# Bounded Collision Force by the Sobolev Norm

Kevin Haninger and Dragoljub Surdilovic Authors affiliated with Fraunhofer
Institut für Produktionsanlagen und Konstruktionstechnik (e-mail:
{kevin.haninger, dragoljub.surdilovic}@ipk.fraunhofer.de). This project has
received funding from the European Union’s Horizon 2020 programme under grant
agreement No 820689

###### Abstract

A robot making contact with an environment or human presents potential safety
risks, including excessive collision force. While experiments on the effect of
robot inertia, relative velocity, and interface stiffness on collision are in
literature, analytical models for maximum collision force are limited to a
simplified mass-spring robot model. This simplified model limits the analysis
of control (force/torque, impedance, or admittance) or compliant robots (joint
and end-effector compliance). Here, the Sobolev norm is adapted to be a system
norm, giving rigorous bounds on the maximum force on a stiffness element in a
general dynamic system, allowing the study of collision with more accurate
models and feedback control. The Sobolev norm can be found through the
$\mathcal{H}_{2}$ norm of a transformed system, allowing efficient
computation, connection with existing control theory, and controller synthesis
to minimize collision force. The Sobolev norm is validated, first
experimentally with an admittance-controlled robot, then in simulation with a
linear flexible-joint robot. It is then used to investigate the impact of
control, joint flexibility and end-effector compliance on collision, and a
trade-off between collision performance and environmental estimation
uncertainty is shown.

##  I Introduction

Mechatronic design for position control is largely standardized - every
position-controlled robot has high drivetrain stiffness with a controller
designed for high-bandwidth tracking performance and disturbance rejection. On
the other hand, mechatronic design for physically interactive robots is more
application-specific [1, 2, 3], where joint stiffness, end-effector
compliance, and control architecture vary significantly. To design interactive
robots, methods to quantify the impact of compliance and control on
performance and safety are needed. Some trade-offs between performance and
robustness have been shown [4], and the role of joint compliance on relaxing
coupled stability conditions investigated [5], but unified design methods are
not yet established.

TABLE I: Collision metrics in literature

Quantity | Name | Domain | Notes, reference
---|---|---|---
$\max\left|f\left(t\right)\right|$ | Maximum Force* | Human, Environment | Most fundamental [6, 7, 8, 9]
$f_{ss}=f(t)|_{t\rightarrow\infty}$ | Steady State Force | Clamped human, Stiffness environment | [6, 10, 11]
$\max\left|f\left(t\right)v\left(t\right)\right|$ | Maximum Power | Human, Environment | Mentioned, but not investigated in [6]
$\int\left|f\left(t\right)v\left(t\right)\right|dt$ | Absolute Energy Transfer* | Human, Environment | Typ. taken over initial penetration [12, 13, 6]
$\Delta_{t}^{-3/2}\left\|a\left(t\right)|\right\|_{2}^{5/2}$ | Head Injury Criterion | Human: Head, unconstrained | 2-norm only over $\Delta_{t}$, [14, 15]

$*$ Also extended to per-unit-area metrics (i.e. pressure, energy flux)

One design requirement for interactive robots robot is establishing safe
contact with a human or environment, in both collision (unintended) or contact
transition (intended). To allow safe robots moving at reasonable speeds in
semi-structured environments, a limit on contact force should be achieved over
possible contacts. Design for contact is challenging, as contact with a stiff
environment introduces broad frequency input to the robot, exciting high-
frequency resonance modes which are often unmodelled. Additionally, the high-
frequency dynamics of the robot can’t be easily reshaped by control due to
controller bandwidth limitations. This makes the intrinsic dynamics of a robot
especially important in collision, motivating the need for physical compliance
on otherwise high-impedance robots[15].

Figure 1: Two inertia robot model with motor inertia $J$, joint flexibility
$K_{jt}$ (e.g. from torque sensor or harmonic gearbox), link inertia $I$, and
compliance at the interface $K_{e}$ (e.g. soft skin, compliant end-effector),
in contact with an environment. Scaled Dirac delta impulses $\delta$ induce
the initial contact velocity.

Compliance can be realized as a flexible joint between the motor and load [16,
17], a compliant covering [18], or a soft end-effector. Choosing the stiffness
of these compliant elements is as much an art as a science, relying on
iterative testing, and realized values in commercial products are often kept
as trade secrets. Often, other important robot characteristics are fixed (e.g.
motor inertia, link inertia), leaving the choice of stiffness and control
architecture as important design choices.

Designing compliance and control to avoid user injury is important for human-
robot applications. Characterizing human injury from robot collision is
experimentally well-established, and several established metrics are shown in
Table I. Early work [19] experimentally established force thresholds for pain.
A line of investigation [20, 21, 15, 22, 23] draws collision metrics from
automotive testing standards and experimentally establishes the impact of the
robot’s reflected inertia, velocity, and impact geometry on these metrics.

Analytically relating robot design to these biomechanic safety limits is
challenging. ISO 15066:2016, which sets guidelines for human-robot collision,
uses a two mass, single stiffness model under perfect elastic collision to
motivate safe robot speed limits [6]. For low-stiffness flexible-joint robots,
it is suggested that link-side dynamics dominate collision force and that
motor inertia can be neglected [21]. While later work [24] elaborates, a wide
range of joint stiffness are used in practice, from series-elastic and
variable stiffness actuators ($20-1000$ Nm/rad [25, 26]), impedance-controlled
flexible-joint robots (e.g. Kuka IIWA $\sim 10$kNm/rad [27]), to a standard
harmonic drive gearbox ($>$15 kNm/rad), suggesting analysis which can
accommodate a range of joint stiffnesses is useful.

For contact with a pure-stiffness environment, there is established literature
for stability [28, 10, 7], and limiting the peak force of a rigid, perfectly
backdriveable robot [8] using inelastic/elastic collision models. Energy-based
planning methods have also been introduced, to bound kinetic energy in robot
links [23], or bound maximum power flow between robot and environment [29].

To quantify the effect of compliance and control on collision force, this
paper adapts the Sobolev norm as a system norm, allowing a rigorous bound on
maximum collision force with an arbitrary environment. The Sobolev norm is
shown to be the $\mathcal{H}_{2}$ norm of a transformed system, allowing
computation with existing methods. This norm is validated experimentally and
in simulation, for pure stiffness as well as inertial environments. The impact
of joint and interface compliance, motor dynamics, and control on a linearized
two-inertia robot model is investigated. Finally, a trade-off between
collision performance and environmental estimation uncertainty is shown.

##  II System Metrics for Interactive Robots

This section motivates the collision model and approaches for translating
collision metrics into system norms.

###  II-A Impact Metrics and Models

Some impact metrics used in experimental literature can be seen in Table I,
where the force and velocity at the interaction port are denoted $f(t)$ and
$v(t)$, and acceleration of an environmental inertia is $a(t)$ (e.g. head
acceleration).

Contact is here modelled with a rigidly coupled robot/environment - neglecting
unilateral contact conditions. The robot model is seen in Figure 1, with two
inertias (motor and link inertia), and two stiffnesses (joint and end-effector
compliance). This model accommodates impedance and admittance-controlled
robots, which respectively measure joint torque at $K_{jt}$ or end-effector
force/torque near $K_{e}$ (wrist force/torque sensor). For a serial
manipulator, this model can be considered as a linearization about a contact
pose.

To model the relative velocity $v_{0}$ between robot and environment at the
moment contact is made, a state-space model could be used with initial
conditions $\dot{\theta}=\dot{q}=v_{0}$. Here, it is convenient to represent
this initial velocity as a Dirac delta input at each robot inertia, scaled by
that inertia’s momentum. This impulse, when applied at $t=0$, induces the
initial velocity $v_{0}$. As any output force is linear in this contact
velocity, a unit velocity $v_{0}=1$ is assumed.

###  II-B Metric Bounds by Norms

Given the model in Figure 1, it is desired to guarantee bounds on the
collision metrics seen in Table I as a function of robot properties. Let the
force of interest be $f(t)=K_{e}(x-q)$, the contact force at the interface
with the environment. Denote the transfer function from $\delta\rightarrow f$
as $G_{f}(s)$. Recall that $max|f(t)|=||f(t)||_{\infty}$, and when there is no
other input to the system ($x=0$, $\tau_{m}=0$, $\delta$ is the Dirac
impulse), $f(t)=\mathscr{L}^{-1}\left\\{G_{f}(s)\right\\}$ [30]. Although
$f(t)$ is, in general, an arbitrary output of the system, it will be here
regarded as the force response to the unit velocity initial condition with a
specified environment and control policy.

If a different linear collision model is written with input
$u(t)\in\mathcal{L}_{\infty}$ (note that an impulse
$\delta\notin\mathcal{L}_{\infty}$), with transfer function $G_{uf}(s)$ for
dynamics $u\rightarrow f$, the force can be bounded as
$\|f(t)\|_{\infty}<\|g_{uf}(t)\|_{1}\|u(t)\|_{\infty}$ [31], but the $1$ norm
of an impulse response $g_{uf}(t)=\mathcal{L}^{-1}\left\\{G_{uf}(s)\right\\}$
is not easily computed, and the conservatism of this norm in collision
analysis is noted [7].

###  II-C Sobolev Spaces

Recall that while norm equivalence is guaranteed over a finite-dimensional
vector space $\mathcal{V}$ (i.e. $\exists
c\,\,s.t.\,\,\left\|x\right\|_{q}<c\left\|x\right\|_{p}\,\,\forall
x\in\mathcal{V}$), this does not hold for uncountably infinite spaces, such as
continuous-time signals $f(t)\in\mathcal{L}_{p}$. So, bounding $\|f(t)\|_{2}$
or $\|f(t)\|_{1}$ does not guarantee a bound on $\|f(t)\|_{\infty}$.

