A. Near-Field Bistatic SAR Geometry

During the paper, one radar node is assumed to be static, while the other node is assumed to be moving with a constant velocity $\vec{v}_{\mathrm{R}}$. A block diagram of the proposed setup is shown in Fig. 1. The origin of the common coordinate frame is set at the position of the first RX antenna of the static radar node, so the position of its TX antenna $\vec{p}_{\text{TX},1}$ and the $n_{\mathrm{A}}$-th RX antenna $\vec{p}_{\text{RX},n_{\mathrm{A}},1}$ can be derived from the known array geometry. Given the tilt angle $\gamma$ and the position of the second radar $\vec{p}_{\mathrm{R},2}=(x_{\mathrm{R}},y_{\mathrm{R}})^{\mathrm{T}}$, its antenna positions can be calculated by

View Source \begin{align*} \vec{p}_{\text{TX},2} &= \begin{pmatrix}\cos \left(\gamma \right) & -\sin \left(\gamma \right)\\ \sin \left(\gamma \right) & \cos \left(\gamma \right) \end{pmatrix} \vec{p}_{\text{TX},1} + \vec{p}_{\mathrm{R},2} \tag{1}\\ \vec{p}_{\text{RX},n_{\mathrm{A}},2} &= \begin{pmatrix}\cos \left(\gamma \right) & -\sin \left(\gamma \right)\\ \sin \left(\gamma \right) & \cos \left(\gamma \right) \end{pmatrix} \vec{p}_{\text{RX},n_{\mathrm{A}},1} + \vec{p}_{\mathrm{R},2}. \tag{2} \end{align*}

$$\begin{align*} \vec{p}_{\mathrm{R},2}(t) = \begin{pmatrix}x_{\mathrm{R},0}\\ y_{\mathrm{R},0} \end{pmatrix} + \vec{v}_{\mathrm{R}} \cdot t, \tag{3} \end{align*}$$ View Source \begin{align*} \vec{p}_{\mathrm{R},2}(t) = \begin{pmatrix}x_{\mathrm{R},0}\\ y_{\mathrm{R},0} \end{pmatrix} + \vec{v}_{\mathrm{R}} \cdot t, \tag{3} \end{align*}

The model parameter descriptions are gathered in Table 1.

B. Signal Model

During the SAR integration time, multiple frames are transmitted and received by both radars simultaneously. Every frame consists of a chirp sequence with $N_{\text{ch}}$ chirps. During every chirp sequence, an unknown constant frequency offset between both reference oscillators is assumed, which leads to a constant time drift between both local time bases. The local time for the $n_{\mathrm{F}}$-th frame is calculated as

View Source \begin{equation*} t_{i,n_{\mathrm{F}}}(t) = \left(1+\delta _{t,i,n_{\mathrm{F}}}\right)t + \Delta \tau _{0,i,n_{\mathrm{F}}}, \tag{4} \end{equation*}

TABLE 1 Geometry Model Parameters

In addition to these systematic errors in the bistatic beat signals, each locally generated TX signal suffers from random phase errors. This behaviour can be modelled chirp-wise by a zero-mean time-dependent phase noise part $\phi _{\text{pn},k,n_{\mathrm{F}},1/2}(t)$ and a constant unknown start phase of each chirp $\Theta _{k,n_{\mathrm{F}},1/2}$ in each radar node, where $k$ indicates the chirp index.

Applying the systematic and stochastic error sources to a FMCW signal model, as was done comprehensively in [30], yields the following expressions for the bistatic beat signal phases from radar 2 to 1

View Source \begin{align*} \Phi _{\mathrm{B},k,n_{\mathrm{F}}}^{2\to 1}&(t,\tau) \\ =&2\pi \left[-f_{0}\delta _{t,n_{\mathrm{F}}} - \mu \tau + \mu \left(\Delta \tau _{0,n_{\mathrm{F}}} + kT_{\text{rep}}\delta _{t,n_{\mathrm{F}}}\right) \right] t \\ &-2\pi \mu \delta _{t,n_{\mathrm{F}}}t^{2} - 2\pi f_{0}\tau + \phi _{\text{pn},k,n_{\mathrm{F}},2}(t-\tau) \\ \ &- \phi _{\text{pn},k,n_{\mathrm{F}},1}(t) + \underbrace{\Theta _{k,n_{\mathrm{F}},2} - \Theta _{k,n_{\mathrm{F}},1}}_{=\Delta \Theta _{k,n_{\mathrm{F}}}} \tag{5} \end{align*}

$$\begin{align*} \Phi _{\mathrm{B},k,n_{\mathrm{F}}}^{1\to 2}&(t,\tau) \\ =&2\pi \left[f_{0}\delta _{t,n_{\mathrm{F}}} - \mu \tau - \mu \left(\Delta \tau _{0,n_{\mathrm{F}}} + kT_{\text{rep}}\delta _{t,n_{\mathrm{F}}}\right) \right] t \\ &-2\pi \mu \delta _{t,n_{\mathrm{F}}}t^{2} -2\pi f_{0}\tau + \phi _{\text{pn},k,n_{\mathrm{F}},1}(t-\tau) \\ &- \phi _{\text{pn},k,n_{\mathrm{F}},2}(t) - \Delta \Theta _{k,n_{\mathrm{F}}}, \tag{6} \end{align*}$$ View Source \begin{align*} \Phi _{\mathrm{B},k,n_{\mathrm{F}}}^{1\to 2}&(t,\tau) \\ =&2\pi \left[f_{0}\delta _{t,n_{\mathrm{F}}} - \mu \tau - \mu \left(\Delta \tau _{0,n_{\mathrm{F}}} + kT_{\text{rep}}\delta _{t,n_{\mathrm{F}}}\right) \right] t \\ &-2\pi \mu \delta _{t,n_{\mathrm{F}}}t^{2} -2\pi f_{0}\tau + \phi _{\text{pn},k,n_{\mathrm{F}},1}(t-\tau) \\ &- \phi _{\text{pn},k,n_{\mathrm{F}},2}(t) - \Delta \Theta _{k,n_{\mathrm{F}}}, \tag{6} \end{align*}

$$\begin{align*} \tau _{\text{LoS},n_{\mathrm{A}}}^{2\to 1} &= \frac{\left\Vert \vec{p}_{\text{RX},n_{\mathrm{A}},1}-\vec{p}_{\text{TX},2}\right\Vert }{\mathrm{c}_{0}} \tag{7}\\ \tau _{\text{LoS},n_{\mathrm{A}}}^{1\to 2} &= \frac{\left\Vert \vec{p}_{\text{RX},n_{\mathrm{A}},2}-\vec{p}_{\text{TX},1}\right\Vert }{\mathrm{c}_{0}} \tag{8} \end{align*}$$ View Source \begin{align*} \tau _{\text{LoS},n_{\mathrm{A}}}^{2\to 1} &= \frac{\left\Vert \vec{p}_{\text{RX},n_{\mathrm{A}},1}-\vec{p}_{\text{TX},2}\right\Vert }{\mathrm{c}_{0}} \tag{7}\\ \tau _{\text{LoS},n_{\mathrm{A}}}^{1\to 2} &= \frac{\left\Vert \vec{p}_{\text{RX},n_{\mathrm{A}},2}-\vec{p}_{\text{TX},1}\right\Vert }{\mathrm{c}_{0}} \tag{8} \end{align*}

