RADAR

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Description

The RADAR (Radio Detection and Ranging) is a Remote Sensor that detects objects by simulating radio wave transmission and reflection. It calculates received signal power using the radar equation, compares each return against thermal noise to determine a signal-to-noise ratio, and reports a detection when a return exceeds a configurable threshold.

The antenna is described by a single physical aperture, from which both the gain and the beamwidth follow. Gain and beamwidth are two descriptions of the same antenna, so they cannot be configured independently: concentrating more power into the beam necessarily narrows the region the sensor can look at. Targets away from the boresight return less power than targets on it, following the shape of the mainlobe.

Every target that clears the detection threshold is a candidate for reporting, and a configurable selection rule decides between them. The sensor reports the range, bearing, cross-section, range rate and Doppler shift of that single target.

The receiver responds across a finite range of powers rather than an unlimited one. A return outside that range either reads at the limit it passed or is discarded entirely, depending on how the sensor is configured. The sensor also carries the fault states common to every sensor, which replace the measurement with the reading a failed receiver would produce.

The sensor automatically adds the orbited celestial body (and the Moon for Earth-orbiting spacecraft) as default targets on initialisation.

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Example Use Cases

• Proximity Detection: Detect nearby spacecraft or debris during rendezvous operations.
• Altimetry: Measure distance to planetary surfaces for landing or orbital analysis.
• Target Tracking: Monitor the range and bearing of a tracked object within the beam.
• Closing Target Discrimination: Pick a manoeuvring target out of a scene where a planet returns far more power but holds a near constant range.

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Module Implementation

The RADAR sensor calculates signal strength using the monostatic radar equation and determines detection by comparing each target return against the noise floor.

Antenna Aperture

The peak gain along the boresight follows from the aperture area measured in wavelengths:

$$G=\eta {\left(\frac{\pi D}{\lambda }\right)}^2$$

where $D$ is the aperture diameter, $\lambda$ is the wavelength, and $\eta$ is the aperture efficiency accounting for the illumination taper, blockage and surface error of a real antenna. A value of $\eta =1$ describes a uniformly illuminated aperture, where a reflector is typically around $0.6$. The gain is reported in decibels as:

$$G_{\text{dB}}=10{\log }_{10}G$$

The half-power beamwidth of the same aperture is:

$${\theta }_b=\frac{70\lambda }{D}$$

with ${\theta }_b$ in degrees. The two expressions describe one antenna, so gain and beamwidth trade against each other. Doubling the aperture raises the gain by a factor of four and halves the beamwidth. Halving the wavelength raises the gain by a factor of four and likewise halves the beamwidth, so a sensor cannot concentrate its power without narrowing where it is able to look.

Quantity	Role	Relationship
Aperture diameter $D$	Configured	Sets gain and beamwidth
Aperture efficiency $\eta$	Configured	Scales gain only
Wavelength $\lambda$	Configured	Sets gain and beamwidth
Gain $G$	Derived	$\eta (\pi D/\lambda {)}^2$
Beamwidth ${\theta }_b$	Derived	$70\lambda /D$
Field of view ${\theta }_{\text{FOV}}$	Derived	$2{\theta }_b$

Field of View

The mainlobe spans twice the half-power beamwidth, so the field of view is:

$${\theta }_{\text{FOV}}=2{\theta }_b$$

Only targets inside this cone are sensed at all. A target at unit direction $\hat{d}$ from the sensor is within the field of view when:

$$\hat{u}\cdot \hat{d}\ge \cos \left(\frac{{\theta }_{\text{FOV}}}{2}\right)=\cos \left({\theta }_b\right)$$

where $\hat{u}$ is the boresight direction of the sensor, including any misalignment applied by an error model.