If a signal is continuous with a continuous derivative, Sobolev spaces can be
used to find absolute bounds on $\|f(t)\|_{\infty}$. Define the Sobolev norm
as

$\displaystyle\left\|f(t)\right\|_{W^{1,p}}$ $\displaystyle=$
$\displaystyle\|f\left(t\right)\|^{p}_{p}+\|\dot{f}\left(t\right)\|^{p}_{p},$
(1)

then the following inequality holds from [32], Theorem 8.8,

$\displaystyle\left\|f\left(t\right)\right\|_{\infty}\leq\gamma\left\|f\left(t\right)\right\|_{W^{1,p}}\,\,\,\forall
f\in W^{1,p},$ (2)

for a fixed constant $\gamma$. The proof is reproduced in the Appendix, with
the intuition that while a signal with a finite square integral can have an
arbitrarily large peak (e.g. a Gaussian distribution as covariance approaches
zero), if the square integral of a signal and its derivative are bounded, a
bound on the absolute signal peak exists.

If the $2-$norm is used and $f(t)$ is scalar,
$\left\|f\left(t\right)\right\|_{W^{1,2}}$ can be expressed in the frequency
domain with Parseval’s Theorem [30], yielding:

$\displaystyle\|f(t)\|_{W^{1,2}}$
$\displaystyle=\|F(j\omega)\|^{2}_{2}+\|j\omega F(j\omega)\|^{2}_{2}$ (3)

where $F\left(j\omega\right)$ is the Fourier transform of $f\left(t\right)$.
For the collision model here, $F(j\omega)=G_{f}(j\omega)$, the transfer
function from $\delta\rightarrow f$.

When the signal $F(j\omega)$ is a rational polynomial with relative degree
two, poles only in the open left half plane, and $\lim_{s\rightarrow
0}|F(s)|<\infty$, the Sobolev norm exists (i.e. is finite), and can be easily
computed from existing signal theory.

These existence conditions restrict the signals which can be analyzed with the
Sobolev norm, but when the force of interest is the force through a stiffness
element between two inertias, this condition is satisfied. As force is
typically measured by strain gauges on a pure stiffness (in both joint torque
sensors and 6-DOF force/torque sensors), this condition is practical for
bounding the measured force on real-world robots.

##  III Mechatronic Design for Collision

This section validates the Sobolev norm for interactive robots in contact with
a pure stiffness, and a stiffness/inertia environment.

###  III-A Pure Stiffness Environment Validation

Contact experiments with a Manutec r3 industrial manipulator under admittance
control, as seen in Figure 2, with details on controller architecture in [10].
The admittance parameters (stiffness, damping, and inertia) are varied over a
set of 64 contact transitions, with ranges shown in Table II.

Figure 2: End-effector of a Manutec r3, in contact with pure-stiffness
environment ($\sim 56kN/m$) as used in experimental validation of the proposed
collision norm.

The model which the Sobolev norm is applied to is a linearized model for the
position-controlled robot, and the outer-loop admittance control parameters
(target mass, damping and stiffness). The environment is identified as a
$56kN/m$ stiffness. This gives a total transfer function:

$\displaystyle F(s)=\frac{G}{1+CAGK_{e}+GC+GKe}\left(Mv_{0}+CX^{d}\right)$ (4)

where $C(s)$ is the position controller, $G(s)$ linearized model from torque
to position, $K_{e}$ environmental stiffness, $A(s)$ the target admittance,
$F(s)$ the interaction force and $X^{d}(s)$ the reference trajectory signal.
Model and controller values used can be seen in Table II. The position
reference is dropped in this analysis because the steady-state force it
induces gives an infinite Sobolev norm, and the transient response (which
contains potentially dangerous force peaks) is assumed to be dominated by the
momentum of the robot (i.e. the $Mv_{0}$ term in (4)).

For each contact experiment, the model was simulated with identical controller
parameters, two example experimental and model force trajectories can be seen
in Figure 3. A good correspondence in the initial force peak, resonant
frequency, and decay rate are achieved with a model response using the
parameters of Table II. Steady-state error between model and experiment arises
from the contribution of the $X^{d}$ reference trajectory, which is not
included in the simulation.

The model is then used to calculate the $\|f(t)\|_{W^{1,2}}$ norm for each
controller configuration, which is compared with the corresponding observed
maximum force from the experiments. Results can be seen in Figure 4, achieving
good correspondence between the Sobolev Norm and observed peak force across a
range of peak force values.

Figure 3: Force response in two contact transitions compared with simulated response (simulation ignores trajectory reference, leading to steady-state error).  TABLE II: Model parameters fit to the Manutec r3 Param | Value | Units/Notes
---|---|---
$G\left(s\right)$ | $\left(308s^{2}+40s\right)^{-1}$ | with $M=308$
$C\left(s\right)$ | $85s+370$ | position control bandwidth $\sim 4$ Hz
$K_{e}$ | $5.6e4$ | $[N/m]$
$A\left(s\right)$ | $\frac{1}{M_{t}\left(s^{2}+2\xi\omega_{t}+\omega_{t}^{2}\right)}$ | Ranges used:$\begin{cases}\omega_{t}&\left[.5,\,1,\,3,\,6,\,8\right]\\\ M_{t}&\left[25-400\right]\end{cases}$
Figure 4: Peak initial force of the experiment compared with Sobolev Norm of
the matching model for various admittance parameters.

###  III-B Inertial Environment Validation

To validate the norm in inertial environments, a simulation was used based on
Figure 1, with the environment being a damped inertia
($f=B_{h}\dot{x}+I_{h}\ddot{x}$). The robot parameters are motivated by the
first joint of the KUKA LWR as identified in [27], and the human parameters
(i.e. $K_{e}$ and environment inertia $H$) from Table A.3 in [6]. These human
parameters are translated into rotational coordinates under an assumed contact
radius of $.2$-$.75$ meters.

TABLE III: Robot and Human Model Parameters Param | Value | Units/Notes
---|---|---
$M$ | 3.19 | Kgm2
$B$ | 24.3 | Nms/rad
$K_{jt}$ | 10 | kNm/rad
$I$ | 4.5 | Kgm2
$V$ | 20.3 | Nms/rad
$K_{e}$ | $\in\\{10r^{2},150r^{2}\\}$ | kNm/rad
$B_{h}$ | $75r^{2}$ | Nms/rad
$I_{h}$ | $\in\\{.6r^{2},175r^{2}\\}$ | Kgm2
$r$ | $\in\\{.2,.75\\}$ | m
Figure 5: Norm validation in simulation with inertial environment.

The results seen in Figure 5 compare the maximum force observed in simulated
collision with the Sobolev norm of the model. Again a good correspondence is
achieved over the range of stiffnesses and inertias expected from humans, with
a largely monotonic relationship. Qualitatively, this norm is sensitive to
damping (just as the $\mathcal{H}_{\infty}$ norm is), and sufficient damping
on the robot model must be included to maintain good predictions.

###  III-C Discussion

The relationships seen in Figures 4 and 5 suggests that although the Sobolev
norm is only an upper bound in theory, it can be useful as a design tool:
within a family of related systems, decreasing the Sobolev norm reduces peak
force. However, the experiments here only demonstrate the Sobolev norm on
systems where parameters of a fixed architecture are varied. Although this is
relevant for mechatronic design (e.g. choosing joint stiffness, interface
compliance, controller gains), additional validation is required before using
the Sobolev norm to compare impact force between arbitrary systems.

##  IV Design and analysis via Sobolev norm

With the Sobolev norm providing a useful design approximation for collision
performance, its compatibility with established $\mathcal{H}_{2}$ theory can
be used to investigate the impact of joint flexibility, interface compliance,
and control on peak collision force.

###  IV-A State Space Model

The system shown in Figure 1 can be directly written in state-space form,
separating the environment by taking it’s velocity $\dot{x}$ as input. For a
pure-stiffness environment, the stiffness of environment can be incorporated
in $K_{e}$ and $\dot{x}=x=0$. The augmented state
$\xi=[\theta,\dot{\theta},q,\dot{q},x]$ and the decomposition
$B=[B_{\delta},B_{\tau},B_{\dot{x}}]$ will be used in the following.

$\displaystyle\frac{d}{dt}\xi$ $\displaystyle=$ $\displaystyle
A\xi+B\left[\begin{array}[]{c}\delta\\\ \tau_{m}\\\ \dot{x}\end{array}\right]$
(8) $\displaystyle w=\left[\begin{array}[]{c}f\\\ \dot{f}\end{array}\right]$
$\displaystyle=$ $\displaystyle C_{w}\xi+D\left[\begin{array}[]{c}\delta\\\
\tau_{m}\\\ \dot{x}\end{array}\right]$ (14)

$\displaystyle\begin{split}\small\left[\begin{array}[]{c|c}A&B\\\ \hline\cr
C_{w}&D_{w}\end{array}\right]=\\\
\left[\begin{array}[]{ccccc|ccc}0&1&0&0&0&0&0&0\\\
\texttt{-}K_{jt}J^{\texttt{-}1}&\texttt{-}BJ^{\texttt{-}1}&K_{jt}J^{\texttt{-}1}&0&0&1&J^{\texttt{-}1}&0\\\
0&0&0&1&0&0&0&0\\\
K_{jt}I^{\texttt{-}1}&0&\texttt{-}\left(K_{jt}\texttt{+}K_{e}\right)I^{\texttt{-}1}&\texttt{-}VI^{\texttt{-}1}&0&1&0&0\\\
0&0&0&0&0&0&0&1\\\ \hline\cr 0&0&K_{e}&0&\texttt{-}K_{e}&0&0&0\\\
0&0&0&K_{e}&0&0&0&\texttt{-}K_{e}\end{array}\right]\end{split}$

###  IV-B Sobolev as an $\mathcal{H}_{2}$ norm

The Sobolev norm of the force response of the impulse collision model can be
written as

$\displaystyle\|f\|_{W^{1,2}}$ $\displaystyle=$
$\displaystyle\|f(t)\|_{2}^{2}+\|\dot{f}(t)\|_{2}^{2}$ (15) $\displaystyle=$
$\displaystyle\int_{0}^{\infty}w^{T}(t)w(t)dt$ (16) $\displaystyle=$
$\displaystyle\|G_{\delta f}(s)\|_{2}^{2},$ (17)

where $G_{\delta f}$ is the transfer function from $\delta$ to $f$. The
$\mathcal{H}_{2}$ norm can be directly calculated through the controllability
or observability Gramian [30], for example

$\displaystyle\|G_{\delta w}\|_{2}^{2}$ $\displaystyle=$
$\displaystyle\mathrm{trace}\left(B_{\delta}^{T}L_{o}B_{\delta}\right)$ (18)
$\displaystyle A^{T}L_{o}+L_{o}A$ $\displaystyle=$ $\displaystyle-
C_{w}^{T}C_{w}.$ (19)

The state feedback controller $\tau_{m}=K\xi$ which minimizes the sum of the
square of the Sobolev norm and a quadratic control penalty can be found
through an LQR problem [30]

$\displaystyle K_{lqr}$
$\displaystyle=\arg\min_{K}\int_{0}^{\infty}w^{T}(t)w(t)+\tau_{m}(t)^{T}R\tau_{m}(t)dt$
(20)
$\displaystyle=\arg\min_{K}\|f(t)\|^{2}_{W^{1,2}}+\int_{0}^{\infty}\tau_{m}(t)^{T}R\tau_{m}(t)dt,$
(21)

with minimum cost of $\mathrm{tr}(B_{\delta}^{T}SB_{\delta})$, where $S$ is
the unique solution to the LQR Ricatti Equation

$\displaystyle A^{T}S+SA-SB_{\tau}R^{-1}B_{\tau}^{T}S+C_{w}^{T}C_{w}=0.$ (22)

Although the LQR controller is not practical (complete state-space information
is rarely available, high-DOF robots are not approximately linear, robot
control must also achieve other objectives), this gives a baseline to compare
the possible impact of control on collision.