$$\begin{align*} \tau _{\ell,n_{\mathrm{A}}}^{2\to 1} &= \frac{\left\Vert \vec{p}_{\mathrm{T},\ell }-\vec{p}_{\text{TX},2}\right\Vert + \left\Vert \vec{p}_{\text{RX},n_{\mathrm{A}},1} - \vec{p}_{\mathrm{T},\ell }\right\Vert }{\mathrm{c}_{0}} \tag{9}\\ \tau _{\ell,n_{\mathrm{A}}}^{1\to 2} &= \frac{\left\Vert \vec{p}_{\mathrm{T},\ell }-\vec{p}_{\text{TX},1}\right\Vert + \left\Vert \vec{p}_{\text{RX},n_{\mathrm{A}},2} - \vec{p}_{\mathrm{T},\ell }\right\Vert }{\mathrm{c}_{0}}, \tag{10} \end{align*}$$ View Source \begin{align*} \tau _{\ell,n_{\mathrm{A}}}^{2\to 1} &= \frac{\left\Vert \vec{p}_{\mathrm{T},\ell }-\vec{p}_{\text{TX},2}\right\Vert + \left\Vert \vec{p}_{\text{RX},n_{\mathrm{A}},1} - \vec{p}_{\mathrm{T},\ell }\right\Vert }{\mathrm{c}_{0}} \tag{9}\\ \tau _{\ell,n_{\mathrm{A}}}^{1\to 2} &= \frac{\left\Vert \vec{p}_{\mathrm{T},\ell }-\vec{p}_{\text{TX},1}\right\Vert + \left\Vert \vec{p}_{\text{RX},n_{\mathrm{A}},2} - \vec{p}_{\mathrm{T},\ell }\right\Vert }{\mathrm{c}_{0}}, \tag{10} \end{align*}

$$\begin{align*} &s_{\mathrm{B},k,n_{\mathrm{A}},n_{\mathrm{F}}}^{2\to 1} (t) = A_{\text{RX},LoS}^{2\to 1}\cdot \mathrm{e}^{\mathrm{j}\Phi _{\mathrm{B},k,n_{\mathrm{F}}}^{2\to 1}(t,\tau _{\text{LoS},n_{\mathrm{A}}}^{2\to 1})} \\ &\quad + \sum _{\ell } A_{\text{RX},\ell }^{2\to 1}\cdot \mathrm{e}^{\mathrm{j}\Phi _{\mathrm{B},k,n_{\mathrm{F}}}^{2\to 1}(t,\tau _{\ell,n_{\mathrm{A}}}^{2\to 1})} + n_{k,n_{\mathrm{A}},n_{\mathrm{F}},1}(t)\tag{11}\\ &s_{\mathrm{B},k,n_{\mathrm{A}},n_{\mathrm{F}}}^{1\to 2} (t) = A_{\text{RX},LoS}^{1\to 2}\cdot \mathrm{e}^{\mathrm{j}\Phi _{\mathrm{B},k,n_{\mathrm{F}}}^{1\to 2}(t,\tau _{\text{LoS},n_{\mathrm{A}}}^{1\to 2})} \\ &\quad + \sum _{\ell } A_{\text{RX},\ell }^{1\to 2}\cdot \mathrm{e}^{\mathrm{j}\Phi _{\mathrm{B},k,n_{\mathrm{F}}}^{1\to 2}(t,\tau _{\ell,n_{\mathrm{A}}}^{1\to 2})} + n_{k,n_{\mathrm{A}},n_{\mathrm{F}},2}(t), \tag{12} \end{align*}$$ View Source \begin{align*} &s_{\mathrm{B},k,n_{\mathrm{A}},n_{\mathrm{F}}}^{2\to 1} (t) = A_{\text{RX},LoS}^{2\to 1}\cdot \mathrm{e}^{\mathrm{j}\Phi _{\mathrm{B},k,n_{\mathrm{F}}}^{2\to 1}(t,\tau _{\text{LoS},n_{\mathrm{A}}}^{2\to 1})} \\ &\quad + \sum _{\ell } A_{\text{RX},\ell }^{2\to 1}\cdot \mathrm{e}^{\mathrm{j}\Phi _{\mathrm{B},k,n_{\mathrm{F}}}^{2\to 1}(t,\tau _{\ell,n_{\mathrm{A}}}^{2\to 1})} + n_{k,n_{\mathrm{A}},n_{\mathrm{F}},1}(t)\tag{11}\\ &s_{\mathrm{B},k,n_{\mathrm{A}},n_{\mathrm{F}}}^{1\to 2} (t) = A_{\text{RX},LoS}^{1\to 2}\cdot \mathrm{e}^{\mathrm{j}\Phi _{\mathrm{B},k,n_{\mathrm{F}}}^{1\to 2}(t,\tau _{\text{LoS},n_{\mathrm{A}}}^{1\to 2})} \\ &\quad + \sum _{\ell } A_{\text{RX},\ell }^{1\to 2}\cdot \mathrm{e}^{\mathrm{j}\Phi _{\mathrm{B},k,n_{\mathrm{F}}}^{1\to 2}(t,\tau _{\ell,n_{\mathrm{A}}}^{1\to 2})} + n_{k,n_{\mathrm{A}},n_{\mathrm{F}},2}(t), \tag{12} \end{align*}

TABLE 2 Radar Signal Model Parameters

Furthermore, if the rate of change of the phase noise term $\phi _{\text{pn},k,n_{\mathrm{F}},1/2}$ is considerably smaller than $\frac{2\pi }{\tau }$, the time shift due to the ToF can be neglected, as shown in [27]. The phase noise differences in both beat signal terms can then be simplified to

View Source \begin{align*} &\phi _{\text{pn},k,n_{\mathrm{F}},2}(t-\tau) - \phi _{\text{pn},k,n_{\mathrm{F}},1}(t) \\ &\approx \phi _{\text{pn},k,n_{\mathrm{F}},2}(t) - \phi _{\text{pn},k,n_{\mathrm{F}},2}(t) \\ &\approx -\left[\phi _{\text{pn},k,n_{\mathrm{F}},1}(t-\tau) - \phi _{\text{pn},k,n_{\mathrm{F}},2}(t)\right]. \tag{13} \end{align*}

The beat frequencies and peak phases that correspond to a given channel path with ToF $\tau$ are given by

