Geometry-Induced Acceleration in Case of a Straight-Line Target Trajectory in Bistatic Radar

Bistatic radar signal processing often assumes a constant velocity for target detection. The bistatic geometry alters, however, the effectively measurable velocity in relation to the target position and motion even if the target has a constant ground speed. Hence, the bistatic geometry induces instantaneous bistatic target acceleration. This is especially the case if a straight-line target trajectory cuts across several closely spaced velocity isolines. The letter tries to quantify this geometry-induced instantaneous bistatic acceleration, in general, for straight-line target trajectories. Thereby, position-dependent acceleration values are derived for a practical use case.

SECTION I. Introduction

A bistatic pulse-Doppler radar estimates the target distance and velocity using the propagation delay and instantaneous phase differential of individual frequency carriers. Due to the motion, the bistatic time delay $\tau (t)$ of the radar return itself becomes time dependent. It is possible to approximate $\tau (t)$ by Taylor expansion at $t=t_0 + \Delta t$ up to the third degree as $$\begin{align*} &\begin{aligned} \tau _0 + \underbrace{\tau ^{\prime }(t_0)}_{\propto \,v_{\mathrm{eff}}(t_0)} \, \Delta t + \frac{1}{2} \, \underbrace{\tau ^{\prime \prime }(t_0)}_{\propto \,a_{\mathrm{eff}}(t_0)} \, (\Delta t)^2 + \frac{1}{6} \, \tau ^{\prime \prime \prime }(t_0) \, (\Delta t)^3 \end{aligned} \tag{1} \end{align*}$$ in which the second-order derivative $\tau ^{\prime \prime }(t_0)$ is often associated with the target acceleration only. However, the geometrical relation between the transmitter and the receiver nodes affects the effectively measurable velocity. The relative ranges and angles vary simultaneously with respect to the individual nodes along the target trajectory. These variations have an instant impact on the effective bistatic velocity $v_{\mathrm{eff}}$. Since the actual bistatic Doppler velocity is an instantaneous time derivative of the bistatic range, its change will induce an instantaneous acceleration $a_{\mathrm{eff}}$ even for a constant target speed over ground.

Pulse-compression schemes with long coherent processing intervals (CPIs) go along with high-velocity resolutions and are especially influenced by such acceleration effects. Malanowski et al. stated in [5] that the effective velocity change within the CPI shall be less than a velocity resolution cell. They further estimated that a velocity of $100\,\mathrm{m}/\mathrm{s}$ caused a single-digit $\mathrm{m}/\mathrm{s}^2$ acceleration in their particular bistatic FM passive radar geometry. Therefore, the assumption of a constant velocity over the CPI is often not justified. Despite this seems to be commonly present in most mono-, bi-, or multistatic radar geometries, the current literature rarely addresses the issue of a geometry-induced instantaneous acceleration. Allen et al. gave a monostatic derivation in [2, p. 693]. This shall be extended to bistatic radar geometries accompanied by practical values.

SECTION II. Bistatic Scenario With a Baseline Crossing Straight-Line Target Trajectory

This section introduces a simple bistatic radar scenario in which a straight-line target trajectory crosses the baseline with a constant ground speed. Then, the instantaneous bistatic velocity and acceleration present at the receiver are calculated.

In Fig. 1, the transmitter node TX and receiver node RX at the positions $(x=\pm L/2, y=0, z=0)$ are separated by a baseline length $L = 40\,\mathrm{km}$. The target starts from the lower left at $(-60\,\mathrm{km},-35\,\mathrm{km},z_0)$ with an additional height offset $z_0 = 2\,\mathrm{km}$. It then moves with a constant ground speed of $\Vert \vec{v}_{\mathrm{trg}} \Vert = 200\,\mathrm{m/s} = 720\,\mathrm{km/h}$ in a $y$-referenced motion angle $\delta _{\mathrm{yx}} = 60^{\circ }$. The baseline is intentionally crossed at $(621\,\mathrm{m}, 0, z_0)$ so that the result is not fully symmetrical.

Fig. 1.

Cartesian  $xy$-plot  of the bistatic geometry with isoranges (cyan) and a straight trajectory (green) crossing the transmitter–receiver baseline.

In a bistatic geometry, the target speed $\Vert \vec{v}_{\mathrm{trg}} \Vert$ relates to the effectively measurable bistatic velocity $v_{\mathrm{eff}}(t)$ such that $$\begin{align*} v_{\mathrm{eff}}(t) &= \Vert \vec{v}_{\mathrm{trg}} \Vert \, \cos \left(\delta _{\mathrm{bi}}(t) \right) \, \cos \left(\beta (t) / 2 \right). \tag{2} \end{align*}$$

The bistatic angle $\beta (t)$ and the velocity angle $\delta _{\mathrm{bi}}(t)$ relative to the bistatic angle bisector are determined by the target position [4, p.120]. A sketch is illustrated in Fig. 2. These angles become time-dependent due to the target motion so that if the target speed $\Vert \vec{v}_{\mathrm{trg}} \Vert$ is constant, (2) will describe a purely geometry-induced change. The instantaneous acceleration has been computed with the finite difference of adjacent effective velocities and approximated with numerical differentiation by $$\begin{align*} a_{\mathrm{eff}}[k] = \frac{v_{\mathrm{eff}}[k] - v_{\mathrm{eff}}[k-1]}{t_{\mathrm{delta}}}. \tag{3} \end{align*}$$ The effective bistatic velocities are shown in Fig. 3. The accelerations from (3) are plotted in Fig. 4 in a step width of $t_{\mathrm{delta}} = 0.1\,\mathrm{s}$ at the indices $k = \lceil t \,/\, t_{\mathrm{delta}} \rceil$ for the scenario and along the target trajectory of Fig. 1, as described above.

