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Physics Engine Documentation

This document describes the physics models implemented in RadarSim's core engine.

Table of Contents

Radar Equation

Standard Monostatic Form

The fundamental radar range equation calculates received power from a target:

$$P_r = \frac{P_t G_t G_r \lambda^2 \sigma}{(4\pi)^3 R^4 L}$$

Where:

Symbol Description Units
$P_r$ Received power W
$P_t$ Transmitted power W
$G_t$ Transmit antenna gain linear
$G_r$ Receive antenna gain linear
$\lambda$ Wavelength ($c/f$) m
$\sigma$ Radar Cross Section (RCS) m²
$R$ Target range m
$L$ System losses linear

Implementation: src/physics/radar_equation.py::_calculate_signal_power_jit()

Signal-to-Noise Ratio (SNR)

$$SNR = \frac{P_r}{P_n} = \frac{P_r}{k T_0 B F_n}$$

Where:

  • $k = 1.380649 \times 10^{-23}$ J/K (Boltzmann constant)
  • $T_0 = 290$ K (Standard reference temperature)
  • $B$ = Noise bandwidth (Hz)
  • $F_n$ = Receiver noise figure (linear)

Implementation: src/physics/radar_equation.py::_calculate_snr_jit()

Antenna Gain Approximation

For parabolic antennas, gain is approximated from beamwidth:

$$G \approx \frac{30,000}{\theta_{az} \times \theta_{el}}$$

Where $\theta$ is the 3-dB beamwidth in degrees.

Reference: Skolnik, "Radar Handbook", 3rd Ed., Eq. 6.9

Atmospheric Attenuation

ITU-R P.676-12 Model

RadarSim implements the ITU-R Recommendation P.676-12 for atmospheric gas attenuation, covering frequencies up to 1000 GHz.

Total Attenuation

$$A_{total} = (\gamma_o + \gamma_w) \times d \times 2$$

Where:

  • $\gamma_o$ = Oxygen specific attenuation (dB/km)
  • $\gamma_w$ = Water vapor specific attenuation (dB/km)
  • $d$ = One-way path length (km)
  • Factor of 2 accounts for two-way radar path

Oxygen Absorption

Key absorption features:

  • 60 GHz complex: Strong O₂ resonance (~15 dB/km at peak)
  • 118.75 GHz line: Secondary O₂ absorption
# Normalized pressure and temperature
rp = pressure_hpa / 1013.25
rt = 288.0 / temperature_k

# 60 GHz peak
if 57 < frequency_ghz < 63:
    gamma_o = 15.0 * rp * (rt ** 0.5)

Water Vapor Absorption

Key absorption lines:

  • 22.235 GHz: Primary H₂O line
  • 183.31 GHz: Strong H₂O line (~30 dB/km in humid conditions)

Implementation: src/physics/atmospheric.py::ITU_R_P676

Rain Attenuation (ITU-R P.838)

RadarSim now implements specific attenuation due to rain:

$$ \gamma_R = k \cdot R^\alpha \quad (\text{dB/km}) $$

Where:

  • $R$: Rain rate (mm/hr)
  • $k, \alpha$: Frequency-dependent coefficients (Horizontal polarization)
Freq (GHz) $k_H$ $\alpha_H$
10 (X-Band) 0.0101 1.276
20 (K-Band) 0.0367 1.154
35 (Ka-Band) 0.0751 1.099

Implementation: src/simulation/engine.py step loop

RCS Fluctuation Models

Swerling Models

RadarSim implements all four Swerling models for realistic RCS fluctuation:

Model Distribution Decorrelation Physical Basis
Swerling I Exponential Scan-to-scan Many equal scatterers
Swerling II Exponential Pulse-to-pulse Many equal, fast variation
Swerling III Chi-squared (4 DOF) Scan-to-scan One dominant + many small
Swerling IV Chi-squared (4 DOF) Pulse-to-pulse Dominant + small, fast