View Source \begin{align*} f_{\mathrm{B},k,n_{\mathrm{F}}}^{2\to 1} =& -f_{0}\delta _{t,n_{\mathrm{F}}} - \mu \tau + \mu (\Delta \tau _{0,n_{\mathrm{F}}} + kT_{\text{rep}}\delta _{t,n_{\mathrm{F}}}) \\ &+ \epsilon _{f,\text{pn},k,n_{\mathrm{F}}} + \epsilon _{f,\text{awgn},k,n_{\mathrm{F}},1}\tag{14}\\ f_{\mathrm{B},k,n_{\mathrm{F}}}^{1\to 2} =& f_{0}\delta _{t,n_{\mathrm{F}}} - \mu \tau - \mu (\Delta \tau _{0,n_{\mathrm{F}}} + kT_{\text{rep}}\delta _{t,n_{\mathrm{F}}}) \\ &- \epsilon _{f,\text{pn},k,n_{\mathrm{F}}} + \epsilon _{f,\text{awgn},k,n_{\mathrm{F}},2}, \tag{15} \end{align*}

$$\begin{align*} \varphi _{\mathrm{B},k,n_{\mathrm{F}}}^{2\to 1} &= -2\pi f_{0} \tau + \Delta \Theta _{k,n_{\mathrm{F}}} + \epsilon _{\varphi,\text{pn},k,n_{\mathrm{F}}} + \epsilon _{\varphi,\text{awgn},k,n_{\mathrm{F}},1}\tag{16}\\ \varphi _{\mathrm{B},k,n_{\mathrm{F}}}^{1\to 2} &= -2\pi f_{0} \tau - \Delta \Theta _{k,n_{\mathrm{F}}} - \epsilon _{\varphi,\text{pn},k,n_{\mathrm{F}}} + \epsilon _{\varphi,\text{awgn},k,n_{\mathrm{F}},2}, \tag{17} \end{align*}$$ View Source \begin{align*} \varphi _{\mathrm{B},k,n_{\mathrm{F}}}^{2\to 1} &= -2\pi f_{0} \tau + \Delta \Theta _{k,n_{\mathrm{F}}} + \epsilon _{\varphi,\text{pn},k,n_{\mathrm{F}}} + \epsilon _{\varphi,\text{awgn},k,n_{\mathrm{F}},1}\tag{16}\\ \varphi _{\mathrm{B},k,n_{\mathrm{F}}}^{1\to 2} &= -2\pi f_{0} \tau - \Delta \Theta _{k,n_{\mathrm{F}}} - \epsilon _{\varphi,\text{pn},k,n_{\mathrm{F}}} + \epsilon _{\varphi,\text{awgn},k,n_{\mathrm{F}},2}, \tag{17} \end{align*}

A. Bistatic SAR Resolution

The resolution of an FMCW radar SAR system can be divided into range and Doppler resolution. As discussed in [6], the point spread function (PSF) of a point target in the imaging scene is increasingly spatially variant as the radar nodes approach the image scene (see Fig. 2). Cardillo derived a gradient-based approach to approximate the spatial-dependent range and cross-range resolution of a bistatic SAR system in [32]. Therefore, this method is introduced and used to quantify the resolution. The bistatic range covers the distance from the TX antenna to the target and from the target to the RX antenna:

View Source \begin{equation*} R_{\mathrm{b}}(\vec{p}_{\mathrm{T}}) = \left\Vert \vec{p}_{\mathrm{T}}-\vec{p}_{\text{TX}}\right\Vert + \left\Vert \vec{p}_{\text{RX}}-\vec{p}_{\mathrm{T}}\right\Vert, \tag{18} \end{equation*}

$$\begin{align*} f_{\mathrm{D}}(\vec{p}_{\mathrm{T}}) &= \frac{1}{\lambda _{0}}\left\langle \vec{v}_{\text{TX}},\vec{n}_{\mathrm{TX,T}}\right\rangle \\ \ &\ \text{with} \quad \vec{n}_{\mathrm{TX,T}} = \frac{\vec{p}_{\mathrm{T}}-\vec{p}_{\text{TX}}}{\left\Vert \vec{p}_{\mathrm{T}}-\vec{p}_{\text{TX}}\right\Vert },\tag{19} \end{align*}$$ View Source \begin{align*} f_{\mathrm{D}}(\vec{p}_{\mathrm{T}}) &= \frac{1}{\lambda _{0}}\left\langle \vec{v}_{\text{TX}},\vec{n}_{\mathrm{TX,T}}\right\rangle \\ \ &\ \text{with} \quad \vec{n}_{\mathrm{TX,T}} = \frac{\vec{p}_{\mathrm{T}}-\vec{p}_{\text{TX}}}{\left\Vert \vec{p}_{\mathrm{T}}-\vec{p}_{\text{TX}}\right\Vert },\tag{19} \end{align*}

$$\begin{align*} \vec{\nabla } R_{\mathrm{b}}= \left(\frac{\partial }{\partial x}R_{\mathrm{b}}, \frac{\partial }{\partial y}R_{\mathrm{b}}\right)^{\mathrm{T}},\tag{20}\\ \vec{\nabla } f_{\mathrm{D}} = \left(\frac{\partial }{\partial x}f_{\mathrm{D}}, \frac{\partial }{\partial y}f_{\mathrm{D}}\right)^{\mathrm{T}}, \tag{21} \end{align*}$$ View Source \begin{align*} \vec{\nabla } R_{\mathrm{b}}= \left(\frac{\partial }{\partial x}R_{\mathrm{b}}, \frac{\partial }{\partial y}R_{\mathrm{b}}\right)^{\mathrm{T}},\tag{20}\\ \vec{\nabla } f_{\mathrm{D}} = \left(\frac{\partial }{\partial x}f_{\mathrm{D}}, \frac{\partial }{\partial y}f_{\mathrm{D}}\right)^{\mathrm{T}}, \tag{21} \end{align*}

$$\begin{equation*} \vec{n}_{\vec{\nabla }R_{\mathrm{b}}} = \frac{\vec{\nabla } R_{\mathrm{b}}}{\left|\vec{\nabla } R_{\mathrm{b}}\right|}\quad \text{and}\quad \vec{n}_{\vec{\nabla }f_{\mathrm{D}}} = \frac{\vec{\nabla } f_{\mathrm{D}}}{\left|\vec{\nabla } f_{\mathrm{D}} \right|} \tag{22} \end{equation*}$$ View Source \begin{equation*} \vec{n}_{\vec{\nabla }R_{\mathrm{b}}} = \frac{\vec{\nabla } R_{\mathrm{b}}}{\left|\vec{\nabla } R_{\mathrm{b}}\right|}\quad \text{and}\quad \vec{n}_{\vec{\nabla }f_{\mathrm{D}}} = \frac{\vec{\nabla } f_{\mathrm{D}}}{\left|\vec{\nabla } f_{\mathrm{D}} \right|} \tag{22} \end{equation*}

$$\begin{equation*} \vec{\delta }_{r} = 0.89\cdot \frac{1/B}{\left|\vec{\nabla } R_{\mathrm{b}}\right|}\cdot \vec{n}_{\vec{\nabla }R_{\mathrm{b}}}, \tag{23} \end{equation*}$$ View Source \begin{equation*} \vec{\delta }_{r} = 0.89\cdot \frac{1/B}{\left|\vec{\nabla } R_{\mathrm{b}}\right|}\cdot \vec{n}_{\vec{\nabla }R_{\mathrm{b}}}, \tag{23} \end{equation*}

$$\begin{equation*} \vec{\delta }_{\mathrm{D}} = 0.89\cdot \frac{1/T_{\text{int}}}{\left|\vec{\nabla } f_{\mathrm{D}} \right|}\cdot \vec{n}_{\vec{\nabla }f_{\mathrm{D}}}, \tag{24} \end{equation*}$$ View Source \begin{equation*} \vec{\delta }_{\mathrm{D}} = 0.89\cdot \frac{1/T_{\text{int}}}{\left|\vec{\nabla } f_{\mathrm{D}} \right|}\cdot \vec{n}_{\vec{\nabla }f_{\mathrm{D}}}, \tag{24} \end{equation*}

$$\begin{equation*} \delta _{\text{xD}} = \frac{\delta _{\mathrm{r}}}{\sin \Theta }. \tag{25} \end{equation*}$$ View Source \begin{equation*} \delta _{\text{xD}} = \frac{\delta _{\mathrm{r}}}{\sin \Theta }. \tag{25} \end{equation*}

$$\begin{equation*} \delta _{\mathrm{cD,nf}} = {0.89}\frac{7\lambda _{0}}{\Psi _{\mathrm{r}}^{2}}. \tag{26} \end{equation*}$$ View Source \begin{equation*} \delta _{\mathrm{cD,nf}} = {0.89}\frac{7\lambda _{0}}{\Psi _{\mathrm{r}}^{2}}. \tag{26} \end{equation*}

Figure 3.