###  IV-C Joint Stiffness Analysis

The impact of control and joint flexibility on collision performance can now
be investigated. The complete model is the system in Figure 1, under
parameters in Table III, in contact with a $K_{e}=30$ kNm/rad environment.
Four different cases can be compared: the uncontrolled system, the system
without motor dynamics omitted (i.e. $J=0,B=0$), the system with torque
control $\tau_{m}=(K_{p}+K_{d}s)K_{jt}(\theta-q)$ ($K_{p}=1.5$, $K_{d}=.2$),
and the system under state feedback with LQR gains from (20) with $R=5$. These
four cases are compared as the joint stiffness changes in Figure 6.

Figure 6: Collision performance of robot with no control ($\tau_{m}=0$), no
motor dynamics ($J=B=0$), torque control and LQR control, with parameters from
Table III at $K_{e}=30$ kNm/rad. Note $y$ axis range.

At low joint stiffnesses, control and motor dynamics make no difference, and
all systems have the same collision performance which is dominated by link
dynamics. As joint stiffness increases, the no control system collision
performance increases to $150\%$ of the no motor case, showing how at higher
joint stiffnesses, the motor dynamics affect the collision performance.
However, the effect of motor dynamics is substantially reduced by the inner-
loop torque control (for both PD or LQR). Although physical compliance reduces
the impact of motor dynamics, it does not need to be very low to achieve good
collision performance if control is used.

LQR and PD torque control present similar behaviors, which can be seen from
the similarity of their expressions. Typical LQR results with $R=5\rightarrow
K_{lqr}=[1.28,\,.03,\,-1.30,\,-.05,\,0]1e4$ and a PD torque controller written
as a similar state feedback gain $K_{pd}=[Kp,\,Kd,\,-Kp,\,-Kd,\,0]K_{jt}$. The
matching of signs (as well as approximate magnitude commonly used on PD
control of joint torque) shows that inner-loop PD torque control is well-
aligned with reducing peak impact force.

###  IV-D Interface Stiffness Analysis

While decreasing the stiffness of the interface $K_{e}$ provides improvements
in collision performance, there is a trade-off with performance. Classical
interpretations of performance in control theory is the ability to reshape the
apparent dynamics of the system (i.e. reshaping the transfer functions of
disturbance rejection, reference tracking, noise sensitivity). For interactive
robots, the apparent dynamics are a direct performance goal as well, although
the range of dynamics required is not as easily formalized.

There is a secondary sense of performance for interactive robots; their
ability to discover actionable information from the environment. Just as
actuator dynamics and non-collocation limit the dynamics which can be
rendered, sensor dynamics and non-collocation limit the information flow from
the environment. As a concrete example, take a robot coming into contact with
a kinematic plane at an uncertain position. While knowledge of this plane’s
location may be important for subsequent behavior, the compliance at the end-
effector can limit the ability to estimate the hidden information of the
environment.

Working this example within the model in Figure 1 as written in (IV-A), this
information transfer can be stated as the covariance of the posterior estimate
$p(\hat{x}|y(0:T))$, where $y(0:T)$ is the history of measurements available.
Impedance- and admittance-controlled robots use motor position measurements,
along with measurements of the force along $K_{jt}$ and $K_{int}$,
respectively which can be written as emission equations on the states seen in
(IV-A).

$\displaystyle y_{imp}$ $\displaystyle=$
$\displaystyle\left[\begin{array}[]{ccccc}1&0&0&0&0\\\
K_{jt}&0&-K_{jt}&0&0\end{array}\right]\xi$ (25) $\displaystyle y_{adm}$
$\displaystyle=$ $\displaystyle\left[\begin{array}[]{ccccc}1&0&0&0&0\\\
0&0&K_{int}&0&-K_{int}\end{array}\right]\xi$ (28)

Under a unified noise model, with process noise covariance
$\Sigma_{w}=\mathtt{diag}([.1,.5,.1,10,15])$, sensor noise
$\Sigma_{v}=\mathtt{diag}([.110])$, the steady-state covariance of the
estimate of environment location $\hat{x}$ can be found in closed form from
standard solutions to the Kalman filter. The impact of $K_{int}$ on this
covariance can be seen in Figure 7, where it is compared with collision
performance. Collision performance monotonically improves as stiffness
decreases, but the quality of the estimation of environment decreases at lower
stiffnesses as well. Low stiffnesses allow torque-sensor noise to dominate the
force arising from the environment.

Figure 7: Effect of interface stiffness $K_{e}$ on collision performance
(blue) and covariance of estimate of environment location under impedance
(top, green) and admittance (bottom, green), parameters from Table III at
$K_{jt}=10$ kNm/rad.

##  V Conclusion

This paper introduced a system norm for collision analysis based on the
Sobolev norm. By bounding the maximum force on a pure stiffness, it provides
useful predictions for human-robot and robot-environment contact transitions.
The norm was validated with experiments of a pure stiffness environment, and
simulation of an inertial environment, and the maximum impact force and
Sobolev norm are shown to correspond well over parametric variation of the
systems. The Sobolev norm of a system is then stated as the $\mathcal{H}_{2}$
norm of a related system, and existing control theory used to establish the
impact of joint stiffness and interface stiffness on impact performance,
towards validating design approximations which are used in practice.

This norm provides the advantage over existing methodologies (perfect elastic
collision of two inertias with single stiffness) by allowing the accounting
for joint stiffness, motor dynamics, and control. Furthermore, collision can
be considered over a wider continuum of environments, also supporting design
for contact transitions with industrial, high-stiffness environments.

## Appendix

The proof of [32], Theorem 8.8 is recreated with additional steps for clarity.
Let $f\left(t\right)\in C^{1}$, a continuous function with a continuous
derivative, and for $1\leq p<\infty$ define the function
$G\left(s\right)=\left|s\right|^{p-1}s$. The function $v=G\left(f\right)$ is
in $C^{1}$ and

$\displaystyle v^{\prime}$ $\displaystyle=G^{\prime}\left(f\right)f^{\prime}$
$\displaystyle=p\left|f\right|^{p-1}f^{\prime}$ $\displaystyle\Rightarrow
v=\int_{-\infty}^{t}p\left|f\left(\tau\right)\right|^{p-1}f^{\prime}\left(\tau\right)d\tau.$

Note that $\left|v\left(t\right)\right|=\left|f\left(t\right)\right|^{p}$, and
can be bounded as

$\displaystyle\left|v\left(t\right)\right|$
$\displaystyle\leq\int_{-\infty}^{t}p\left|f\left(\tau\right)\right|^{p-1}\left|f^{\prime}\left(\tau\right)\right|d\tau,$
$\displaystyle\Rightarrow\left|f\left(t\right)\right|^{p}\leq
p\left\|f\right\|_{p}^{p-1}\left\|f^{\prime}\right\|_{p}.$

Manipulation of this term with a Sobolev norm
$\left\|f\right\|_{W^{1,2}}=\left\|f\right\|_{p}^{p}+\left\|f^{\prime}\right\|_{p}^{p}=\alpha^{p}$,
and noting $p^{1/p}\leq e^{1/e}\,\,\forall p>1$ gives

$\left|f\left(t\right)\right|\leq\gamma\alpha$

for a positive constant $\gamma$ which does not depend on $f(t)$.

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# Estimation of Bistatic Radar Detection Performance Under Discrete Clutter
Conditions Using Stochastic Geometry

Shobha Sundar Ram and Gourab Ghatak
Indraprastha Institute of Information Technology Delhi, India

###### Abstract

We propose a metric called the bistatic radar detection coverage probability
to evaluate the detection performance of a bistatic radar under discrete
clutter conditions. Such conditions are commonly encountered in indoor and
outdoor environments where passive radars receivers are deployed with
opportunistic illuminators. Backscatter and multipath from the radar
environment give rise to ghost targets and point clutter responses in the
radar signatures resulting in deterioration in the detection performance. In
our work, we model the clutter points as a Poisson point process to account
for the diversity in their number and spatial distribution. Using stochastic
geometry formulations we provide an analytical framework to estimate the
probability that the signal to clutter and noise ratio from a target at any
particular position in the bistatic radar plane is above a predefined
threshold. Using the metric, we derive key radar system perspectives regarding
the radar performance under noise and clutter limited conditions; the range at
which the bistatic radar framework can be approximated to a monostatic
framework; and the optimal radar transmitted power and bandwidth. Our
theoretical results are experimentally validated with Monte Carlo simulations.