Setting the field of view directly resizes the aperture to the one that produces the requested mainlobe, which moves the gain to match:

$$D=\frac{140\lambda }{{\theta }_{\text{FOV}}}$$

Boresight Misalignment

The boresight the sensor looks along is not necessarily the one it was mounted with. An error model can pitch it about the local right axis by an angle $\alpha$, representing mechanical tolerance, thermal distortion or vibration induced jitter:

$${\mathbf{R}}_{\text{eff}}=\mathbf{R}{\mathbf{R}}_{\text{right}}(\alpha )$$

where $\mathbf{R}$ is the mounted rotation of the sensor. Every part of the measurement is taken against this misaligned frame rather than the mounted one, so a misalignment moves what the sensor reports rather than simply being recorded alongside it:

• The field of view test is applied to the misaligned boresight, so a target near the edge of the mainlobe is lost once $\alpha$ carries it outside.
• The off-boresight angle $\varphi$ driving the mainlobe taper is measured from the misaligned boresight. A target the sensor was mounted to look straight at therefore returns a fraction $L(\alpha {)}^2$ of the power it would otherwise return.
• The bearing is resolved in the misaligned frame. Because the misalignment is a pitch about the right axis, a target on the mounted boresight is reported at an elevation of $\alpha$ and an azimuth of zero.

A sensor with $\alpha =0$ measures against its mounted frame exactly, with no rotation composed onto it.

Mainlobe Pattern

The gain only reaches its peak along the boresight. The mainlobe is approximated by a Gaussian in the off-boresight angle $\varphi$:

$$L(\varphi )=\exp \left(-4\ln 2{\left(\frac{\varphi }{{\theta }_b}\right)}^2\right)$$

This falls to $L=1/2$ at $\varphi ={\theta }_b/2$, which is the definition of the half-power beamwidth. The same antenna both transmits and receives, so a target is attenuated by the pattern on the way out and again on the way back, giving a two-way factor of $L(\varphi {)}^2$. A target half a beamwidth off the boresight therefore returns a quarter of the power that the same target would return on it.

Illuminated Cross-Section

For a spherical target, the radar cross-section is approximated as the geometric cross-section of the disc it presents. Only the illuminated part of a target returns a signal, so the effective cross-section is limited by the patch the beam covers at the range of the target. The radius of that patch is:

$$r_b=R\tan \left(\frac{{\theta }_b}{2}\right)$$

and the effective cross-section is the disc of whichever is smaller, the target or the patch:

$${\sigma }_e=\pi \min {\left(\frac{d}{2},r_b\right)}^2$$

where $d$ is the target diameter and $R$ is the range to the target surface. This gives two distinct regimes:

Target inside the beam. The whole disc returns, so the cross-section is independent of range:

$${\sigma }_e=\pi {\left(\frac{d}{2}\right)}^2$$

Target over-filling the beam. The beam is fully occupied by the target and the illuminated patch grows with range:

$${\sigma }_e=\pi R^2{\tan }^2\left(\frac{{\theta }_b}{2}\right)$$

Radar Equation

The received power from a target is calculated using the monostatic radar equation, with the effective cross-section and the two-way pattern factor applied:

$$P_r=\frac{P_t{G}^2{\lambda }^2{\sigma }_{e}L(\varphi {)}^2}{(4\pi {)}^3{R}^4}$$

where $P_t$ is the transmit power. Because the effective cross-section is range dependent when the beam is over-filled, the range law differs between the two regimes. For a target inside the beam the return falls with the fourth power of range, as expected of a point target. For a target over-filling the beam, the illuminated patch grows with $R^2$ and cancels two of those powers:

$$P_r=\frac{P_t{G}^2{\lambda }^2\pi {\tan }^2({\theta }_b/2)L(\varphi {)}^2}{(4\pi {)}^3{R}^2}$$

A target only scatters back the power that reached it, so the return is bounded by the transmitter that illuminated it:

$$P_r\le P_t$$

This bound is reached only at ranges short enough that the far-field form of the radar equation no longer applies, such as a target large enough to enclose the sensor.

Receiver Range

The receiver responds across a finite range of powers, between the weakest return it can lift out of its own front end and the level at which it saturates:

Limit	Default	Role
$P_{\min }$	$0$ W	The floor the receiver produces on its own, below which a return cannot be resolved
$P_{\max }$	$10^{-3}$ W	The level the front end saturates at, above which a return cannot be read

Each return is brought into that range before anything else is done with it:

$$P_r^{\prime }=\min \left(\max \left(P_r,P_{\min }\right),P_{\max }\right)$$

A return below the floor therefore reads at the floor, and a return above the ceiling reads at the ceiling. In both cases the sensor reports a signal that the target did not return, which is what a receiver driven beyond the range it can measure does. Everything downstream uses $P_r^{\prime }$ in place of $P_r$, so the detection test, the total signal and the signal-to-noise ratio all describe the bounded value. The default floor of zero leaves the lower limit inert until it is configured.