Comparison of cross-Doppler resolution between the analytical formulas of equations (25) and (26) and the numerical results from a back-projection evaluation for point targets swept along the x (a) and y (b) axis referring to the scene of Fig. 2. Also monostatic cross-Doppler resolution derived from the back-projection images is plotted.

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B. Impact of Self-Localization Errors

To assess the required self-localization accuracy of the radar-based positioning algorithm for the BiSAR processing, the influence of erroneous trajectory estimation on the imaging performance needs to be investigated. For automotive SAR scenarios, the effects of constant position and velocity estimation errors on the SAR imaging performace are discussed in [19], [35]. While moderate positional offsets have a negligible influence on the processing results, errors in the estimated velocity lead to a displacement of the estimated target position in the direction of the Doppler gradient $\vec{n}_{\vec{\nabla }f_{\mathrm{D}}}$ and a slight target power loss due to defocusing. Both works assume far-field condition, which are not present in the scenarios discussed in this paper. Hence, the influence of velocity estimation errors on the imaging quality of near-field SAR heavily depends on the constellation and cannot be easily addressed by a closed-form analytical solution. To quantify the effect of erroneous velocities for scenarios within the scope of this work, the constellation given in Fig. 2 with a target at position 1 and at position 5 is simulated for a range of velocity errors. These errors are divided into a component along the trajectory $\varepsilon _{v,\text{traj}}$ and a component perpendicular to the linear trajectory $\varepsilon _{v,\text{xtraj}}$. Normalizing to the nominal velocity magnitude $|\vec{v} |$ yields $\delta _{v,\text{traj}} = \varepsilon _{v,\text{traj}}/|\vec{v} |$ and $\delta _{v,\text{xtraj}} = \varepsilon _{v,\text{xtraj}}/|\vec{v} |$, providing better comparability.

The coherent integration loss, which is the relationship between the actual targets peak power and the peak power of a perfectly focused target, is used as a metric to assess SAR image quality. The result of this simulation is shown in Fig. 4. It indicates that a velocity error component along the trajectory significantly affects defocusing compared to an error perpendicular to the trajectory direction. Additionally, the degree of defocusing strongly depends on the target's position. These findings provide an indication of the required precision of the proposed particle filter's tracking results in the measurement evaluation section. While a higher error perpendicular to the trajectory is permissible, the integration loss due to a velocity error in the trajectory direction decreases significantly faster, especially for target 5 with a larger bistatic angle.

Figure 4.

Influence of the velocity estimation error on the scenario shown in Fig. 2. The metric of the coherent integration loss is calculated for targets 1 and 5.

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C. Bistatic RCS

To complete the brief theoretical discussion on near-field bistatic SAR, emphasis will be placed on the differences in scattering characteristics of complex targets in monostatic and bistatic SAR measurements, which are described by their RCS values.

The bistatic RCS is a measure for the scattered energy of an illuminated target into the direction of a receiver that is spatially separated from the transmitter's position and can be calculated by

View Source \begin{equation*} \sigma = \lim _{r\to \infty } 4\pi r^{2} \frac{\left|E_{\mathrm{s}}\right|}{\left|E_{\mathrm{i}}\right|}, \tag{27} \end{equation*}

Figure 5.

Bistatic RCS simulation of a bicycle model using CST Studio Suit 2024 ray-tracing tool. The targets was excited from an azimuth angle of 45°, and the corresponding bistatic RCS was observed over a range from −90° to 90° (b). The observed rays for the monostatic case (45°) (a), and for a bistatic obervation angle of −45° (c) are exemplarily shown.

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In addition to this simulation, several other publications support the information gain of bistatic radar imaging compared to only monostatic observation. In [7], it is shown that geometries with weak monostatic scattering behavior often yield large bistatic returns. A practical use case is given in [40], where improvements in the detection of drones with multistatic radar networks are investigated.

A. Theory

The basic equations behind the PF originate in the Bayes filter algorithm. This algorithm generally has two quantities: the observations $\vec{z}_{D_{i}}$, containing the measurement along a sub-aperture $D_{i}$, and the system's current state $\vec{x}_{D_{i}}$, which consists in this case of the relative pose of the moving radar at the start of the sub-trajectory $D_{i}$ and its vectorial velocity. Both quantities are assumed to be time-dependent random variables, each described by a stochastic process. The sub-aperture $D_{i}$ indicates that for each filter iteration, $N_{\text{sub}}$ chirp measurements recorded during the movement are processed coherently instead of using each radar measurement on its own, as was done in our previous work [29].

The conditional probability density function (PDF) of the state vector, in [41] referred to as belief

View Source \begin{equation*} \text{bel}\left(\vec{x}_{D_{i}}\right) = p\left(\vec{x}_{D_{i}}\vert \vec{z}_{D_{0}:D_{i}}\right), \tag{28} \end{equation*}

$$\begin{equation*} \overline{\text{bel}}\left(\vec{x}_{D_{i}}\right) = \int p\left(\vec{x}_{D_{i}}\vert \vec{x}_{D_{i-1}}\right) \text{bel}\left(\vec{x}_{D_{i-1}}\right) \mathrm{d}\vec{x}_{D_{i-1}}. \tag{29} \end{equation*}$$ View Source \begin{equation*} \overline{\text{bel}}\left(\vec{x}_{D_{i}}\right) = \int p\left(\vec{x}_{D_{i}}\vert \vec{x}_{D_{i-1}}\right) \text{bel}\left(\vec{x}_{D_{i-1}}\right) \mathrm{d}\vec{x}_{D_{i-1}}. \tag{29} \end{equation*}

$$\begin{equation*} \text{bel}\left(\vec{x}_{D_{i}}\right) = \eta \cdot p\left(\vec{z}_{D_{i}}\vert \vec{x}_{D_{i}}\right) \cdot \overline{\text{bel}}\left(\vec{x}_{D_{i}}\right), \tag{30} \end{equation*}$$ View Source \begin{equation*} \text{bel}\left(\vec{x}_{D_{i}}\right) = \eta \cdot p\left(\vec{z}_{D_{i}}\vert \vec{x}_{D_{i}}\right) \cdot \overline{\text{bel}}\left(\vec{x}_{D_{i}}\right), \tag{30} \end{equation*}

This basic algorithm is fundamental to a variety of stochastic filters that differ mainly in their way of representing the densities $\overline{\text{bel}}(\vec{x}_{D_{i}})$ and $\text{bel}(\vec{x}_{D_{i}})$. Probably the most prominent implementation of Bayes' rule is the Kalman filter and its derivates, which assume Gaussian PDFs. Therefore, only two parameters are required to represent the density functions, which makes them very efficient. If the models for the state dynamics or the observations tend to differ too strongly from a Gaussian representation, alternatives to the Kalman filter are required. Especially for the use case of phase-based localization, the measurement functions tend to become multimodal due to the $2\pi$ ambiguity of phase measurements [31]. Therefore, the PF has proven to be very performant in dealing with multimodal PDFs, as it represents the belief by a set of weighted particles