Implementation

def apply_swerling_fluctuation(rcs_mean: float, model: int) -> float:
    if model == 1 or model == 2:
        # Exponential distribution (Rayleigh amplitude)
        return np.random.exponential(rcs_mean)
    elif model == 3 or model == 4:
        # Chi-squared with 4 DOF
        return np.random.gamma(2, rcs_mean / 2)
    else:
        return rcs_mean  # Swerling 0 (non-fluctuating)

Implementation: src/physics/rcs.py::SwerlingModel

Terrain & Line-of-Sight

4/3 Earth Model

To account for atmospheric refraction, RadarSim uses the standard 4/3 effective Earth radius model:

$$R_e' = \frac{4}{3} R_e = 8495 \text{ km}$$

Where $R_e = 6371$ km is the mean Earth radius.

Line-of-Sight Algorithm

The LOS check uses raymarching with 100+ steps:

for i in range(num_steps):
    # Interpolate position along ray
    t = i / num_steps
    ray_x = radar_x + t * (target_x - radar_x)
    ray_y = radar_y + t * (target_y - radar_y)
    
    # Calculate ray height with Earth curvature
    ray_alt = radar_alt + t * (target_alt - radar_alt)
    ray_alt -= earth_curvature_drop(distance * t)
    
    # Check against terrain
    terrain_height = terrain_map.get_elevation(ray_x, ray_y)
    if ray_alt < terrain_height:
        return False  # Blocked by terrain

Radar Horizon

Maximum detection range limited by horizon:

$$R_{horizon} = \sqrt{2 R_e' h}$$

For radar height $h = 100$ m: $R_{horizon} \approx 41$ km (one-way)

Implementation: src/physics/terrain.py::TerrainMap

Monopulse Tracking

Sum/Difference Patterns

RadarSim implements amplitude-comparison monopulse for sub-beamwidth angular accuracy.

$$ \Sigma(\theta) = 2 \cos\left(\frac{\pi d}{\lambda}\sin\theta\right) \cdot \text{sinc}\left(\frac{\pi D}{\lambda}\sin\theta\right) $$

$$ \Delta(\theta) = 2j \sin\left(\frac{\pi d}{\lambda}\sin\theta\right) \cdot \text{sinc}\left(\frac{\pi D}{\lambda}\sin\theta\right) $$

Error Signal & Accuracy

The error signal $\epsilon = \Delta / \Sigma$ provides a linear response near boresight, allowing for precise angle estimation.

Theoretical Accuracy: $$ \sigma_\theta = \frac{\theta_{3dB}}{k_m \sqrt{2 \cdot SNR}} $$ Where $k_m \approx 1.6$ is the monopulse slope.

Implementation: src/tracking/monopulse.py

Ambiguity Analysis

Maximum Unambiguous Values

  • Range: $R_{max} = c / (2 \cdot PRF)$
  • Velocity: $V_{max} = (\lambda \cdot PRF) / 4$

Blind Speeds

Targets traveling at multiples of the blind speed will appear stationary (zero Doppler) due to aliasing: $$ V_{blind} = n \cdot \frac{\lambda \cdot PRF}{2} $$

Implementation: src/ui/analysis/ambiguity_plot.py

Environmental Clutter

RadarSim models clutter as a statistical process that degrades the effective SNR: $SNR_{eff} = SNR - CNR$.

Clutter Models

Type Distribution Typical $\sigma^0$
Ground Weibull -15 to -25 dB
Sea GIT Model (Douglas Sea State) -20 to -45 dB
Rain Marshall-Palmer Variable ($Z=200R^{1.6}$)

Implementation: src/physics/clutter.py

References

  1. Skolnik, M.I. (2008). Radar Handbook, 3rd Edition. McGraw-Hill.

    • Chapter 2: Radar equation fundamentals
    • Chapter 6: Antenna theory
    • Chapter 7: RCS models

  2. ITU-R P.676-12 (2019). Attenuation by atmospheric gases. International Telecommunication Union.

  3. Barton, D.K. (2005). Radar System Analysis and Modeling. Artech House.

    • Swerling models derivation

  4. Blake, L.V. (1980). Radar Range-Performance Analysis. Artech House.

    • 4/3 Earth model origins

Document generated for RadarSim v1.0.0