###### Index Terms:

stochastic geometry, bistatic radar detection, Monte Carlo simulations, indoor
clutter, Poisson point process

##  I Introduction

Historically, bistatic radars were researched for military applications - for
tracking missiles, for spotting certain types of targets whose radar cross-
section or Doppler information is enhanced in a bistatic/multi-static
configuration or for detecting targets in a forward scattering scenario - when
they interrupt the direct path between a radar transmitter and receiver [1, 2,
3, 4]. However, there are some significant disadvantages in bistatic systems.
First, the synchronization between the radar transmitter and receiver is
challenging. Second, the view of the bistatic radar may be more limited than a
monostatic radar since a target has to be visible to both the transmitting and
the receiving antennas. Finally, the localization and Doppler estimation
algorithms are significantly more complex in a bistatic scenario when compared
to monostatic radar [3]. Despite these limitations, recently, there has been a
revival of interest in the research and development of bistatic radar
specifically in the context of microwave and millimeter wave integrated
sensing and communication systems (ISAC). For example, radar sensing using
wireless access points as opportunistic illuminators, with passive, cheap and
undetectable receivers, have been researched for detecting, tracking and
monitoring human activities in indoor environments [5, 6]. Another example is
millimeter based vehicle-to-everything (V2X) communication systems that are
increasingly incorporating sensing functionality within their communication
protocols [7]. There are several advantages to these ISAC systems: first, dual
functionality is realized with common hardware and spectrum resources [8, 9].
Second, in some cases, the incorporation of the localization functionality
facilitates improvement in communication metrics.

Indoor wireless and localization systems encounter significant discrete
clutter scatterers such as furniture and walls with strong specular
backscatter in the radar signatures. Further, they give rise to multipath off
target backscatter which manifests as ghost targets in the radar signatures
[10]. Similarly, in outdoor automotive scenarios, reflections from roads,
railings and surrounding environments give rise to significant point clutter
responses in the radar signatures. While there has been statistical analysis
of bistatic radar detection performance due to surface clutter [11], there is
little research into quantifying the performance due to discrete clutter since
there can be considerable variation in the environmental conditions. In this
work, we propose a metric called bistatic radar detection coverage probability
($\operatorname{\mathcal{P}_{DC}^{Bi}}$) to quantify the radar detection
performance based on the signal to clutter and noise ratio (SCNR) using
stochastic geometry (SG) principles. $\operatorname{\mathcal{P}_{DC}^{Bi}}$
specifies the fraction of targets in the bistatic region-of-interest which
have SCNR above a predefined threshold. In [12, 13] we used SG to derive the
monostatic radar detection coverage probability to quantify the performance of
a monostatic radar under discrete clutter conditions. The metric accounted for
wide variation in the radar, target and clutter conditions and provided system
perspectives on radar deployment and optimization of bandwidth, pulse width,
maximum power and gains of transmitting and receiving antennas. In this work,
we use SG principles to model the distribution of the discrete clutter
scatterers as a Poisson point process (PPP) similar to [14]. The PPP is a
natural choice since it is a completely random process where each clutter
point is stochastically independent to the location of all the other clutter
points. Thus it accounts for the diversity in the number and spatial
distribution of the clutter scatterers around the radar. With this model, we
consider each radar deployment as an independent instance of an underlying
spatial stochastic process. The PPP assumption leads to elegant expressions of
the SCNR and enables the radar operator to dimension the deployments and
derive useful insights not only throughout different spatial locations in a
particular deployment, but also across different deployments with modified
parameters.

We use the classical bistatic radar geometry framework proposed by [15] where
the bistatic range is identified as the geometric mean ($\kappa$) of the one-
way propagation distances between the radar transmitter and receiver with
respect to the target. An important important parameter in bistatic radar is
the baseline length ($L$) between the transmitter and receiver. In our work,
we derive the SCNR for scenarios where $L\leq 2\kappa$ by considering the
returns from a single target and from clutter scatterers within the same
clutter resolution cell as the target. We consider two types of clutter
resolution cells - beamwidth limited clutter resolution cell which is the
region in the bistatic plane within the main lobes of both the transmitting
and receiving antennas; and the range limited resolution cell which is the
region defined by the beamwidth of either antenna and the range resolution
bin. Based on the SCNR, our main contribution in this paper is an analytical
framework that quantifies the radar performance as a function of radar
parameters (such as beamwidth, transmitted power, bandwidth), clutter
parameters (clutter distribution, and clutter cross-section) and target
parameters (target cross-section). Second, we provide key system insights into
radar performance under noise and clutter limited conditions; the maximum
power at which detection performance saturates; the optimal radar bandwidth
for best detection performance; and the bistatic radar range as a function of
baseline length, at which the bistatic radar approximates to monostatic radar.
We experimentally validate the analytical results with Monte Carlo
simulations. The paper is organized as follows. In the following section, we
present the bistatic radar geometry followed by theorems to compute
$\operatorname{\mathcal{P}_{DC}^{Bi}}$ for both beamwidth limited and range
limited radar resolution cells along with their corollaries. Then in
Section.III, we present the radar detection performance as a function of
radar, target and clutter parameters. Finally, we conclude the paper with a
summary of the insights gained by the study in Section.IV

_Notation:_ We have used the following notation in this paper. Random
variables are indicated by bold face fonts while regular parameters and
realizations of random variables are written using regular font.

##  II Theory

In this paper, we follow the North referenced bistatic geometry configuration
popularized by Jackson [15]. We consider a two-dimensional Cartesian
coordinate space with the bistatic radar transmitter and receiver at
$(\pm\frac{L}{2},0)$ respectively as shown in Fig.1(a). We consider a target
at ($r_{t},\operatorname{\mathbf{\theta_{t}}}$) where the distance between the
target and the transmitter and receiver are $R_{tx}$ and $R_{rx}$ respectively
and the bistatic radar angle is $\beta$. We assume that the target is at a
fixed bistatic range $\kappa$ with respect to the radar where $\kappa$ is the
geometric mean of $R_{tx}$ and $R_{rx}$ ($\kappa=\sqrt{R_{tx}R_{rx}}$). The
angular position of the target, $\operatorname{\mathbf{\theta_{t}}}$, is a
uniform random variable between $[0,2\pi)$. Note that the locus of all the
points of $\theta_{t}$ from $[0,2\pi)$ for a fixed $\kappa$ and $L$ forms a
Cassini oval [3] as shown in Fig.1(b).

(a)

(b)

Figure 1: Histograms of sine of bistatic angle ($\sin\beta$) and minimum
normalized one-way range ($R_{min}/\kappa$).

There are three cases: the co-site region ($L<2\kappa$), the lemiscate region
($L=2\kappa$) and when $L>2\kappa$, the oval breaks into two parts, centered
around either transmitter (when $R_{rx}>>R_{tx}$) or the receiver (when
$R_{tx}>>R_{rx}$). We will consider the first two cases in this work. For a
fixed $\kappa$ and baseline length ($L$) and a specific realization of
$\theta_{t}$, the polar radial position of the target $r_{t}$ is determined by
solving the quadratic equation from the triangle law of cosines given by

$\displaystyle\left(r_{t}^{2}+\frac{L^{2}}{4}\right)^{2}-r_{t}^{2}L^{2}\cos^{2}\theta_{t}=\kappa^{4}.$
(1)

Then we can determine $R_{tx}$, $R_{rx}$ and $\beta$ from the equations below
-

$\displaystyle R_{tx}^{2}$
$\displaystyle=\left(r_{t}^{2}+\frac{L^{2}}{4}\right)+r_{t}L\cos\theta_{t}$
(2) $\displaystyle R_{rx}^{2}$
$\displaystyle=\left(r_{t}^{2}+\frac{L^{2}}{4}\right)-r_{t}L\cos\theta_{t}$
(3) $\displaystyle\cos\beta$
$\displaystyle=\frac{\left(R_{tx}^{2}+R_{rx}^{2}-L^{2}\right)}{2\kappa^{2}}.$
(4)

Some comments regarding the bistatic radar geometry: First, note that when
$L=0$, we obtain $R_{tx}=R_{rx}=\kappa$, and the bistatic radar geometry
becomes a monostatic case and the Cassini ovals become concentric circles.
Second, when $\theta_{t}=0$ and for low values of $\kappa$, we obtain the
forward scattering scenario when the target is directly between the transmit
and receiver. Third, we get the maximum $\beta$ for $\theta_{t}=\frac{\pi}{2}$
when $R_{tx}=R_{rx}$. Based on equations (1) to (4), we obtain

$\displaystyle\sin\beta^{max}=\sqrt{\frac{L^{2}}{\kappa^{2}}-\frac{L^{4}}{4\kappa^{4}}}$
(5)

When we consider the co-site scenario where $L<2\kappa$, then the above
expression for $\sin\beta^{max}\approx\frac{L}{\kappa}$. On the other hand, in
the lemiscate scenario where $L=2\kappa$, (5) becomes $\sqrt{3}$. In the third
scenario when $L>2\kappa$, we cannot obtain $\beta^{max}$ since the Cassini
ovals break into two separate regions as discussed earlier.

###  II-A Signal to Clutter and Noise Ratio (SCNR)

We assume that the transmitted power from the radar is $P_{tx}$ and the gains
of the transmitting and receiving antennas are $G_{tx}(\theta)$ and
$G_{rx}(\theta)$ with half-power beamwidths, $\Delta\theta_{tx}$ and
$\Delta\theta_{rx}$ respectively. We assume that the target is in the far-
field of both the transmitter and receiver. Then the two-way propagation from
the bistatic radar transmitter to the target and then back to the receiver is
a function of the product of $R_{tx}$ and $R_{rx}$ and indicated as
$\operatorname{\mathcal{H}(\kappa)}$. In the simplest scenario, where there is
line-of-sight (LOS) propagation, $\operatorname{\mathcal{H}(\kappa)}$ is

$\displaystyle\operatorname{\mathcal{H}(\kappa)}=\frac{\lambda^{2}}{(4\pi)^{3}R_{tx}^{2}R_{rx}^{2}}=\frac{\lambda^{2}}{(4\pi)^{3}\kappa^{4}},$
(6)

where $\lambda$ is the wavelength corresponding to the carrier frequency. Then
the received power from the target is given by the bistatic radar range
equation given by

$\displaystyle\textbf{S}(\kappa)=P_{tx}G_{tx}(\operatorname{\mathbf{\theta_{t}}})G_{rx}(\operatorname{\mathbf{\theta_{t}}})\mathbf{\sigma_{t}}\operatorname{\mathcal{H}(\kappa)},$
(7)

where $\mathbf{\sigma_{t}}$ is the bistatic radar cross-section. We assume
that the target is within the main lobe of both the transmitter and receiver
antennas and that the gain pattern is uniform within the lobes. The maximum
gains of the transmitting and receiving lobes are inversely proportional to
their respective beamwidths
$G_{tx}^{max}G_{rx}^{max}=\frac{A_{0}}{\operatorname{\Delta\theta_{tx}\Delta\theta_{rx}}}$.
The constant of proportionality, $A_{0}$, is directly related to factors
related to the impedance matching efficiency, dielectric and conductor loss
and the aperture efficiency and typically determined through empirical
studies. Thus, (7) can be written as

$\displaystyle\textbf{S}(\kappa)=P_{tx}G_{tx}^{max}G_{rx}^{max}\mathbf{\sigma_{t}}\operatorname{\mathcal{H}(\kappa)}=\frac{P_{tx}A_{0}\mathbf{\sigma_{t}}\operatorname{\mathcal{H}(\kappa)}}{\operatorname{\Delta\theta_{tx}\Delta\theta_{rx}}}.$
(8)

Studies have shown that the bistatic $\sigma_{t}$ follows a similar
distribution to monostatic radar cross-section [1] which are typically
expressed in terms of one of four Swerling models. Specifically, microwave and
millimeter wave targets such as humans and vehicles are modeled as Swerling-1
models since they are slow moving with several scattering centers of similar
strength [16, 17]. The Swerling-1 model follows an exponential distribution
with a mean radar cross-section $\operatorname{\sigma_{t_{avg}}}$ as shown
below.