View Source \begin{equation*} \mathcal {X}_{D_{i}} = \left\lbrace \vec{x}_{D_{i}}^{\;n_{\mathrm{P}}}, w_{D_{i}}^{n_{\mathrm{P}}}\right\rbrace _{n_{\mathrm{P}}=1}^{N_{\mathrm{P}}}, \tag{31} \end{equation*}

$$\begin{equation*} w_{D_{i}}^{n_{\mathrm{P}}} \propto w_{D_{i-1}}^{n_{\mathrm{P}}}\cdot \frac{p\left(\vec{z}_{D_{i}}\vert \vec{x}_{D_{i}}^{\;n_{\mathrm{P}}}\right)p\left(\vec{x}_{D_{i}}^{\;n_{\mathrm{P}}}\vert \vec{x}_{D_{i-1}}^{\;n_{\mathrm{P}}}\right)}{q\left(\vec{x}_{D_{i}}^{\;n_{\mathrm{P}}}\vert \vec{x}_{D_{i-1}}^{\;n_{\mathrm{P}}};\vec{z_{D_{i}}}\right)}, \tag{32} \end{equation*}$$ View Source \begin{equation*} w_{D_{i}}^{n_{\mathrm{P}}} \propto w_{D_{i-1}}^{n_{\mathrm{P}}}\cdot \frac{p\left(\vec{z}_{D_{i}}\vert \vec{x}_{D_{i}}^{\;n_{\mathrm{P}}}\right)p\left(\vec{x}_{D_{i}}^{\;n_{\mathrm{P}}}\vert \vec{x}_{D_{i-1}}^{\;n_{\mathrm{P}}}\right)}{q\left(\vec{x}_{D_{i}}^{\;n_{\mathrm{P}}}\vert \vec{x}_{D_{i-1}}^{\;n_{\mathrm{P}}};\vec{z_{D_{i}}}\right)}, \tag{32} \end{equation*}

$$\begin{equation*} w_{D_{i}}^{n_{\mathrm{P}}} \propto w_{D_{i-1}}^{n_{\mathrm{P}}}\cdot p\left(\vec{z}_{D_{i}}\vert \vec{x}_{D_{i}}^{\;n_{\mathrm{P}}}\right). \tag{33} \end{equation*}$$ View Source \begin{equation*} w_{D_{i}}^{n_{\mathrm{P}}} \propto w_{D_{i-1}}^{n_{\mathrm{P}}}\cdot p\left(\vec{z}_{D_{i}}\vert \vec{x}_{D_{i}}^{\;n_{\mathrm{P}}}\right). \tag{33} \end{equation*}

$$\begin{equation*} \text{bel}(\vec{x}_{D_{i}}) \approx \sum _{n_{\mathrm{P}}=1}^{N_{\mathrm{P}}} w_{D_{i}}^{n_{\mathrm{P}}} \delta \left(\vec{x}_{D_{i}} - \vec{x}_{D_{i}}^{\;n_{\mathrm{P}}}\right), \tag{34} \end{equation*}$$ View Source \begin{equation*} \text{bel}(\vec{x}_{D_{i}}) \approx \sum _{n_{\mathrm{P}}=1}^{N_{\mathrm{P}}} w_{D_{i}}^{n_{\mathrm{P}}} \delta \left(\vec{x}_{D_{i}} - \vec{x}_{D_{i}}^{\;n_{\mathrm{P}}}\right), \tag{34} \end{equation*}

$$\begin{equation*} N_{\mathrm{P,eff,}D_{i}} \approx \frac{1}{\sum _{n_{\mathrm{P}}=1}^{N_{\mathrm{P}}}\left(w_{D_{i}}^{n_{\mathrm{P}}}\right)^{2}}. \tag{35} \end{equation*}$$ View Source \begin{equation*} N_{\mathrm{P,eff,}D_{i}} \approx \frac{1}{\sum _{n_{\mathrm{P}}=1}^{N_{\mathrm{P}}}\left(w_{D_{i}}^{n_{\mathrm{P}}}\right)^{2}}. \tag{35} \end{equation*}

B. Implementation

First, the state vector is defined as

View Source \begin{equation*} \vec{x}_{D_{i}} = \begin{pmatrix}x_{\mathrm{R},D_{i}} & y_{\mathrm{R},D_{i}} & \dot{x}_{\mathrm{R},D_{i}} & \dot{y}_{\mathrm{R},D_{i}} & \gamma _{D_{i}} \end{pmatrix}^{\mathrm{T}}. \tag{36} \end{equation*}

Figure 6.

Bistatic radar constellation with two-channel SIMO radars, where three sub-apertures along the radar trajectory are sketched. In this example, four measurements per sub-aperture are taken and are processed during one filter iteration. The different LoS path distances are drawn.

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To predict the particles of the $i$-th sub-aperture, a model with constant velocity and a constant tilt angle is applied. Therefore, a particle from the $i-1$-th iteration is transferred to the current iteration by applying the linear equation

View Source \begin{equation*} \vec{x}_{D_{i}}^{\;n_{\mathrm{P}}} = \underbrace{\begin{pmatrix}1 & 0 & \Delta T & 0 & 0 \\ 0 & 1 & 0 & \Delta T & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{pmatrix}}_{\mathbf {F}\left[\Delta T\right]} \vec{x}_{D_{i-1}}^{\;n_{\mathrm{P}}} + \vec{w}_{D_{i}}, \tag{37} \end{equation*}

$$\begin{align*} &\mathbf {P_{\mathrm{D_{i}}}} = \vec{\alpha }_{\mathrm{P}}\cdot \frac{\cdot N_{\mathrm{P}}}{N_{\mathrm{P,eff,}D_{i}}} \\ & \cdot \text{diag}\begin{pmatrix}\sigma _{x_{\mathrm{R}},D_{i}}^{2} & \sigma _{y_{\mathrm{R}},D_{i}}^{2} & \sigma _{\dot{x}_{\mathrm{R}},D_{i}}^{2} & \sigma _{\dot{y}_{\mathrm{R}},D_{i}}^{2} & \sigma _{\gamma,,D_{i}}^{2} \end{pmatrix}, \tag{38} \end{align*}$$ View Source \begin{align*} &\mathbf {P_{\mathrm{D_{i}}}} = \vec{\alpha }_{\mathrm{P}}\cdot \frac{\cdot N_{\mathrm{P}}}{N_{\mathrm{P,eff,}D_{i}}} \\ & \cdot \text{diag}\begin{pmatrix}\sigma _{x_{\mathrm{R}},D_{i}}^{2} & \sigma _{y_{\mathrm{R}},D_{i}}^{2} & \sigma _{\dot{x}_{\mathrm{R}},D_{i}}^{2} & \sigma _{\dot{y}_{\mathrm{R}},D_{i}}^{2} & \sigma _{\gamma,,D_{i}}^{2} \end{pmatrix}, \tag{38} \end{align*}