Fig. 2.

Bistatic angles $\beta$, $\delta _{\mathrm{bi}}$, $\delta _{\mathrm{yx}}$, and simulation plane $x^{\prime }y^{\prime }$ at $z_0 = 2\,\mathrm{km}$.

Fig. 3.

Effective instantaneous velocity at points of maximum acceleration with $v_{\mathrm{eff}} \approx \pm 95\,\mathrm{m/s}$ for the baseline-crossing target trajectory.

Fig. 4.

Effective instantaneous acceleration along the baseline-crossing trajectory as average of adjacent values in simulated steps of $t_{\mathrm{delta}} = 0.1\,\mathrm{s}$.

The resulting effective velocity in Fig. 3 has its strongest derivative equivalent to a maximum induced acceleration at two positions close to the bistatic foci marked with ${1}$ and ${2}$. These two points are in a bistatic range of $R_\mathrm{bi\_2} \approx L/2 \, + \, 3\,\mathrm{km}$ at which the accelerations of $-1.95\,\mathrm{m}/\mathrm{s}^2$ and $-2.07\,\mathrm{m}/\mathrm{s}^2$ are induced, as shown in Fig. 4. In both cases, the distance to the closest node is about $10\,\mathrm{km}$. The estimated third-order derivative of range (“jerk”) remains below 0.04 $\mathrm{m}/\mathrm{s}^3$ over the trajectory aside from the points of maximal acceleration. This approach already provides a good estimate and the results were fairly independent of further reduced simulation step widths.

SECTION III. Bistatic Geometry-Induced Acceleration

The downside of a single-trajectory-based analysis is that the estimated maximal acceleration depends on the chosen target trajectory. This section suggests, therefore, a more generic evaluation method for the bistatic scenario presented in Fig. 1 so that a set of multiple parallel trajectories are considered with an associated $y$-referenced motion angle of $\delta _{\mathrm{yx}} = 60^{\circ }$ at arbitrary $(x^{\prime },y^{\prime },z=z_0)$ locations. The target position height offset $z_0=2\,\mathrm{km}$ is defined above a flat $xy$ reference plane in which the transmitter and receiver nodes were placed (see Fig. 2). Thereby, a constant target ground speed $\Vert\vec v_\mathrm{trg}\Vert = 200\,\mathrm{m/s}$ is maintained. It was chosen to perform a 2.5-D simulation, which eases the illustration of the results, in particular, close to the bistatic foci. This simulation approach treats the geometry as a scalar field $R_{\mathrm{bi\_2}}(x^{\prime },y^{\prime }) \in \mathbb {R}^1$ that represents the bistatic range as half traveled distance including $L/2$ as $$\begin{align*} R_{\mathrm{bi\_2}}(x, y) = \frac{R_{\mathrm{tx\_trg}}(x, y, z_0) + R_{\mathrm{rx\_trg}}(x, y, z_0)}{2}. \tag{4} \end{align*}$$ Equation (4) is evaluated on a mesh grid with a rectangular spacing of $150\,\mathrm{m}$. The acceleration field $A_{\mathrm{eff}}$ can then be derived with the nabla operator $\vec{\nabla }$ as double-directional derivative of $R_{\mathrm{bi\_2}}$ $$\begin{align*} A_{\mathrm{eff}}(x, y) &= -\vec{\nabla }\underbrace{\left(-\vec{\nabla }\, R_{\mathrm{bi\_2}} \cdot \vec{v}_{\mathrm{trg}} \right)}_{= \, V_{\mathrm{eff}}} \cdot \, \vec{v}_{\mathrm{trg}}. \tag{5} \end{align*}$$

Finally, the relations obtained from (5) depend on the effective velocity isolines $V_{\mathrm{eff}}(x^{\prime }, y^{\prime })$, as illustrated in Fig. 5, for motion angle $\delta _{\mathrm{yx}} = 60^{\circ }$. A close density will cause high induced instantaneous acceleration areas of $A_{\mathrm{eff}}(x^{\prime }, y^{\prime })$ shown in Fig. 6.

Fig. 5.

Two-dimensional effectively measurable velocity field $V_{\mathrm{eff}}$ in $\mathrm{m}/\mathrm{s}$ for $\delta _{\mathrm{yx}} = 60^{\circ }$, $\Vert \vec{v}_{\mathrm{trg}} \Vert = 200\,\mathrm{m}/\mathrm{s}$ at $z_0 = 2\,\mathrm{km}$ and $L= 40\,\mathrm{km}$.