$\displaystyle\mathcal{P}(\sigma_{t})=\frac{1}{\operatorname{\sigma_{t_{avg}}}}exp\left(\frac{-\sigma_{t}}{\operatorname{\sigma_{t_{avg}}}}\right).$
(9)

In the bistatic radar plane, we consider discrete clutter scatterers, located
at $(r_{c},\theta_{c})$, that represent both direct reflections and multipath
from environmental artefacts. For example, in an indoor environment, the
target is the human while the multipath from the walls, ceilings and furniture
give rise to ghost targets and clutter. Based on the room geometry and the
movement of the human, there can be considerable variation in the strength,
spatial distribution and the number of point clutter responses. Similarly, in
automotive radar, the multipath reflections from guard railings and
surrounding buildings give rise to discrete point clutter responses in the
radar data. Again, there is considerable randomness in the strength and
spatial distribution of these point clutter responses. Hence, we model the
distribution of these clutter scatterers as a Poisson point process (PPP) -
where the number of clutter scatterers in each realization follows a Poisson
distribution and the positions of these scatterers follow a uniform
distribution. This assumption is consistent with those adopted in previous
works [14, 12, 13]. The generalized Weibull model [18] has been frequently
used to describe the distribution of the radar cross-section of clutter. Here
the distribution is a function of the average bistatic radar cross-section,
$\operatorname{\sigma_{c_{avg}}}$ and the shape parameter, $\alpha$ as shown
in

$\displaystyle\mathcal{P}(\sigma_{c})=\frac{\alpha}{\operatorname{\sigma_{c_{avg}}}}\left(\frac{\sigma_{c}}{\operatorname{\sigma_{c_{avg}}}}\right)^{\alpha-1}\exp\left(-\left(\frac{\sigma_{c}}{\operatorname{\sigma_{c_{avg}}}}\right)^{\alpha}\right),$
(10)

The shape parameter can vary from zero - corresponding to Rayleigh
distribution, to one corresponding to exponential distribution and is
determined from empirical studies. In scenarios, where the reflectivity of
some scatterers dominate the others resulting in strong discrete clutter
characteristics, the distribution is most similar to an exponential form [19].
Therefore, in this work we assume that $\mathcal{P}(\sigma_{c})$ follows

$\displaystyle\mathcal{P}(\sigma_{c})=\frac{1}{\operatorname{\sigma_{c_{avg}}}}\exp\left(-\frac{\sigma_{c}}{\operatorname{\sigma_{c_{avg}}}}\right).$
(11)

We, further, assume that the noise, target and clutter statistics do not
change appreciably during the coherent processing interval of the radar. These
conditions are generally met for microwave or millimeter-wave radars [20, 21].
In order to determine the clutter returns at the bistatic radar, we
specifically consider those clutter scatterers that fall within the same radar
resolution cell as the target. There are two types of resolution cells as
shown in the figure below: the beamwidth limited resolution cell corresponding
to the two-dimensional region that falls within the main lobes of both the
transmitting and receiving antennas; and the range limited cell for a radar
waveform of $\Delta\tau$ pulse width. The former’s cell size for a target at
$(r_{t},\theta_{t})$ is given by

$\displaystyle
A_{c_{\theta}}=\frac{R_{tx}\Delta\theta_{t}R_{rx}\Delta\theta_{r}}{\sin\beta}=\frac{\kappa^{2}\operatorname{\Delta\theta_{tx}\Delta\theta_{rx}}}{\sin\beta}.$
(12)

For fixed $\kappa$ and antenna beamwidths, the area resolution cell is
inversely proportional to $\beta$.

(a)

(b)

Figure 2: (a) Beamwidth limited clutter resolution cell ($A_{c_{\theta}}$)and
(b) range limited clutter resolution cell ($A_{c_{r}}$)

In other words, in the limiting scenario when the bistatic radar system
resembles a monostatic case (when $\beta=0$), the beams from both the
transmitter and receiver are parallel and hence intersection area or $A_{c}$
becomes infinite. The latter’s cell size is given by

$\displaystyle A_{c_{r}}=\frac{c\Delta\tau
R_{min}\Delta\theta_{rx}}{2\cos^{2}\beta},$ (13)

where $R_{min}$ is the minimum of $R_{tx}$ and $R_{rx}$. All of the clutter
scatterers within the same resolution cell as the target give rise to clutter
returns, $\mathbf{C}$, at the radar receiver given by

$\displaystyle\mathbf{C}(\kappa)=\operatorname{\sum_{c\in\Phi\cap\mathbf{A_{c}}}}P_{tx}G_{tx}(\operatorname{\mathbf{\theta_{c}}})G_{rx}(\operatorname{\mathbf{\theta_{c}}})\mathbf{\sigma_{c}}\operatorname{\mathcal{H}(\kappa_{c})}.$
(14)

Here $\theta_{c}$ and $\kappa_{c}$ are the polar angle and bistatic range of
each clutter scatterer respectively and $\mathbf{A_{c}}$ is the bistatic radar
resolution cell that could correspond to either (12) or (13). Since
$\mathbf{\theta_{t}}$ is a random variable, $\mathbf{A_{c}}$ which is a
function of $\mathbf{\theta_{t}}$ is also a random variable. For a given noise
of the radar receiver, $N_{s}=K_{B}T_{s}BW$ where $K_{B},T_{s}$ and $BW$ are
the Boltzmann constant, system noise temperature and bandwidth respectively,
the signal to clutter and noise ratio is given by
$\mathbf{SCNR}(\kappa)=\frac{\mathbf{S}(\kappa)}{\mathbf{C}(\kappa)+N_{s}}$

###  II-B Bistatic radar detection coverage probability

Classical radar detection framework is based on binary hypothesis testing
derived from the Neyman-Pearson (NP) theorem. Here, the basic problem is to
decide, on the basis of the radar receiver measurement, between the
_alternative hypothesis_ ($\mathcal{H}_{1}$) where the target echo is present
along with noise and clutter in the measurement, and the _null hypothesis_
($\mathcal{H}_{0}$) where only noise and clutter are present in the
measurement. Based on the decisions, there are three possible outcomes - a
true detection, a missed detection when the signal along with noise and
clutter falls below the threshold and a false alarm when the clutter and noise
fall above the threshold. The separation between the PDFs of $\mathcal{H}_{0}$
and $\mathcal{H}_{1}$ is a function of the mean signal to clutter and noise
ratio (SCNR). In other words, when the SCNR is high, it is far easier to
obtain a high probability of detection while maintaining a low probability of
false alarm. In this work, we propose a new metric called the radar detection
coverage probability that is directly based on the SCNR in the region-of-
interest. It is analogous to the radar detection coverage probability that was
derived for monostatic radar scenarios in our previous works [12, 13].

######  Definition 1.

The bistatic radar detection coverage probability
($\operatorname{\mathcal{P}_{DC}^{Bi}}$) is defined as the probability that
the SCNR for a single target at a bistatic range ($\kappa$) is above a
predefined threhold, $\gamma$ i.e
$\operatorname{\mathcal{P}_{DC}^{Bi}}(\kappa)\triangleq\mathbb{P}\left(\mathbf{SCNR}(\kappa)\geq\gamma\right)$

######  Theorem 1.

The $\operatorname{\mathcal{P}_{DC}^{Bi}}$ for a given bistatic range $\kappa$
where $L<2\kappa$ corresponding to co-site conditions, with exponential
clutter statistics, for a beamlimited radar resolution cell, for a fixed
transmitted power and transmitter and receiver beamwdiths, is given by

$\displaystyle\operatorname{\mathcal{P}_{DC}^{Bi}}$
$\displaystyle=exp\left(-\frac{\gamma
N_{s}\operatorname{\Delta\theta_{tx}\Delta\theta_{rx}}}{\operatorname{\operatorname{\sigma_{t_{avg}}}P_{tx}A_{0}\operatorname{\mathcal{H}(\kappa)}}}-\frac{\rho\kappa^{3}\operatorname{\Delta\theta_{tx}\Delta\theta_{rx}}\gamma\operatorname{\sigma_{c_{avg}}}}{L(\operatorname{\sigma_{t_{avg}}}+\gamma\operatorname{\sigma_{c_{avg}}})}\right)$
(15)

###### Proof.

For a bistatic radar with a target at bistatic range $\kappa$, the
$\mathbf{SCNR}$ is a function of several random variables such as the target
cross-section, the position of target, the number and spatial distribution of
the discrete clutter scatterers and their radar cross-section as shown below.