Where the sensor is set to ignore signals outside this range, a return beyond either limit is discarded instead of being held at it:

$$P_r^{\prime }=\{\begin{array}{ll}{P}_r& P_{\min }\le P_r\le P_{\max }\\ 0& \text{otherwise}\end{array}$$

A discarded target is treated as though it was never sensed. It contributes nothing to the total signal, it cannot be detected, and it is not a candidate for reporting, so no range, bearing, cross-section or rate is reported for it. This is the stricter of the two behaviours: a measurement the receiver could not have made properly is thrown away rather than reported at a level it never read.

Thermal Noise

The system thermal noise power is:

$$N=k_{B}TB$$

where $k_B$ is Boltzmann’s constant, $T$ is the system temperature, and $B$ is the receiver bandwidth.

Detection

A threshold detector fires on a single return rather than on the sum of the scene, so each target is compared against the noise floor on its own. Target $i$ is detected when:

$$P_{r,i}^{\prime }\ge N\cdot 10^{\Gamma /10}$$

where $\Gamma$ is the detection threshold in decibels. An ideal receiver with $N=0$ leaves the ratio undefined and detects nothing.

Signal-to-Noise Ratio

The total signal is the sum of every return the receiver was able to read:

$$P_{\text{total}}=\sum _i{P}_{r,i}^{\prime }$$

The reported signal-to-noise ratio describes the target being reported rather than the scene as a whole, so that it and the reported range refer to the same target:

$$\text{SNR}=10{\log }_{10}\left(\frac{P_{r,\text{reported}}^{\prime }}{N}\right)\text{ dB}$$

Target Selection

Every detected target is a candidate for reporting, and one of three rules decides between them:

Rule	Selection	Description
Strongest signal	$\arg {\max }_i{P}_{r,i}$	The target returning the most power, which is what a sensor without tracking logic sees
Nearest range	$\arg {\min }_i{R}_i$	The closest target, regardless of how strongly it returned
Fastest closing	$\arg {\min }_i{\dot{R}}_i$	The target with the most negative range rate

When no target clears the detection threshold, the strongest return is reported regardless, so that the sensor still describes what it can see even though it declares no detection.

Range Measurement

The range is measured to the surface of the target rather than its centre:

$$R=\left|{\mathbf{r}}_{\text{target}}-{\mathbf{r}}_{\text{sensor}}\right|-\frac{d}{2}$$

The reported measurement carries noise applied as a fraction of the true range:

$$R_m=R\left(1+ϵ\right)$$

where $ϵ$ is drawn uniformly within $\pm$ the configured distance noise fraction.

Bearing

The direction to the reported target is resolved into the effective sensor frame, which carries any boresight misalignment and has the boresight along its up axis. With $d_r$, $d_f$ and $d_u$ the components of $\hat{d}$ along the right, forward and up axes of that frame, the bearing is reported as an azimuth and an elevation:

$$A=\arctan \left(\frac{d_r}{d_u}\right)$$ $$E=\arctan \left(\frac{d_f}{\sqrt{d_r^2+d_u^2}}\right)$$

Both angles are zero when the target lies along the boresight.

Range Rate and Doppler Shift

The range rate is recovered by differencing the true range of the target between successive captures:

$$\dot{R}=\frac{R(t)-R(t-\Delta t)}{\Delta t}$$

It is positive when the target is moving away from the sensor. The equivalent Doppler shift imposed on the carrier is:

$$f_d=-\frac{2\dot{R}}{\lambda }$$

The factor of two accounts for the shift being applied once on the path out to the target and again on the path back. By convention the sign is opposite to the range rate, so the Doppler shift is positive for a closing target.

Signal Triggering

Where the sensor is placed behind a signal trigger, it only measures while that trigger has fired. An untriggered sensor is not transmitting, so it has nothing to receive, and the output holds every value from the last capture it took rather than describing a scene it is not illuminating. This is resolved ahead of the fault state below, since a sensor that is not measuring cannot report a faulty measurement either.

