Z., J.X., J.Z., Z.H., B.M. and H.G.; Visualization, Qidong Zhang; Supervision, G.L. and H.G.; Project administration, G.L. and H.G.; Funding acquisition, G.L. and H.G. All authors have read and agreed to the published version of the manuscript.

Data Availability Statement

The original contributions presented in this study are included in the article. Further inquiries can be directed to the corresponding author.

Conflicts of Interest

The authors declare no conflicts of interest.

Footnotes

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

Figures and Tables

Figure 1 Geometric model of ultra-short baseline passive positioning.

Figure 2 Frequency $f_p$ to be measured corresponding to different carrier frequencies $f_c$ and distances $R_{10}$.

Figure 3 Positioning method workflow.

Figure 4 $\delta$ under different SNR conditions at a range of 10 km.

Figure 5 SNR is 0 dB, $\delta$ at different ranges.

Figure 6 $\delta$ with varying SNR for different $\xi$ within a range of 10 km.

Figure 7 The SNR is 0 dB, the $\delta$ of different $\xi$ varies within different ranges.

Figure 8 Signal processing and fitting results under continuous signal transmission cycle of NRS.

Figure 9 Signal processing and fitting results under NRS emission of non-periodic and discontinuous signals.

Figure 10 The influence of SS algorithm order p on RE.

Figure 11 Comparison of results under the same distance but different SNR.

Figure 12 Comparison of results under the same SNR but different distances.

Figure 13 The CRLB results of the proposed algorithm and SAMDP theory RE.

Figure 14 Comparison between the proposed algorithm and SAMDP experimental simulation results.

Figure 15 UAV-borne experimental platform.

Figure 16 Track fitting and positioning results: (a) UAV received signal position display and track fitting; (b) Real target position and estimated position display.

Figure 17 Single frequency signal fitting measurement results.

Table 1

Frequency analysis parameter settings.

Parameter	Value
NRS Signal Frequency ($f_p$)	18 GHz, 35 GHz
UAV Speed (v)	20 m/s
Short Baseline Spacing (d)	1 m
Positioning Target Range	5−50 km
Positioning Target Azimuth	10 m

Table 2

Parameter settings for RE analysis.

Parameter	Value
SNR	$-5$–5 dB
Accumulated Duration (T)	1 s, 4 s, 10 s
Accumulated Interval ($\Delta t$)	1 ms

Table 3

Parameter settings for NRS.

Parameter	Value
Carrier Frequency	35 GHz
Pulse Width	10 μs
Signal Bandwidth	10 MHz
Range	10 km
Azimuth	40 m
Speed	0 m/s

Table 4

Parameter settings for UAV.

Parameter	Value
Speed	20 m/s
Short Baseline Spacing	1 m
Accumulated Duration	4 s
Accumulated Interval	1 ms

Table 5

Parameter settings.

Parameter	Value
Carrier Frequency of NRS	12,000 MHz
Short Baseline Spacing	0.518 m
UAV Speed	2.09 m/s
Accumulated Duration	150 s
Accumulated Interval	0.1 s

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