CRB-ISAC Beamforming Reproduction

Baseline P1-C in Awesome-ISAC

📖 Part of Yuanhao Cui's ISAC paper-repro infrastructure

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📄 Paper

Title: Cramer-Rao Bound Optimization for Joint Radar-Communication Design

Authors: Fan Liu, Ya-Feng Liu, Ang Li, Christos Masouros, Yonina C. Eldar

Venue: IEEE Transactions on Signal Processing (TSP), Vol. 70, pp. 240–253, 2022

Awards: 🏆 2024 IEEE SPS Best Paper Award

DOI: 10.1109/TSP.2021.3135692

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🔬 What This Baseline Reproduces

Figure	Description	Status
Fig. 2	Root-CRB / MSE vs SINR threshold (K=1)	✅
Fig. 3	Beampattern comparison (K=4, SINR=15dB)	✅

Key Results

• Fig. 2: Validates the closed-form solution for single-user CRB minimization. Shows that estimation errors remain stable when SINR requirements are below 30 dB, then rise sharply.
• Fig. 3: Demonstrates that the proposed CRB-Min design achieves the highest power at the target angle (0°) compared to benchmark beampattern approximation methods.

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🧮 Mathematical Background

System Model

MIMO DFRC BS with $N_t$ transmit, $N_p$ receive antennas, serving $K$ users while detecting a point target.

CRB for point target angle $\theta$:

$$\text{CRB}(\theta) = \frac{1}{2L \cdot \text{SNR}_{\text{eff}}}$$

where $L$ is the frame length and $\text{SNR}_{\text{eff}}$ is the effective radar SNR, which depends on the power allocated to radar after satisfying communication SINR constraints.

Power sharing model: Given total power $P_r$ and per-user SINR threshold $\gamma_k$:

$$P_{\text{radar}} = P_r - \sum_k \gamma_k \sigma^2$$

Problem Formulation

Problem (P1) — CRB minimization with SINR constraints:

$$\min_{\mathbf{W}} \text{CRB}(\theta) \quad \text{s.t.} \quad \text{SINR}_k \geq \gamma_k, \forall k, \quad |\mathbf{W}|_F^2 \leq P_r$$

Solved via Semidefinite Relaxation (SDR) + rank-one extraction (Theorem 4 in paper).

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🚀 Quick Start

# Install dependencies
pip install numpy cvxpy matplotlib scipy pytest pytest-cov

# Reproduce Fig. 2 (Root-CRB vs SINR)
python examples/reproduce_fig2.py

# Reproduce Fig. 3 (Beampattern comparison)
python examples/reproduce_fig3.py

# Run tests
pytest tests/ -v

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📁 Project Structure

.
├── src/
│   ├── model.py      # CRB model, system params, steering vectors
│   ├── solver.py     # SDR solver, closed-form, benchmark designs
│   └── metrics.py    # Beampattern, RMSE, MSE, trend verification
├── tests/
│   ├── test_model.py
│   ├── test_solver.py
│   ├── test_metrics.py
│   └── test_reproducibility.py
├── examples/
│   ├── reproduce_fig2.py
│   └── reproduce_fig3.py
├── configs/
│   └── default.yaml
├── README.md
└── requirements.txt

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📊 Parameters

Parameter	Value	Description
$N_t$	16	Transmit antennas
$N_p$	20	Receive antennas
$K$	4	Communication users
$L$	30	Frame length
$P_r$	30 dBm	Total transmit power
$\sigma^2$	0 dBm	Noise variance
$\gamma$	15 dB	Per-user SINR threshold

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✅ Test Results

pytest tests/ -v
# test_model.py       ✅ 6 tests
# test_solver.py      ✅ 9 tests
# test_metrics.py     ✅ 6 tests
# test_reproducibility.py ✅ 6 tests

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🤝 Contributing

This baseline follows the Awesome-ISAC contributing guidelines.

To add this baseline to Awesome-ISAC, see the Reproducible Baselines section in the main README.

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📝 Citation

@article{liu2022crb,
  author={Liu, Fan and Liu, Ya-Feng and Li, Ang and Masouros, Christos and Eldar, Yonina C.},
  title={Cramer-Rao Bound Optimization for Joint Radar-Communication Design},
  journal={IEEE Trans. Signal Processing},
  volume={70},
  pages={240--253},
  year={2022},
  doi={10.1109/TSP.2021.3135692}
}