$\displaystyle\mathbf{SCNR}(\kappa)$
$\displaystyle=\frac{P_{tx}A_{0}\mathbf{\sigma_{t}}\operatorname{\mathcal{H}(\kappa)}/\operatorname{\Delta\theta_{tx}\Delta\theta_{rx}}}{\operatorname{\sum_{c\in\Phi\cap\mathbf{A_{c}}}}P_{tx}A_{0}\mathbf{\sigma_{c}}\operatorname{\mathcal{H}(\kappa_{c})}/\operatorname{\Delta\theta_{tx}\Delta\theta_{rx}}+N_{s}}$
(16)
$\displaystyle=\frac{\mathbf{\sigma_{t}}}{\operatorname{\sum_{c\in\Phi\cap\mathbf{A_{c}}}}\frac{\mathbf{\sigma_{c}}\operatorname{\mathcal{H}(\kappa_{c})}}{\operatorname{\mathcal{H}(\kappa)}}+\frac{N_{s}\operatorname{\Delta\theta_{tx}\Delta\theta_{rx}}}{P_{tx}A_{0}\operatorname{\mathcal{H}(\kappa)}}}$
(17)

Since
$\operatorname{\mathcal{P}_{DC}^{Bi}}=\mathbb{P}\left(\mathbf{SCNR}(\kappa)\geq\gamma\right)$,
we can write (16) as

$\displaystyle\mathcal{P}\left(\mathbf{\sigma_{t}}\geq\operatorname{\sum_{c\in\Phi\cap\mathbf{A_{c}}}}\frac{\gamma\mathbf{\sigma_{c}}\operatorname{\mathcal{H}(\kappa_{c})}}{\operatorname{\mathcal{H}(\kappa)}}+\frac{\gamma
N_{s}\operatorname{\Delta\theta_{tx}\Delta\theta_{rx}}}{P_{tx}A_{0}\operatorname{\mathcal{H}(\kappa)}}\right).$
(18)

As per (9), $\mathbf{\sigma_{t}}$ follows the Swerling-1 model. Hence, (18)
can be expanded to

$\displaystyle\operatorname{\mathcal{P}_{DC}^{Bi}}$
$\displaystyle=exp\left(\operatorname{\sum_{c\in\Phi\cap\mathbf{A_{c}}}}\frac{-\gamma\mathbf{\sigma_{c}}\operatorname{\mathcal{H}(\kappa_{c})}}{\operatorname{\sigma_{t_{avg}}}\operatorname{\mathcal{H}(\kappa)}}-\frac{\gamma
N_{s}\operatorname{\Delta\theta_{tx}\Delta\theta_{rx}}}{\operatorname{\operatorname{\sigma_{t_{avg}}}P_{tx}A_{0}\operatorname{\mathcal{H}(\kappa)}}}\right)$
(19) $\displaystyle=exp\left(\frac{-\gamma
N_{s}\operatorname{\Delta\theta_{tx}\Delta\theta_{rx}}}{\operatorname{\operatorname{\sigma_{t_{avg}}}P_{tx}A_{0}\operatorname{\mathcal{H}(\kappa)}}}\right)I(\kappa).$
(20)

In the above expression, we have separated the
$\operatorname{\mathcal{P}_{DC}^{Bi}}$ into two terms. The first expression
which is entirely a function of constants shows the effect of signal to noise
ratio on the performance of the bistatic radar, while the second term
$I(\kappa)$ shows the effect of signal to clutter ratio on the performance of
the system. Since the exponent of sum of terms can be written as a product of
exponents, $I(\kappa)$ can be written as

$\displaystyle
I=\operatorname{\underset{\mathbf{\sigma_{c}},c\in\Phi\cap\mathbf{A_{c}}}{\operatorname{\mathbb{E}}}}\left[\operatorname{\prod_{c\in\Phi\cap\mathbf{A_{c}}}}exp\left(\frac{-\gamma\mathbf{\sigma_{c}}}{\operatorname{\sigma_{t_{avg}}}}\right)\right],$
(21)

where $\mathbf{A_{c}}$, the radar range resolution cell, is a function of
$\kappa$ and the position of the target $\mathbf{\theta_{t}}$. Also, due to
the close proximity of the clutter points to the target, we have assumed that
$\operatorname{\mathcal{H}(\kappa)}\approx\operatorname{\mathcal{H}(\kappa_{c})}$.
Based on the principles of stochastic geometry, we use the probability
generating functional (PGFL) of a homogeneous PPP [22], to rewrite,
$I(\kappa)$ as

$\displaystyle I$
$\displaystyle=exp\left(-\underset{\mathbf{\sigma_{c}},c}{\operatorname{\mathbb{E}}}\left[\iint_{\mathbf{r_{c},\phi_{c}}}\rho\left(1-exp\left(\frac{-\gamma\mathbf{\sigma_{c}}}{\operatorname{\sigma_{t_{avg}}}}\right)\right)d(\vec{x}_{c})\right]\right)$
(22)
$\displaystyle=exp\left(-\operatorname{\underset{\sigma_{c},\mathbf{A_{c}}}{\operatorname{\mathbb{E}}}}\left[\left(1-exp\left(\frac{-\gamma\mathbf{\sigma_{c}}}{\operatorname{\sigma_{t_{avg}}}}\right)\right)\rho\mathbf{A_{c}}\right]\right).$
(23)

The integral in (22) has to be evaluated in the region of beamwidth limited
clutter resolution cell $A_{c_{\theta}}$ as shown in Fig.2(a). If we assume
that the clutter scatter statistics are constant within that cell, then the
integral can be further reduced to

$\displaystyle
I=exp\left(-\operatorname{\underset{\sigma_{c},\mathbf{A_{c}}}{\operatorname{\mathbb{E}}}}\left[\left(1-exp\left(\frac{-\gamma\mathbf{\sigma_{c}}}{\operatorname{\sigma_{t_{avg}}}}\right)\right)\rho\frac{\kappa^{2}\operatorname{\Delta\theta_{tx}\Delta\theta_{rx}}}{\sin\mathbf{\beta}}\right]\right)$
(24)

Now the angular position of the target $\mathbf{\theta_{t}}$ is a uniform
random variable from $[0,2\pi)$. We perform a Monte Carlo simulation to study
the distribution of the corresponding $\sin\beta$ for 10000 trials of
$\theta_{t}$ for different ratios of $\kappa/L$. The resulting distribution is
shown in Fig.3(a) where we observe that the $\sin\beta$ takes the value of
$\sin\beta^{max}$ with a high probability. Hence, we can further reduce it to

$\displaystyle I$
$\displaystyle=exp\left(-\underset{\mathbf{\sigma_{c}}}{\operatorname{\mathbb{E}}}\left[1-exp\left(\frac{-\gamma\mathbf{\sigma_{c}}}{\operatorname{\sigma_{t_{avg}}}}\right)\right]\frac{\rho\kappa^{2}\operatorname{\Delta\theta_{tx}\Delta\theta_{rx}}}{\sin\beta^{max}}\right)$
(25)
$\displaystyle=exp\left(-\underset{\mathbf{\sigma_{c}}}{\operatorname{\mathbb{E}}}\left[1-exp\left(\frac{-\gamma\mathbf{\sigma_{c}}}{\operatorname{\sigma_{t_{avg}}}}\right)\right]\Upsilon(\kappa)\right)$
(26)

Since $\Upsilon(\kappa)$ is a constant independent of $\sigma_{c}$, it can be
pulled out of the integral for computing the expectation as shown below.

$\displaystyle
I(\kappa)=exp\left(-\Upsilon(\kappa)\int_{0}^{\infty}\left(1-exp\left(\frac{-\gamma\mathbf{\sigma_{c}}}{\operatorname{\sigma_{t_{avg}}}}\right)\right)\mathcal{P}(\sigma_{c})d\sigma_{c}\right)$
(27)
$\displaystyle=exp\left(-\frac{\Upsilon(\kappa)\gamma\operatorname{\sigma_{c_{avg}}}}{\operatorname{\sigma_{t_{avg}}}+\gamma\operatorname{\sigma_{c_{avg}}}}\right)=exp\left(-\frac{\rho\kappa^{3}\operatorname{\Delta\theta_{tx}\Delta\theta_{rx}}\operatorname{\sigma_{c_{avg}}}}{L(\operatorname{\sigma_{t_{avg}}}+\gamma\operatorname{\sigma_{c_{avg}}})}\right)$
(28)

The integral in (27) integrates to (28). We substitute $I(\kappa)$ in (20) to
prove the theorem in (15). ∎

Now, we discuss the key insights obtained from the theorem. The
$\operatorname{\mathcal{P}_{DC}^{Bi}}$ consists of an exponential expression
consisting of two parts. The first part shows the effect of the signal to
noise ratio while the second part shows the effect of the signal to clutter
ratio. In other words, when clutter is absent ($\rho=0$) we obtain the
performance of the radar solely under noise limited conditions. Similarly when
$N_{s}=0$, we obtain solely the effect of clutter on the radar performance.
_Noise Limited Conditions:_ We will first discuss the performance of the radar
under noise-limited conditions. The $\operatorname{\mathcal{P}_{DC}^{Bi}}$
increases with increase in transmitted power, $P_{tx}$; reduced radar antenna
beamwidths $\operatorname{\Delta\theta_{tx}\Delta\theta_{rx}}$; and reduced
radar receiver noise. Further, as expected, the increase in target’s bistatic
range causes the performance to deteriorate. In LOS conditions, this
deterioration is at the rate of $\kappa^{4}$. The increase in the mean target
cross-section, on the other hand, improves the performance. The findings from
this expression are consistent from those obtained from the Frii’s bistatic
radar range equation and not very dissimilar from the monostatic radar
conditions.
_Clutter Limited Conditions:_ Next, we discuss the second term where we study
how the presence of discrete clutter scatterers affects the performance of the
radar. First, we note that, unlike the noise limited conditions, the
performance is independent of the transmitted power since an increase in
$P_{tx}$ causes a proportionate increase in both target and clutter returns.
Second, the reduction in the radar antenna beamwidths,
$\operatorname{\Delta\theta_{tx}\Delta\theta_{rx}}$, improves the detection
performance in a similar manner to the noise limited conditions. This is
because the clutter resolution cell size reduces for lower beamwidths
resulting in lower returns. In LOS conditions, the
$\operatorname{\mathcal{P}_{DC}^{Bi}}$ deteriorates at the rate of
$\kappa^{3}$ instead of $\kappa^{4}$ (as is the case of noise limited
conditions). This is similar to the monostatic radar scenarios where the
signal to clutter ratios deteriorated at the rate of the third power of the
monostatic range. Next, the farther the target is from the radar, the effect
of the clutter returns begin to dominate over the effect of noise due to the
increase in the clutter resolution cell size.