A sensor with no trigger attached is always treated as triggered, so the gate is inert unless it is configured.

Where a power model is attached, capture is also refused while the solved node voltage sits below the brownout floor of that model. The last measurement is held in the same way as an untriggered sensor.

Fault States

The sensor carries the fault states common to every sensor. A fault replaces the measurement with the reading a failed receiver would produce, and stands in place of a capture rather than being applied on top of one, so nothing about the scene in front of the sensor reaches the output.

State	Reported signal	Behaviour
Nominal	$P_{r,\text{reported}}^{\prime }$	The sensor measures the scene as described above
Stuck	Unchanged	Every value from the last capture before the fault is held indefinitely
Max	$P_{\max }$	The receiver reads at its ceiling
Random	$P_{\min }+u\left(P_{\max }-P_{\min }\right)$	The receiver reads anywhere across its range, with $u$ drawn uniformly on $[0,1)$
Off	$0$	The receiver reads nothing

In every faulted state other than stuck, the faulted signal is still measured against the noise floor, so the ratio and the detection follow from it exactly as they would from a real return:

$$\text{SNR}=10{\log }_{10}\left(\frac{P_{\text{fault}}}{N}\right)\text{ dB},\text{detected when }{P}_{\text{fault}}\ge N\cdot 10^{\Gamma /10}$$

A saturated receiver therefore declares a detection with nothing in front of it, which is the false alarm such a failure produces in practice. No target is described in any faulted state, since a faulted sensor is not resolving one, so the reported range, bearing, cross-section, range rate and Doppler shift are all zero. The noise floor is still reported, as it belongs to the receiver rather than to the scene.

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Assumptions/Limitations

• The radar cross-section uses a spherical approximation based on target diameter; material properties, shape and aspect are not modelled.
• Gain and beamwidth are both derived from the aperture and cannot be set independently of one another.
• An aperture smaller than a wavelength cannot concentrate the beam, so the gain is floored at unity and radiates isotropically. The beamwidth is likewise bounded, since an arbitrarily large aperture would drive it to zero.
• The mainlobe is approximated by a Gaussian. Sidelobes are not modelled, so a target outside the mainlobe returns nothing rather than a small return.
• Boresight misalignment is a single pitch about the right axis. Roll and yaw misalignments are not modelled, so a target on the mounted boresight always moves in elevation and never in azimuth.
• The illuminated patch is measured against the half-power beamwidth, which approximates integrating the pattern across the surface of an extended target.
• The link budget contains no general loss term for radome, feed or atmospheric losses, and no separate receiver noise figure. The system temperature absorbs both antenna and receiver noise.
• Detection is deterministic against a fixed threshold; probability of detection and false alarm rate are not modelled.
• Saturation is a hard limit rather than a compression curve, so no gradual roll-off is applied as a return approaches the ceiling of the receiver.
• The range of the receiver bounds each return individually. The total signal is the sum of the bounded returns, so a scene containing several saturating targets can report a total above that ceiling.
• The lower limit of the receiver defaults to zero, which leaves it inert until it is configured.
• A faulted sensor reports a signal but never a target, so a fault presents downstream as a detection with no range behind it.
• A sensor held by an untriggered signal, or one stuck in fault, leaves its previous output in place rather than clearing it, so a stale reading is indistinguishable from a fresh one on the values alone.
• The bandwidth sets the noise floor only. It does not set a range resolution, so targets at similar ranges are not merged into one return.
• Only the range carries measurement noise. The bearing, range rate and Doppler shift are reported exactly.
• The range rate requires two captures of the same target before it has an interval to difference over, and stays zero until then.
• Received power is bounded by transmitted power, but near-field effects are not otherwise modelled; the radar equation assumes far-field conditions.
• Targets must be added to the sensor’s target list; only listed objects are considered for detection.
• Line-of-sight checks are performed against celestial bodies; occluded targets are not detected.
• Multiple targets within the beam contribute to the total signal, but detection and reporting act on individual returns.
• The sensor must be in an operational state to capture data, and must have fired its signal trigger where one is attached.