Technical Summary: Deep Learning based Three-stage Solution for ISAC Beamforming Optimization

1. Problem Statement

The paper addresses the challenge of optimizing beamforming in a Full-Duplex (FD) Integrated Sensing and Communication (ISAC) system. In this scenario, a Base Station (BS) equipped with Uniform Linear Arrays (ULAs) simultaneously communicates with multiple single-antenna users and performs radar sensing on multiple targets using the same time-frequency resources.

The core objective is to maximize the sum communication rate of all users while adhering to two critical constraints:

1. Transmit Power Constraint: The total power budget must not exceed a maximum limit ($P_{max}$).
2. Sensing Rate Constraint: Each target must achieve a minimum sensing rate ($\Gamma_{sen}^{req}$), which serves as a proxy for target detection probability.

The authors identify limitations in existing Deep Learning (DL) approaches for beamforming. Directly mapping Channel State Information (CSI) to beamforming vectors suffers from:

• High Dimensionality: The optimization variable space grows rapidly with the number of antennas and users, increasing computational complexity and training difficulty.
• Lack of Interpretability: Directly optimizing vectors makes it difficult to understand or control the underlying beamforming behavior (e.g., directionality and interference patterns).
• Complexity: Traditional convex optimization methods become computationally prohibitive in high-dimensional, non-convex scenarios.

2. Methodology: The Three-Stage Framework

To overcome these limitations, the authors propose a three-stage solution that decomposes the beamforming optimization problem into three distinct sub-modules. This framework transforms the problem from direct vector optimization to a more structured process involving feature extraction, beampattern optimization, and vector reconstruction.

Module 1: Unsupervised Learning-based CSI Feature Extraction

• Goal: To compress high-dimensional, variable CSI data into fixed-size latent features while preserving essential information.
• Technique: An Autoencoder (AE) is employed.
  • Input: The CSI dataset includes downlink channels ($h_k$), uplink channels ($g_{l,t}$), radar channels ($g_{l,r}$), and residual self-interference channels ($H_{SI}$). These complex matrices are vectorized and separated into real and imaginary parts to form a real-valued input vector.
  • Process: The encoder maps the high-dimensional CSI to a low-dimensional latent feature vector ($f$). The decoder attempts to reconstruct the original CSI from $f$.
  • Training: The network is trained unsupervised by minimizing the Mean Square Error (MSE) between the original and reconstructed CSI.
  • Usage: During the testing/optimization phase, only the encoder is used to extract features from new CSI inputs.

Module 2: Reinforcement Learning (RL) based Beampattern Optimization

• Goal: To search for the optimal beampattern (power distribution over angles) based on the extracted features, rather than directly searching for beamforming vectors.
• Technique: An Advantage Actor-Critic (A2C) algorithm is used.
  • State ($s_t$): Composed of the extracted CSI features ($f_t$) and the sum communication rate from the previous time slot ($R_{com}^{t-1}$).
  • Action ($a_t$): Instead of outputting complex beamforming vectors, the agent outputs beampattern samples. The action space consists of $A$ discrete gain values sampled across the angular domain (-90° to 90°) for three specific beampatterns:
    1. Communication beampattern ($p_w$)
    2. Sensing beampattern ($p_v$)
    3. Receive beampattern ($p_u$)
  • Reward ($r$): A piecewise function that rewards the sum of communication rates if both communication and sensing rate constraints are met. It assigns 0 if sensing is met but communication is not, and -1 if either constraint is violated.
  • Advantage: Optimizing in the beampattern domain reduces the action space size (fixed at $A$) regardless of the number of antennas or users, improving training stability and generalization.

Module 3: Supervised Learning-based Beamforming Reconstruction

• Goal: To reconstruct the actual transmit and receive beamforming vectors from the optimized beampatterns generated by the RL agent.
• Technique: A supervised Deep Neural Network (DNN) approximates the inverse function $G^{-1}$.
  • Process: Since the analytical inverse of the beampattern-to-vector mapping is intractable (due to the complex response surface), a DNN is trained to learn this mapping.
  • Dataset: Generated by randomly creating beamforming vectors, normalizing them to meet power constraints, calculating their corresponding beampatterns, and pairing them as (Input: Beampattern, Output: Beamforming Vector).
  • Usage: In the testing stage, the pre-trained reconstruction network takes the beampattern output from the RL agent and instantly generates the corresponding beamforming vectors for implementation.

3. Key Contributions

The paper claims the following contributions:

1. Problem Formulation: A joint optimization problem is formulated for a full-duplex ISAC system to maximize sum communication rate under sensing rate and power constraints. The problem is transformed from beamforming vector optimization to beampattern optimization via array response vectors.
2. Three-Stage Framework: The design of a novel framework that decomposes the problem into:
   • Unsupervised CSI feature extraction (using AE).
   • RL-based beampattern optimization (using A2C).
   • Supervised beamforming reconstruction (using DNN).
3. Performance and Interpretability: The proposed solution enhances generalization through feature extraction, improves training stability by optimizing in the beampattern domain, and offers better interpretability compared to direct vector optimization.

4. Numerical Results

Simulations were conducted in a scenario with $N_t = N_r = 4$ antennas, $K=2$ users, and $L=1$ target. The results highlight:

• Feature Size: A latent feature size of 16 offered the best trade-off between compression and performance. Smaller sizes (e.g., 4) caused information loss, while larger sizes (e.g., 64) led to overfitting or slower convergence.
• Beampattern Resolution: A sample size of 720 points provided the highest performance, though 360 offered a good balance between convergence speed and reward.
• Ablation Study:
  • Case 0 (Baseline): Direct RL on CSI to beamforming vectors. Converged around 170 episodes with a reward of ~800.
  • Case 1 (Feature Extraction only): Converged faster (~130 episodes) with comparable performance, proving the efficiency of feature compression.
  • Case 2 (Beampattern Optimization only): Learned slower but eventually surpassed the baseline (reward >850), validating that beampatterns are more intuitive for RL agents to optimize.
  • Case 3 (Full Three-Stage): Achieved the best performance (reward >900), combining the efficiency of feature extraction with the structural benefits of beampattern optimization.

5. Significance and Claims

The paper concludes that the proposed three-stage solution outperforms baseline RL algorithms that directly optimize beamforming vectors. The significance lies in the shift from optimizing high-dimensional, non-interpretable vectors to optimizing structured, intuitive beampatterns. By decoupling the optimization of the beampattern from the generation of the beamforming vectors, the framework achieves:

• Reduced Complexity: Lower dimensionality in the RL action space.
• Improved Stability: More stable training dynamics.
• Interpretability: The intermediate beampattern representation allows for better understanding of the system's spatial filtering behavior.

The authors assert that this approach is particularly advantageous as the number of antennas and users increases, where traditional convex optimization becomes too complex and direct DL mapping becomes difficult to train and interpret.