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Signal Processing Documentation

Overview

RadarSim implements standard radar signal processing algorithms including detection, filtering, and imaging.

Table of Contents

1. CFAR Detection

File: src/signal/cfar.py

Cell-Averaging CFAR (CA-CFAR)

┌─────────────────────────────────────────────────────────┐
│ Ref Cells │ Guard │ CUT │ Guard │ Ref Cells │
│   (N/2)   │  (G)  │     │  (G)  │   (N/2)   │
└─────────────────────────────────────────────────────────┘

Threshold: $$T = \alpha \cdot \frac{1}{N} \sum_{i=1}^{N} x_i$$

Where $\alpha$ is set by desired $P_{fa}$: $$\alpha = N \cdot (P_{fa}^{-1/N} - 1)$$

A-Scope Visualization

The A-Scope (Phase 23) shows CFAR cells on mouse hover:

  • Red line: Cell Under Test (CUT)
  • Grey region: Guard cells
  • Green region: Reference cells

2. Pulse Compression

Linear Frequency Modulated (LFM) Chirp

RadarSim generates LFM waveforms for improved range resolution:

$$ s(t) = A \cdot \exp\left(j2\pi \left(f_0 t + \frac{\beta t^2}{2}\right)\right) $$

Where $\beta = B/T$ is the chirp rate.

Matched Filter

The matched filter output maximizes SNR: $y(t) = x(t) * s^*(-t)$.

Pulse Compression Ratio: $PCR = B \cdot T$ (e.g., $100 = 20$ dB gain).

Implementation: src/signal/waveforms.py::LFMWaveform

3. Doppler Processing

Doppler Shift

For monostatic radar with radial velocity $v_r$:

$$ f_d = \frac{2 v_r}{\lambda} = \frac{2 v_r f_0}{c} $$

Velocity Resolution & Aliasing

  • Resolution: $\Delta v = \lambda / (2 \cdot N \cdot T_{PRI})$
  • Max Unambiguous Velocity: $v_{max} = \pm (PRF \cdot \lambda) / 4$

Implementation: src/physics/radar_equation.py

4. Range-Doppler Maps

Generation Process

  1. Collect CPI: Gather N pulses of raw I/Q data.
  2. Range FFT: FFT along fast-time (range bins).
  3. Doppler FFT: FFT along slow-time (pulses).
  4. CFAR: Apply 2D CFAR detection.

Implementation: src/ui/range_doppler.py

5. MTI Filtering

File: src/simulation/engine.py

Velocity Threshold

$$V_{threshold} = 2 \text{ m/s}$$ (default)

Targets with radial velocity below threshold are rejected as clutter.

Logic:

if abs(target_velocity_radial) < mti_threshold:
    target.is_detected = False  # Clutter

6. Clutter Models

File: src/physics/clutter.py

Ground Clutter (Weibull)

$$P(\sigma) = \frac{k}{\lambda}\left(\frac{\sigma}{\lambda}\right)^{k-1} \exp\left(-\left(\frac{\sigma}{\lambda}\right)^k\right)$$

Sea Clutter (GIT Model)

Uses Georgia Tech Research Institute (GIT) empirical model based on Douglas Sea State:

$$ \sigma^0 = f(SS, \theta_{grazing}, \lambda, pol) $$

Accounts for:

  • Sea State (0-6)
  • Grazing angle
  • Wind direction (implied)

Rain Clutter (Marshall-Palmer)

$$Z = 200 \cdot R^{1.6}$$ (Z-R relationship)

$$\eta = \frac{\pi^5 |K|^2}{\lambda^4} Z$$

7. SAR Image Formation

File: src/advanced/sar_isar.py

Range Resolution

$$\delta_r = \frac{c}{2B}$$

Where $B$ = bandwidth [Hz].

Azimuth Resolution

$$\delta_a = \frac{D}{2}$$

Where $D$ = antenna length [m].

Algorithms Implemented

Algorithm Description
Range-Doppler (RDA) Standard SAR processing
Backprojection (BPA) Time-domain focusing
Omega-K (ωK) Wavenumber domain
Chirp Scaling (CSA) Efficient, no interpolation

Key Equations

Doppler Rate: $$K_a = -\frac{2v^2}{\lambda R_0}$$

Range Cell Migration: $$\Delta R = \frac{\lambda^2 R_0 f_d^2}{8v^2}$$

Stolt Interpolation: $$K_r' = \sqrt{(k_0 + K_r)^2 - K_a^2} - k_0$$

8. Probability of Detection

File: src/physics/metrics.py

Albersheim's Equation

Required SNR for given $P_d$ and $P_{fa}$:

$$SNR = A + 0.12AB + 1.7B$$

Where:

  • $A = \ln(0.62 / P_{fa})$
  • $B = \ln(P_d / (1 - P_d))$

Swerling Detection Probability

For Swerling I targets: $$P_d = \exp\left(-\frac{T}{1 + SNR}\right)$$

9. B-Scope ECM Strobe Visualization

File: src/ui/b_scope.py

When jamming is detected, vertical "noise bars" appear at the jammer azimuth:

    Jammer at 30°
         │
         ▼
┌────────║─────────────────────────────────┐
│████████║████████                         │  Range
│████████║████████                         │   ↑
│████████║████████  ← Yellow strobe        │
│████████║████████                         │
└────────║─────────────────────────────────┘
         Azimuth →

Implementation:

  • LinearRegionItem pool for up to 5 simultaneous jammers
  • Yellow semi-transparent brush (255, 255, 0, 60)
  • Strobe width: 5° centered on jammer azimuth

10. ROC Curve Analysis

File: src/ui/analysis/roc_curve.py

Receiver Operating Characteristic

Plots $P_d$ vs $P_{fa}$ for different SNR values:

$$P_d = f(P_{fa}, SNR, \text{Swerling Model})$$

Swerling Model Selection

Model Fluctuation Type
0 Non-fluctuating
1 Slow, Rayleigh
3 Slow, Chi-square

Operating Point Display

Shows current radar operating point on ROC curve.

11. SNR Histogram Analysis

File: src/ui/analysis/snr_histogram.py

Real-time distribution of detection SNR values:

Zone SNR Range Color
Weak < 10 dB Red
Moderate 10-20 dB Yellow
Strong > 20 dB Green

Statistics Displayed

  • Mean SNR
  • Median SNR
  • Standard Deviation
  • Total Detections

References

  1. Richards, M.A. - Fundamentals of Radar Signal Processing, 2014
  2. Cumming & Wong - Digital Processing of SAR Data, 2005
  3. Rohling, H. - "Radar CFAR Thresholding", IEEE TAES, 1983
  4. Swerling, P. - "Probability of Detection", IRE Trans, 1960