$$\begin{equation*} \tilde{f}_{n_{\mathrm{A}},D_{i},n_{\text{sub}}}^{\text{RT}} = -\mu \left(\tau _{n_{\mathrm{A}},D_{i},n_{\text{sub}}}^{2\to 1} + \tau _{n_{\mathrm{A}},D_{i},n_{\text{sub}}}^{1\to 2}\right) \tag{39} \end{equation*}$$ View Source \begin{equation*} \tilde{f}_{n_{\mathrm{A}},D_{i},n_{\text{sub}}}^{\text{RT}} = -\mu \left(\tau _{n_{\mathrm{A}},D_{i},n_{\text{sub}}}^{2\to 1} + \tau _{n_{\mathrm{A}},D_{i},n_{\text{sub}}}^{1\to 2}\right) \tag{39} \end{equation*}

$$\begin{equation*} \tilde{\varphi }_{n_{\mathrm{A}},D_{i},n_{\text{sub}}}^{\text{RT}} = -2\pi f_{0} \left(\tau _{n_{\mathrm{A}},D_{i},n_{\text{sub}}}^{2\to 1} + \tau _{n_{\mathrm{A}},D_{i},n_{\text{sub}}}^{1\to 2}\right), \tag{40} \end{equation*}$$ View Source \begin{equation*} \tilde{\varphi }_{n_{\mathrm{A}},D_{i},n_{\text{sub}}}^{\text{RT}} = -2\pi f_{0} \left(\tau _{n_{\mathrm{A}},D_{i},n_{\text{sub}}}^{2\to 1} + \tau _{n_{\mathrm{A}},D_{i},n_{\text{sub}}}^{1\to 2}\right), \tag{40} \end{equation*}

The round-trip distances can be calculated from the measured frequencies using (39) by

View Source \begin{equation*} \tilde{d}_{n_{\mathrm{A}},D_{i},n_{\text{sub}}}^{\text{RT}} = -\mathrm{c}_{0}\frac{\tilde{f}_{n_{\mathrm{A}},D_{i},n_{\text{sub}}}^{\text{RT}}}{\mu }. \tag{41} \end{equation*}

$$\begin{align*} &p\left(\vec{f}_{D_{i}}^{\;\text{RT}}\vert \vec{x}_{D_{i}}^{\;n_{\mathrm{P}}}\right) = \sum _{n_{\text{sub}}=0}^{N_{\text{sub}}-1} \sum _{n_{\mathrm{A}}=0}^{N_{\text{RX}}-1} \\ &\quad \frac{1}{\sqrt{2\pi \sigma _{d_{\text{rt}}}^{2}}}\exp \left\lbrace -\frac{\left(d_{n_{\mathrm{A}},n_{\text{sub}}}^{\text{RT}}\left(\vec{x}_{D_{i}}^{\;n_{\mathrm{P}}}\right)-\tilde{d}_{n_{\mathrm{A}},D_{i},n_{\text{sub}}}^{\text{RT}}\right)^{2}}{2\sigma _{d_{\text{rt}}}^{2}}\right\rbrace, \tag{42} \end{align*}$$ View Source \begin{align*} &p\left(\vec{f}_{D_{i}}^{\;\text{RT}}\vert \vec{x}_{D_{i}}^{\;n_{\mathrm{P}}}\right) = \sum _{n_{\text{sub}}=0}^{N_{\text{sub}}-1} \sum _{n_{\mathrm{A}}=0}^{N_{\text{RX}}-1} \\ &\quad \frac{1}{\sqrt{2\pi \sigma _{d_{\text{rt}}}^{2}}}\exp \left\lbrace -\frac{\left(d_{n_{\mathrm{A}},n_{\text{sub}}}^{\text{RT}}\left(\vec{x}_{D_{i}}^{\;n_{\mathrm{P}}}\right)-\tilde{d}_{n_{\mathrm{A}},D_{i},n_{\text{sub}}}^{\text{RT}}\right)^{2}}{2\sigma _{d_{\text{rt}}}^{2}}\right\rbrace, \tag{42} \end{align*}

$$\begin{align*} &p\left(\vec{\varphi }_{D_{i}}^{\;\text{RT}}\vert \vec{x}_{D_{i}}^{\;n_{\mathrm{P}}}\right) =\bigg \vert \sum _{n_{\text{sub}}=0}^{N_{\text{sub}}-1} \sum _{n_{\mathrm{A}}=0}^{N_{\text{RX}}-1} \\ &\quad \exp \left\lbrace \mathrm{j}\left[k d_{n_{\mathrm{A}},n_{\text{sub}}}^{\text{RT}}\left(\vec{x}_{D_{i}}^{\;n_{\mathrm{P}}}\right) + \tilde{\varphi }_{n_{\mathrm{A}},D_{i},n_{\text{sub}}}^{\text{RT}}\right]\right\rbrace \bigg \vert, \tag{43} \end{align*}$$ View Source \begin{align*} &p\left(\vec{\varphi }_{D_{i}}^{\;\text{RT}}\vert \vec{x}_{D_{i}}^{\;n_{\mathrm{P}}}\right) =\bigg \vert \sum _{n_{\text{sub}}=0}^{N_{\text{sub}}-1} \sum _{n_{\mathrm{A}}=0}^{N_{\text{RX}}-1} \\ &\quad \exp \left\lbrace \mathrm{j}\left[k d_{n_{\mathrm{A}},n_{\text{sub}}}^{\text{RT}}\left(\vec{x}_{D_{i}}^{\;n_{\mathrm{P}}}\right) + \tilde{\varphi }_{n_{\mathrm{A}},D_{i},n_{\text{sub}}}^{\text{RT}}\right]\right\rbrace \bigg \vert, \tag{43} \end{align*}

$$\begin{align*} &p\left(\vec{\varphi }_{D_{i}}^{\;\mathrm{2\to 1}}, \vec{\varphi }_{D_{i}}^{\;\mathrm{1\to 2}}\vert \vec{x}_{D_{i}}^{\;n_{\mathrm{P}}}\right) = \sum _{n_{\text{sub}}=0}^{N_{\text{sub}}-1} \\ &\ \left|\sum _{n_{\mathrm{A}}=0}^{N_{\text{RX}}-1} \exp \left\lbrace \mathrm{j}\left[k d_{n_{\mathrm{A}},n_{\text{sub}}}^{\mathrm{2\to 1}}\left(\vec{x}_{D_{i}}^{\;n_{\mathrm{P}}}\right) + \tilde{\varphi }_{n_{\mathrm{A}},D_{i},n_{\text{sub}}}^{\mathrm{2\to 1}}\right]\right\rbrace \right| \\ &\ \cdot \left|\sum _{n_{\mathrm{A}}=0}^{N_{\text{RX}}-1} \exp \left\lbrace \mathrm{j}\left[k d_{n_{\mathrm{A}},n_{\text{sub}}}^{\mathrm{1\to 2}}\left(\vec{x}_{D_{i}}^{\;n_{\mathrm{P}}}\right) + \tilde{\varphi }_{n_{\mathrm{A}},D_{i},n_{\text{sub}}}^{\mathrm{1\to 2}}\right]\right\rbrace \right|. \tag{44} \end{align*}$$ View Source \begin{align*} &p\left(\vec{\varphi }_{D_{i}}^{\;\mathrm{2\to 1}}, \vec{\varphi }_{D_{i}}^{\;\mathrm{1\to 2}}\vert \vec{x}_{D_{i}}^{\;n_{\mathrm{P}}}\right) = \sum _{n_{\text{sub}}=0}^{N_{\text{sub}}-1} \\ &\ \left|\sum _{n_{\mathrm{A}}=0}^{N_{\text{RX}}-1} \exp \left\lbrace \mathrm{j}\left[k d_{n_{\mathrm{A}},n_{\text{sub}}}^{\mathrm{2\to 1}}\left(\vec{x}_{D_{i}}^{\;n_{\mathrm{P}}}\right) + \tilde{\varphi }_{n_{\mathrm{A}},D_{i},n_{\text{sub}}}^{\mathrm{2\to 1}}\right]\right\rbrace \right| \\ &\ \cdot \left|\sum _{n_{\mathrm{A}}=0}^{N_{\text{RX}}-1} \exp \left\lbrace \mathrm{j}\left[k d_{n_{\mathrm{A}},n_{\text{sub}}}^{\mathrm{1\to 2}}\left(\vec{x}_{D_{i}}^{\;n_{\mathrm{P}}}\right) + \tilde{\varphi }_{n_{\mathrm{A}},D_{i},n_{\text{sub}}}^{\mathrm{1\to 2}}\right]\right\rbrace \right|. \tag{44} \end{align*}