Fig. 6.

Geometry-induced acceleration at $z_0 = 2\,\mathrm{km}$ in $\mathrm{m}/\mathrm{s}^2$ for $\Vert \vec{v}_{\mathrm{trg}}\Vert = 200\,\mathrm{m/s}$ and $\delta _{\mathrm{yx}} = 60^{\circ }$ that causes the tilted asymmetry .

SECTION IV. Brief Interpretation

The 2-D field $A_{\mathrm{eff}}$ in Fig. 6 can be used to analyze the maximum values for a given motion angle $\delta _{\mathrm{yx}}$ of $\vec{v}_{\mathrm{trg}}$. The target starting position in combination with $\delta _{\mathrm{yx}}$ determines the actual passed location and, subsequently, the maximal induced acceleration. Its extrema were confirmed to occur in close vicinity to the bistatic foci and mainly orthogonal to the motion angle $\delta _{\mathrm{yx}}$. It is noteworthy that this effect occurs in a bistatic configuration also between the nodes close to the baseline. Further, a large area orthogonal to the motion direction exceeds an induced acceleration of $0.5\,\mathrm{m}/\mathrm{s}^2$, which is already $1.8\,(\mathrm{km}/\mathrm{h}) \, / \, \mathrm{s}$. A more detailed view on the induced acceleration values in 2-D proximity of the receiver node is shown in Fig. 7. The induced acceleration in extension from the nodes and relative to the middle of the baseline is not strongly influenced by the baseline length $L$. If continuity and equality of mixed partials are assumed, the expression shown in (5) can further be formulated as $$\begin{align*} A_{\mathrm{eff}}(x, y) &= \Vert \vec{v}_{\mathrm{trg}} \, \Vert ^2 \begin{bmatrix}\sin ^2(\delta _{\mathrm{yx}}) \\ \sin (2 \, \delta _{\mathrm{yx}}) \\ \cos ^2(\delta _{\mathrm{yx}}) \end{bmatrix} \cdot \begin{bmatrix}\partial _{xx} \\ \partial _{y}\partial _{x} \\ \partial _{yy} \end{bmatrix} R_{\mathrm{bi\_2}}(x,y) \tag{6} \end{align*}$$ so that the instantaneous acceleration depends on the angles between the gradient vectors and the motion angle $\delta _{\mathrm{yx}}$. It also relates to $\Vert \vec{v}_{\mathrm{trg}} \Vert ^2$. Hence, the magnitude of the instantaneous geometry-induced acceleration will scale quadratically with the target speed $\Vert \vec{v}_{\mathrm{trg}} \Vert$. Other motion angles $\delta _{\mathrm{yx}}$ can further lead to higher induced acceleration values because the angular relation to the gradient vector changes. This is illustrated for a $30^{\circ }$ motion angle in Fig. 8, whereby the increase is particularly noticeable in the baseline region. The height offset $z_0$ widens the shape close to the foci but decreases the maximal acceleration on the trajectory in Fig. 1 by less than $0.05\,\mathrm{m}/\mathrm{s}^2$.

Fig. 7.

Induced acceleration close to the bistatic receiver node (X) in $\mathrm{m}/\mathrm{s}^2$ at $z_0 = 2\,\mathrm{km}$ for $\Vert \vec{v}_{\mathrm{trg}}\Vert = 200\,\mathrm{m/s}$ and $\delta _{\mathrm{yx}} = 60^{\circ }$.

Fig. 8.

Geometry-induced acceleration in $\mathrm{m}/\mathrm{s}^2$ and now computed at $z_0 = 2\,\mathrm{km}$ for $\delta _{\mathrm{yx}} = 30^{\circ }$, which increases in the baseline region.

Another but similar effect is the receiver slew rate caused by a steered antenna tracking a target motion [1, p. 611]. An accelerated target might, in addition, exhibit severe three-digit acceleration values. This will certainly require an adaptive approach at least in passive radar processing [3], [6]. If the target is maneuvering, the analysis needs to consider chained angular relations of (2) or (6) that can lead to higher order derivatives. The impact will depend on the deviation over the CPI, and this will determine the need for an adaptive processing approach. Even though a bistatic geometry has been used, the effect is present in mono- and multistatic geometries as well.

SECTION V. Conclusion

This letter has shown that the radar geometry may induce a noticeable and a kind of counterintuitive instantaneous acceleration. This is even present for targets with a simple straight-line target trajectory. The assumption of a constant target velocity can, thereby, be problematic for long coherent integration applications with a high-velocity resolution. Surprisingly, the target-trajectory-based evaluation indicated that even a very small jerk as derivative of the acceleration can be present. Practical considerations might need to consider different target altitudes and maneuvers. This letter has shown a generic analysis method for such acceleration effects as a function of the target velocity and the bistatic geometry. This model was verified through a numerical simulation and the results were presented as isolines on Cartesian $xy$-plots for selected evaluation sets. Future analysis could include maneuvering and accelerating targets, which may involve more complex bistatic geometry-induced acceleration and higher order components.