######  Corollary 1.1.

In LOS conditions, the bistatic range, $\tilde{\kappa}$, at which the
transition between the noise limited radar performance to clutter limited
radar performance occurs at

$\displaystyle\tilde{\kappa}=\frac{\rho\operatorname{\sigma_{c_{avg}}}\operatorname{\sigma_{t_{avg}}}P_{tx}A_{0}\lambda^{2}}{L(\operatorname{\sigma_{t_{avg}}}+\gamma\operatorname{\sigma_{c_{avg}}})N_{s}}.$
(29)

We prove the above corollary by equating the two terms within the exponent in
(15) at $\tilde{\kappa}$. Further, we assume LOS propagation factor for
$\operatorname{\mathcal{H}(\kappa)}$, from (6), as shown below.

$\displaystyle\frac{\gamma
N_{s}\operatorname{\Delta\theta_{tx}\Delta\theta_{rx}}(4\pi)^{3}\tilde{\kappa}^{4}}{\operatorname{\sigma_{t_{avg}}}P_{tx}A_{0}\lambda^{2}}=\frac{\rho\tilde{\kappa}^{3}\operatorname{\Delta\theta_{tx}\Delta\theta_{rx}}\operatorname{\sigma_{c_{avg}}}}{L(\operatorname{\sigma_{t_{avg}}}+\gamma\operatorname{\sigma_{c_{avg}}})}$
(30)

######  Corollary 1.2.

Converse to the previous corollary, the maximum power beyond which there will
be no further improvement in the radar detection performance due to the
transition from noise limited to clutter limited conditions for a fixed
bistatic range is

$\displaystyle
P_{tx}^{max}=\frac{L\kappa(\operatorname{\sigma_{t_{avg}}}+\gamma\operatorname{\sigma_{c_{avg}}})N_{s}}{\rho\operatorname{\sigma_{c_{avg}}}\operatorname{\sigma_{t_{avg}}}A_{0}\lambda^{2}}$
(31)

As the ratio $\kappa/L$ increases, the bistatic radar scenario begins to
strongly resemble the monostatic radar scenario. As a result the resolution
cell becomes larger. In fact, in the limiting scenario when $\kappa/L$ tends
to infinity, the beams of the transmitter and receiver antennas are parallel
and the resolution cell is infinite in size. At this point, it is more
beneficial to consider other types of clutter resolution cells instead of the
beamwidth resolution cell. In current literature, there is no specific value
of $\kappa/L$ to mark the transition from bistatic radar to monostatic radar.

######  Corollary 1.3.

In the absence of noise, the bistatic range, $\overline{\kappa}$, at which the
transition from bistatic to monostatic radar scenario occurs when

$\displaystyle\overline{\kappa}=\left(\frac{L(\operatorname{\sigma_{t_{avg}}}+\gamma\operatorname{\sigma_{c_{avg}}})}{\rho\operatorname{\Delta\theta_{tx}\Delta\theta_{rx}}\operatorname{\sigma_{c_{avg}}}}\right)^{1/3}.$
(32)

We show the above corollary by identifying $\overline{\kappa}$ as the value at
which $\operatorname{\mathcal{P}_{DC}^{Bi}}$ reduces to 36.7% ($e^{-1}$) of
the maximum possible value as shown below

$\displaystyle
exp\left(-\frac{\rho(\overline{\kappa})^{3}\operatorname{\Delta\theta_{tx}\Delta\theta_{rx}}\operatorname{\sigma_{c_{avg}}}}{L(\operatorname{\sigma_{t_{avg}}}+\gamma\operatorname{\sigma_{c_{avg}}})}\right)=exp(-1)$
(33)

Increase in the density of the clutter returns, $\rho$, gives causes a
deterioration in $\operatorname{\mathcal{P}_{DC}^{Bi}}$ and hence the radar
detection performance. However, the same is not true for the mean clutter
cross-section $\operatorname{\sigma_{c_{avg}}}$.

######  Corollary 1.4.

For large values of $\operatorname{\sigma_{c_{avg}}}$, the radar detection
performance metric, $\operatorname{\mathcal{P}_{DC}^{Bi}}$, is independent of
the clutter cross-section as shown below.

$\displaystyle\lim_{\operatorname{\sigma_{c_{avg}}}\to\infty}exp\left(-\frac{\rho\kappa^{3}\operatorname{\Delta\theta_{tx}\Delta\theta_{rx}}\operatorname{\sigma_{c_{avg}}}}{L(\operatorname{\sigma_{t_{avg}}}+\gamma\operatorname{\sigma_{c_{avg}}})}\right)=exp\left(-\frac{\rho\kappa^{3}\operatorname{\Delta\theta_{tx}\Delta\theta_{rx}}}{L\gamma}\right)$
(34)

######  Theorem 2.

The probability of detection coverage for a given bistatic range $\kappa$
where $L=2\kappa$ corresponding to lemiscate conditions, with exponential
clutter statistics, for a beamlimited radar resolution cell, for a fixed
transmitted power and transmitter and receiver beamwdiths, is given by

$\displaystyle\operatorname{\mathcal{P}_{DC}^{Bi}}$
$\displaystyle=exp\left(-\frac{\gamma
N_{s}\operatorname{\Delta\theta_{tx}\Delta\theta_{rx}}}{\operatorname{\operatorname{\sigma_{t_{avg}}}P_{tx}A_{0}\operatorname{\mathcal{H}(\kappa)}}}-\frac{\rho\kappa^{2}\operatorname{\Delta\theta_{tx}\Delta\theta_{rx}}\gamma\operatorname{\sigma_{c_{avg}}}}{\sqrt{3}(\operatorname{\sigma_{t_{avg}}}+\gamma\operatorname{\sigma_{c_{avg}}})}\right)$
(35)

###### Proof.

The proof for this theorem follows the identical steps of the previous theorem
from (16) to (26). In the final step, since $\sin\beta^{max}=\sqrt{3}$, we
obtain

$\displaystyle\Upsilon(\kappa)=\frac{\rho\kappa^{2}\operatorname{\Delta\theta_{tx}\Delta\theta_{rx}}\gamma\operatorname{\sigma_{c_{avg}}}}{\sqrt{3}(\operatorname{\sigma_{t_{avg}}}+\gamma\operatorname{\sigma_{c_{avg}}})}$
(36)

Substituting (36) in (26), we prove (35). ∎

The performances of the radar under noise limited and clutter limited
conditions in lemiscate scenario do not differ significantly from the co-site
scenario. Hence, we do not include a separate discussion on the effect of
radar, target and clutter parameters on the detection performance.

######  Theorem 3.

The probability of detection coverage for a given bistatic range $\kappa$ and
antenna beamwidth $\Delta\theta_{tx}=\Delta\theta_{rx}=\Delta\Theta$ and
$L\leq 2\kappa$ (co-site/lemiscate regions) for a range limited radar
resolution cell is given by

$\displaystyle\begin{split}\operatorname{\mathcal{P}_{DC}^{Bi}}=exp\left(\frac{-\gamma
N_{s}\operatorname{\Delta\theta_{tx}\Delta\theta_{rx}}}{\operatorname{\operatorname{\sigma_{t_{avg}}}P_{tx}A_{0}\operatorname{\mathcal{H}(\kappa)}}}\right)\\\
exp\left(-\frac{\rho
c\Delta\tau\Delta\Theta\kappa^{2}\operatorname{\sigma_{c_{avg}}}}{2(\kappa_{m}+\sqrt{\kappa_{m}^{2}-L^{2}})(\operatorname{\sigma_{t_{avg}}}+\gamma\operatorname{\sigma_{c_{avg}}})}\right)\end{split}$
(37)

###### Proof.

We begin with similar steps to the proof of the first theorem by considering
steps (16) to (22). Instead of considering the beamwidth limited resolution
cell, we consider the range limited resolution cell given in (13). If we
assume that the radar beamwidths for the transmitter and receiver are equal
$\Delta\theta_{tx}=\Delta\theta_{rx}=\Delta\Theta$, then $A_{c_{r}}$ is a
function of the minimum one-way range, $R_{min}:min(R_{tx},R_{rx})$. In order
to find $R_{min}$, we perform a Monte Carlo simulation with 2000 trials where
the polar angle $\theta_{t}$ is drawn from a uniform distribution from
$[0,2\pi)$ and compute the corresponding $R_{min}$ normalized by $\kappa$. The
resulting distribution of the normalized $R_{min}$ is presented in Fig.3(b).

(a)

(b)

Figure 3: Histograms of sine of bistatic angle ($\sin\beta$) and minimum
normalized one-way range ($R_{min}/\kappa$).

The simulation result shows that $R_{min}$ varies very closely with respect to
$\kappa$ ranging from 0.95 to 1. Hence, without loss of accuracy, we use
$\kappa$ in the expression for $A_{c_{r}}$. Further based on the previous
study of the distribution of $\sin\beta$ presented in Fig.3(a), we use
$\cos\beta^{max}$ also for $A_{c_{r}}$. The resulting expression for
$A_{c_{r}}$ is given by

$\displaystyle
A_{c_{r}}=\frac{c\Delta\tau\Delta\Theta\kappa^{2}}{2(\kappa_{m}+\sqrt{\kappa_{m}^{2}-L^{2}})}.$
(38)

Substituting this in (22), we obtain

$\displaystyle
I=exp\left(\underset{\mathbf{\sigma_{c}}}{\operatorname{\mathbb{E}}}\left[1-exp\left(\frac{-\gamma\mathbf{\sigma_{c}}}{\operatorname{\sigma_{t_{avg}}}}\right)\frac{\rho
c\Delta\tau\Delta\Theta\kappa^{2}}{2(\kappa_{m}+\sqrt{\kappa_{m}^{2}-L^{2}})}\right]\right)$
(39)

For an exponential distribution of $\sigma_{c}$, we obtain

$\displaystyle I=exp\left(-\frac{\rho
c\Delta\tau\Delta\Theta\kappa^{2}\operatorname{\sigma_{c_{avg}}}}{2(\kappa_{m}+\sqrt{\kappa_{m}^{2}-L^{2}})(\operatorname{\sigma_{t_{avg}}}+\gamma\operatorname{\sigma_{c_{avg}}})}\right).$
(40)

Substituting this back in (20), we prove (37). ∎

Similar to the corollaries drawn from Theorem 1, we can derive
$\tilde{\kappa}$ where the radar detection scenario transitions from noise
limited conditions to clutter limited conditions; and the $P_{tx}^{max}$ which
is the maximum power beyond which there will be no further improvement in the
radar performance due to high clutter returns. Also, as shown previously in
Corollary 1.4, when $\operatorname{\sigma_{c_{avg}}}$ is high, there is no
further deterioration in the performance of the radar. Instead, the detection
is affected more by the clutter density $\rho$ . However, there is a unique
inference for conditions of the radar bandwidth in the range limited scenario.
For high values of radar bandwidth, the noise power at the radar receiver
increases due to $N_{s}=K_{B}T_{s}BW^{max}$. However, the clutter returns
reduce since $A_{c_{r}}$ is a function of the pulse width $\Delta\tau$ which
is inversely proportional to bandwidth.