The combined measurement function is given by

View Source \begin{align*} p\left(\vec{z}_{D_{i}}\vert \vec{x}_{D_{i}}^{\;n_{\mathrm{P}}}\right) =& \eta \cdot p\left(\vec{f}_{D_{i}}^{\;\text{RT}}\vert \vec{x}_{D_{i}}^{\;n_{\mathrm{P}}}\right) \cdot p\left(\vec{\varphi }_{D_{i}}^{\;\text{RT}}\vert \vec{x}_{D_{i}}^{\;n_{\mathrm{P}}}\right)^{2}\\ & \cdot p\left(\vec{\varphi }_{D_{i}}^{\;\mathrm{2\to 1}}, \vec{\varphi }_{D_{i}}^{\;\mathrm{1\to 2}}\vert \vec{x}_{D_{i}}^{\;n_{\mathrm{P}}}\right)^{2}, \tag{45} \end{align*}

Figure 7.

Maximum cuts of the unnormalized five-dimensional measurement function given in equation (45) showing slices of position $xy$ (a), velocity $\dot{x}\dot{y}$ (b), and tilt angle dimension $\gamma$ (c+d). In the simulated scenario, the actual state was given by $\vec{x}= (-0.68\,\text{m} \quad {4.63}\,\text{m} \quad {0.15}\,\text{m/s} {-0.26}\,\text{m/s} \quad {168.1}^{\circ })^{\mathrm{T}}$, which corresponds to the start position of the scenario in Fig. 2 in the static radars coordinate system.

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The general bootstrap PF algorithm can be implemented with the above derived set of equations. As previously mentioned, the first frame is processed by the localization scheme from [27]. This gives a coarse estimate of the state, from which the initial particle set is drawn. Subsequently, the state model of (37) is applied to all particles to predict the next state. This step is followed by a recalculation of the weights using (33) and (45).

To further decrease sample impoverishment, the effective number of particles is compared to a threshold in every filter iteration, and a resampling step is conducted if it falls below this threshold. A systematic resampling approach is used, as it offers high computational efficiency and a low sampling variance [41]. The threshold is set to $N_{\mathrm{P,eff},D_{i}} \leq \frac{N_{\mathrm{P}}}{2}$.

The state estimate at the $i$-th iteration can be received by calculating the expected value of the weighted particle set

View Source \begin{equation*} \hat{\vec{x}}_{D_{i}} = {\mathbb {E}}\left[\mathcal {X}_{D_{i}}\right] = \frac{\sum _{n_{\mathrm{P}}} \vec{x}_{D_{i}}^{\;n_{\mathrm{P}}}\cdot w_{D_{i}}^{n_{\mathrm{P}}}}{\sum _{n_{\mathrm{p}}} w_{D_{i}}^{n_{\mathrm{P}}}}. \tag{46} \end{equation*}

C. Localization Results

The described PF algorithm with the filter parameters shown in Table 3 is validated using measurements taken in the laboratory based on the simulation scenario in Fig. 2. The coordinate system used for localization corresponds to the one spanned by radar node 1 as described in the geometry part of Section II. The ground truth information of the filter state is provided by an optical reference system with a 3D root mean square positioning error of 1.0 mm. Therefore, each radar node is equipped with optical markers to obtain its position and orientation. Radar 2 is mounted on a linear actuator to generate SAR trajectories with constant velocities. In this setup, five different measurements were taken with a metal rod at different target positions indicated in Fig. 8. The applied radar parameters are given in Table 4. The filter was evaluated by 16 runs per dataset to show that the PF converges to a global best estimate. The filter estimates of dataset number 1 are plotted in Fig. 9, in which the spread between single filter runs is shown by the 95% confidence interval.

TABLE 3 Filter Parameters

TABLE 4 Radar Parameters

Figure 8.

Indoor scenario for bistatic SAR measurements with PF-based trajectory estimation. The lab is equipped with an optical tracking system to provide ground truth data. A messing rod serves as a target and was deployed at different positions (1-5), which are marked by green dots.

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Figure 9.

Evaluation of 16 filter runs. The transparent area around the respective mean values shows the 95% confidence interval. The dashed lines show the ground truth data received by the optical reference system.

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To assess the filter's localization performance, the root mean square errors (RMSE) for position, orientation, and relative velocity offset in the trajectory and cross-trajectory directions are calculated over all time samples and data sets. The recorded radar raw data is also processed using the chirp sequence-based approach from our previous work in [27] to demonstrate the achieved improvement. Additionally, the particle filter position estimates over time are fitted to a linear trajectory model, which further enhances self-localization given a linear movement with constant velocity. The results are shown in Table 5. A significant improvement in positioning accuracy compared to chirp sequence-based processing is achieved by the proposed algorithm. By evaluating the velocity estimates using the results given in Fig. 4 of Section III-B, an acceptable integration loss of less then 3dB in the SAR images can be expected, especially for the linearly fitted estimates.

TABLE 5 Evaluation of the Self-Localization Results and Comparison to [27]

A. Bistatic Synchronization

As there are still synchronization errors in the bistatic radar data, further processing steps must be carried out frame-wise before creating the bistatic images. To eliminate the time drift and offset, as well as the phase errors described in the signal model in Section II, the steps from [27] are conducted for every chirp sequence, using the localization results gained by the PF processing.

The relative frequency offset $\delta _{t,n_{\mathrm{F}}}$, also referred as time drift, can be estimated from the linear change of the LoS beat frequency peak along the slow time dimension. With the knowledge of the relative radar positions from the PF processing, the frame-wise initial time offset $\Delta \tau _{0,n_{\mathrm{F}}}$, which can be considered as constant distance offset in range dimension, can be calculated by comparing the LoS beat distance with the geometry estimates. To achieve phase coherence between the bistatic nodes, the LO phase offsets between both radars $\Delta \Theta _{k,n_{\mathrm{F}}}$ need to be estimated for every chirp in each frame. This is done be a comparing the LoS peak phases of both bistatic signals. Subsequently, a $\pi$-unwrapping along the slow-time dimension must be executed for the complete bistatic SAR data due to the reduced ambiguity. A detailed description is given in [27], where also a method for the compensation of the fast-time phase noise $\phi _{\text{pn},k,n_{\mathrm{F}},1/2}$ is given which we apply to the bistatic dataset as well.

This yields the synchronized bistatic signals from radar 2 to radar 1 $\tilde{s}_{\mathrm{B},k,n_{\mathrm{A}},n_{\mathrm{F}}}^{2\to 1}(t)$ and from radar 1 to radar 2 $\tilde{s}_{\mathrm{B},k,n_{\mathrm{A}},n_{\mathrm{F}}}^{1\to 2}(t)$. Additionally, the monostatic signal of the mobile radar $s_{\mathrm{B},k,n_{\mathrm{A}},n_{\mathrm{F}}}^{2\to 2}(t)$ can be used to create a monostatic SAR image.

B. Back-Projection

The SAR images are then calculated using the back-projection algorithm. Therefore, the fast-time fourier transform of the beat signals is performed, giving

View Source \begin{align*} S_{\mathrm{B},k,n_{\mathrm{A}},n_{\mathrm{F}}}^{1\to 2/2\to 1}(f) &= \mathcal {F}\left\lbrace \tilde{s}_{\mathrm{B},k,n_{\mathrm{A}},n_{\mathrm{F}}}^{1\to 2/2\to 1}(t)\right\rbrace \tag{47}\\ S_{\mathrm{B},k,n_{\mathrm{A}},n_{\mathrm{F}}}^{2\to 2}(f) &= \mathcal {F}\left\lbrace s_{\mathrm{B},k,n_{\mathrm{A}},n_{\mathrm{F}}}^{2\to 2}(t)\right\rbrace. \tag{48} \end{align*}

$$\begin{align*} d_{\text{hyp},k,n_{\mathrm{A}},n_{\mathrm{F}}}^{i\to j}(x,y) =& \left\Vert \vec{p}(x,y) - \vec{p}_{\text{TX},i,k,n_{\mathrm{F}}}\right\Vert \\ &+ \left\Vert \vec{p}_{\text{RX},j,k,n_{\mathrm{A}},n_{\mathrm{F}}} - \vec{p}(x,y)\right\Vert, \tag{49} \end{align*}$$ View Source \begin{align*} d_{\text{hyp},k,n_{\mathrm{A}},n_{\mathrm{F}}}^{i\to j}(x,y) =& \left\Vert \vec{p}(x,y) - \vec{p}_{\text{TX},i,k,n_{\mathrm{F}}}\right\Vert \\ &+ \left\Vert \vec{p}_{\text{RX},j,k,n_{\mathrm{A}},n_{\mathrm{F}}} - \vec{p}(x,y)\right\Vert, \tag{49} \end{align*}

$$\begin{equation*} f_{\mathrm{B},k,n_{\mathrm{A}},n_{\mathrm{F}}}^{i\to j}(x,y) = -\mu \frac{d_{\text{hyp},k,n_{\mathrm{A}},n_{\mathrm{F}}}^{i\to j}(x,y)}{\mathrm{c}_{0}}. \tag{50} \end{equation*}$$ View Source \begin{equation*} f_{\mathrm{B},k,n_{\mathrm{A}},n_{\mathrm{F}}}^{i\to j}(x,y) = -\mu \frac{d_{\text{hyp},k,n_{\mathrm{A}},n_{\mathrm{F}}}^{i\to j}(x,y)}{\mathrm{c}_{0}}. \tag{50} \end{equation*}

$$\begin{align*} I_{\text{SAR},n_{\mathrm{A}}}^{i\to j} (x,y) =& \sum _{n_{\mathrm{F}}=0}^{N_{\mathrm{F}}-1}\sum _{k=0}^{N_{\text{ch}}-1} S_{\mathrm{B},k,n_{\mathrm{A}},n_{\mathrm{F}}}^{i\to j}(f_{\mathrm{B},k,n_{\mathrm{A}},n_{\mathrm{F}}}^{i\to j}(x,y))\\ & \cdot \mathrm{e}^{2\pi \mathrm{j}f_{\mathrm{c}}\frac{d_{\text{hyp},k,n_{\mathrm{A}},n_{\mathrm{F}}}^{i\to j}(x,y)}{\mathrm{c}_{0}}}, \tag{51} \end{align*}$$ View Source \begin{align*} I_{\text{SAR},n_{\mathrm{A}}}^{i\to j} (x,y) =& \sum _{n_{\mathrm{F}}=0}^{N_{\mathrm{F}}-1}\sum _{k=0}^{N_{\text{ch}}-1} S_{\mathrm{B},k,n_{\mathrm{A}},n_{\mathrm{F}}}^{i\to j}(f_{\mathrm{B},k,n_{\mathrm{A}},n_{\mathrm{F}}}^{i\to j}(x,y))\\ & \cdot \mathrm{e}^{2\pi \mathrm{j}f_{\mathrm{c}}\frac{d_{\text{hyp},k,n_{\mathrm{A}},n_{\mathrm{F}}}^{i\to j}(x,y)}{\mathrm{c}_{0}}}, \tag{51} \end{align*}

C. Bistatic PSF

First, an investigation of the bistatic near-field PSFs is conducted to validate the performance of the trajectory estimation and synchronization by comparing the images to the ideal simulations. Thus, the scenario shown in Fig. 2 is adopted for this measurement. Fig. 8 shows a picture of the laboratory setup. To obtain a point-like target, a small metal rod with a diameter of 2.0cm is used. The results of these measurements are illustrated in Fig. 10. The extend of the PSFs is comparable to the ideal simulation results. The peak positions are slightly shifted from the actual target positions by a few centimeters due to residual localization and synchronization errors, which is acceptable for most applications. A significant difference between the simulation and the measurement can be seen in the image of the bottom right target (number 5), where a lot of signal energy is spread over the bottom right corner. This effect is due to the direct path interference (DPI), but it can be suppressed by advanced DPI suppression techniques [47].

Figure 10.

Bistatic SISO SAR images $I_{\text{SAR},0}^{2\to 1} (x,y)$ of a metal rod target at five different postions. The triangles indicate the ground truth positions of the targets, measured by the optical reference system. The dashed cyan line shows the TX antennas positions and the magenta circle the stationary RX antenna position.

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D. Complex Target Scene

After validating the performance of the direct path processing to have a sufficient quality of coherence in time and space, the differences in the imaging performance of complex targets between monostatic and bistatic SAR recordings are investigated. Due to the results of Section IV, constellations where radar 2 moves towards radar 1 where chosen, as this increases the localization performance. To increase the signal-to-noise ratio (SNR), all RX channels of the receiving radar are summed up coherently. The resulting SIMO SAR images are calculated as

View Source \begin{equation*} I_{\text{SAR}}^{i\to j} (x,y) = \sum _{n_{\mathrm{A}}=0}^{N_{\text{RX}}-1} I_{\text{SAR},n_{\mathrm{A}}}^{i\to j} (x,y). \tag{52} \end{equation*}

Figure 11.

Bistatic outdoor SAR imaging scene with a bicycle and two scooters (a). The resulting bistatic (b) and monostatic (c) images are plotted together with the radar positions.

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Figure 12.

SAR measurement scene of a person in front of the bistatic constellation (a) with the bistatic (b) and monostatic (c) images.

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