######  Corollary 3.1.

The radar bandwidth for optimal detection performance for an impulse radar
with a pulse width of $\Delta\tau=1/BW^{max}$ and noise power is given by

$\displaystyle BW^{max}=\left(\frac{\rho
c\operatorname{\sigma_{c_{avg}}}\operatorname{\sigma_{t_{avg}}}P_{tx}A_{0}\operatorname{\mathcal{H}(\kappa)}\kappa}{2(1-\frac{L^{2}}{\kappa^{2}})(\operatorname{\sigma_{t_{avg}}}+\gamma\operatorname{\sigma_{c_{avg}}})\Delta\Theta
K_{B}T_{s}}\right)^{1/2}$ (41)

The corollary is proved by equating the first order derivative of
$\operatorname{\mathcal{P}_{DC}^{Bi}}$ in (37) with respect to $BW$ to 0.

##  III Results

In this section, we present the results from the theorem and its corollaries.
They are validated with Monte Carlo simulations. The experimental set up for
the Monte Carlo simulations consist of a two-dimensional Cartesian space
spanning $X:-100m:+100m$ and $Y:-100m:+100m$ with the bistatic radar located
at $(\pm 2.5m,0)$. For each realization of the Monte Carlo simulation, we
assume a single target to be located at bistatic range $\kappa$ at a random
realization of $\theta_{t}$ drawn from a uniform distribution between
$[0,2\pi)$ and a radar cross-section drawn from the exponential distribution
with a mean $\operatorname{\sigma_{t_{avg}}}$. Using the bistatic radar
geometry concepts presented in Section II, we compute $R_{tx},R_{rx}$ and
$\beta$. Then for a given transmitter power, radar antenna beamwidths and
wavelength, we compute the received power at the radar for every realization
assuming LOS conditions using (7). Then we model the discrete clutter
scatterers as a PPP - where for each realization of the Monte Carlo
simulation, the number of clutter scatterers are drawn from a Poisson
distribution of mean ($\rho\times$ total area of the Cartesian space) and the
position is a uniform random variable in the two-dimensional space. The cross-
section of each clutter scatterer is drawn from an exponential distribution of
mean $\operatorname{\sigma_{c_{avg}}}$. We determine if the scatterer is
within the beamwidth limited resolution cell and then compute the total
clutter returns using (14). Then for a given system noise temperature and
bandwidth, we compute the noise power of the radar system and the SCNR of each
realization. We determine the $\operatorname{\mathcal{P}_{DC}^{Bi}}$ by
summing the total number of realizations for which the SCNR is above the
threshold. The radar, target and clutter parameters that have been used in the
simulations are summarized in Table.I.

TABLE I: Radar, target and clutter parameters Parameter |  Values |  Parameter |  Values
---|---|---|---
$L$ |  5m |  $\operatorname{\sigma_{t_{avg}}}$ |  $1m^{2}$
$P_{tx}$ |  10W |  $\theta_{t}$ |  $\mathcal{U}[0,2\pi)$
$\operatorname{\sigma_{c_{avg}}}$ |  $1m^{2}$ |  $T_{s}$ |  300K
$\Delta\Theta$ |  $5^{\circ}$ |  $BW$ |  2GHz
$\lambda$ |  5mm |  $\rho$ |  $0.001/m^{2}$

Figure.4(a) shows the natural logarithm of
$\operatorname{\mathcal{P}_{DC}^{Bi}}$ as a function of $\kappa$ under noise
limited and clutter limited conditions. In the noise limited conditions,
$\rho$ is set to zero while $T_{s}$ is set at 300K. We observe that
$ln(\operatorname{\mathcal{P}_{DC}^{Bi}})$ decays at the rate of the fourth
power of $\kappa$. On the other hand, when $T_{s}$ is set at 0 and $\rho$ is
$0.001/m^{2}$, then $ln(\operatorname{\mathcal{P}_{DC}^{Bi}})$ decays at the
rate of the third power of $\kappa$. When both noise and clutter conditions
exist, then the performance follows the clutter response till $\tilde{\kappa}$
after which it follows the noise response. We use corollary 1.1 to compute
$\tilde{\kappa}$ and it is verified in the figure.

(a) $ln(\operatorname{\mathcal{P}_{DC}^{Bi}})$ vs. $ln(\kappa)$ for varying
$\rho$

(b) $\operatorname{\mathcal{P}_{DC}^{Bi}}$ vs. $\kappa$ for varying
$\Delta\Theta$

(c) $\operatorname{\mathcal{P}_{DC}^{Bi}}$ v. $P_{tx}$ for varying $\kappa$

(d) $\operatorname{\mathcal{P}_{DC}^{Bi}}$ vs.
$\operatorname{\sigma_{c_{avg}}}$ for varying $\rho$

(e) $\operatorname{\mathcal{P}_{DC}^{Bi}}$ vs. $T_{s}$ for varying $\rho$

(f) $\operatorname{\mathcal{P}_{DC}^{Bi}}$ vs. $BW$ for varying $\kappa$

Figure 4: Bistatic radar detection coverage probability
($\operatorname{\mathcal{P}_{DC}^{Bi}}$) as a function of radar, target and
clutter parameters.

Next, we consider the effect of $P_{tx}$ on the radar detection performance in
Fig.4(c). We compute the $\operatorname{\mathcal{P}_{DC}^{Bi}}$ for three
different values of $\kappa$. For each case, we observe that the
$\operatorname{\mathcal{P}_{DC}^{Bi}}$ improves with increase in $P_{tx}$
under the noise limited conditions. However, beyond a certain maximum value of
the $P_{tx}$, there is no further significant improvement in the detection
performance due to the high clutter returns. The maximum value of the $P_{tx}$
at which this occurs is computed from corollary 1.2 and corroborates the
results shown in the figure. Next, we study the effect of the clutter cross-
section and the clutter density. We fix $\operatorname{\sigma_{t_{avg}}}$ at
$1m^{2}$ while $\operatorname{\sigma_{c_{avg}}}$ is varied from $0.1m^{2}$ to
$10m^{2}$. We study the variation for three different values of $\rho$. We
observe that the detection performance improves with increase in
$\operatorname{\sigma_{c_{avg}}}$ till
$\operatorname{\sigma_{c_{avg}}}>\operatorname{\sigma_{t_{avg}}}$. This is
consistent with corollary 1.4. However, the performance deteriorates
significantly with increase in $\rho$. In other words, the radar performance
is greatly affected by the _number_ rather than the _strength_ of the clutter
scatterers. Next, we examine the impact of the radar performance for different
antenna beamwidths in Fig.4(b). We assume that beamwidth for both transmitter
and receiver antennas are equal to $\Delta\Theta$. We observe the
$\operatorname{\mathcal{P}_{DC}^{Bi}}$ fall with increase in $\kappa$ and for
increase in $\Delta\Theta$. The increase in radar gain is beneficial in both
the noise limited and clutter limited scenarios. In the noise limited
scenario, the increase in gain results in stronger returns from the target. In
the clutter limited scenario, the reduced beamwidth results in smaller clutter
resolution cells resulting in weaker clutter returns. The view graph also
indicates the values of $\overline{\kappa}$ at which the
$\operatorname{\mathcal{P}_{DC}^{Bi}}$ falls to 36.7% of the maximum value
corroborating with corollary 1.3. We can assume that at this bistatic range,
the radar system can be approximated to monostatic system due to high
$\kappa/L$. Finally, we study the effect of $T_{s}$ on the radar detection
performance for different values of $\rho$. As anticipated, the detection
performance falls for increase in $T_{s}$ and increase in $\rho$. However, the
slopes of the three curves in the view graph also indicates that for greater
values of $\rho$, the change in the radar detection performance is less
significantly impacted by increase in $T_{s}$. This is because for greater
clutter, the radar operates in the clutter limited scenario and is independent
of $T_{s}$. All the results that have been discussed so far are for the
beamwidth limited clutter. Several of these trends also apply to the range
limited clutter and hence not presented due to space constraints. Instead, we
show the impact of radar bandwidth on the detection performance in Fig.4(f)
for different values of $\kappa$. The result shows that the performance
improves with increase in $BW$ due to reduction in $A_{c_{r}}$ till it reaches
a maximum after which the detection performance deteriorates due to increase
in noise. The optimal bandwidth is indicated in the view graph and
corroborates with corollary 3.1.

##  IV Conclusion

We use stochastic geometry formulations to analyze bistatic radar detection
performance through a metric - $\operatorname{\mathcal{P}_{DC}^{Bi}}$. This
metric quantifies the average area of the radar field-of-view over which the
SCNR is above a predefined threshold. It is a function of radar parameters
such as transmitted power, antenna gains, pulse width and noise as well as
target and clutter parameters. Through this metric, we obtain key system
insights including the bistatic range at which the radar transitions from
noise to clutter limited conditions; and approximates to monostatic behavior;
as well as the maximum transmitted power and bandwidth for realizing peak
detection performance. Due to the flexibility afforded by the analytical
framework, it can be used for The results are validated with Monte Carlo
simulations.

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