Ping-Pong Sampling in mmWave MIMO

• Ping-Pong sampling is a two-sided, alternating pilot protocol that autonomously refines beam alignment in mmWave MIMO channels without needing explicit feedback.
• The approach employs LSTM-based recurrent neural networks and DNNs to iteratively update pilot vectors and optimize beamformers and RIS phases for enhanced SNR.
• Empirical evaluations demonstrate that with only 8 pilot exchanges, the method reduces overhead and outperforms traditional channel estimation techniques, even under low SNR conditions.

Ping-Pong sampling refers to a two-sided, alternating pilot signaling and active sensing protocol for beam alignment in millimeter-wave (mmWave) MIMO systems, where the transmitter (Tx) and receiver (Rx) iteratively and autonomously adapt their measurement vectors over multiple rounds without the need for explicit feedback. This class of strategies is particularly suited for scenarios in which both ends of a link are equipped with large antenna arrays but only a limited number of RF chains, and the channel is modeled as high-dimensional and structured by multipath sparsity. The ping-pong protocol facilitates rapid, robust, and interpretable joint beam alignment, and can be further generalized to incorporate reconfigurable intelligent surfaces (RIS) for joint beam and reflection coefficient optimization (Jiang et al., 2023).

1. System Model and Problem Formulation

The basic system model considers a point-to-point mmWave MIMO channel with a transmitter (“agent A”, Tx) of $N_t$ antennas and one RF chain, and a receiver (“agent B”, Rx) of $N_r$ antennas and one RF chain, both configured as ULAs (Uniform Linear Arrays) with half-wavelength spacing. The narrowband, block-fading channel matrix from Rx to Tx is $\mathbf{G} \in \mathbb{C}^{N_t \times N_r}$, remaining constant over the full sensing and data transmission interval. The channel is represented as a sparse sum of a few multipath components: $$\mathbf{G} = \sum_{\ell=1}^L \alpha_\ell\,\mathbf{a}_t(\phi^{\rm t}_\ell)\mathbf{a}_r(\phi^{\rm r}_\ell)^H,$$ where $\alpha_\ell \sim \mathcal{CN}(0,1)$, $\mathbf{a}_t(\cdot), \mathbf{a}_r(\cdot)$ are standard array response vectors, and $L$ is the number of paths. The objective is to identify transmit and receive beamformers $(\mathbf{f}^*, \mathbf{w}^*)$ that maximize the receive SNR: $$\max_{\|\mathbf f\|=\|\mathbf w\|=1} |\mathbf{w}^H\mathbf{G}^H\mathbf{f}|^2.$$ With perfect CSI, the optimal solution is given by the principal singular vectors of $\mathbf{G}$.

An RIS-assisted extension models the effective MIMO channel as

$N_r$0

where $N_r$1 and $N_r$2 represent the Tx–RIS and RIS–Rx sub-channels and $N_r$3 is the vector of unit-modulus RIS phase shifts.

2. Ping-Pong Pilot Transmission Protocol

The core of ping-pong sampling is an $N_r$4-round pilot exchange protocol. In each round $N_r$5, two steps occur:

• Ping (Tx → Rx): The transmitter sends a scalar pilot $N_r$6 using pilot beamformer $N_r$7. The receiver observes:

$N_r$8

applying its current sensing vector $N_r$9.

• Pong (Rx → Tx): Based on all observations $\mathbf{G} \in \mathbb{C}^{N_t \times N_r}$0, Rx selects its transmit sensing vector $\mathbf{G} \in \mathbb{C}^{N_t \times N_r}$1, transmits back a pilot, and Tx observes:

$\mathbf{G} \in \mathbb{C}^{N_t \times N_r}$2

using its current receive vector $\mathbf{G} \in \mathbb{C}^{N_t \times N_r}$3.

Each agent thus accumulates a personal sequence of pilot measurements and autonomously adapts its sensing protocol for future rounds. Notably, there is no explicit feedback channel—adaptation is based solely on local measurements. The formal protocol is captured by update rules ($\mathbf{G} \in \mathbb{C}^{N_t \times N_r}$4, $\mathbf{G} \in \mathbb{C}^{N_t \times N_r}$5, $\mathbf{G} \in \mathbb{C}^{N_t \times N_r}$6, $\mathbf{G} \in \mathbb{C}^{N_t \times N_r}$7), where each side generates its next sensing vector based on history.

3. Deep Active Sensing Framework

To operationalize the adaptive policy mappings in ping-pong sampling, each side employs a dedicated LSTM-based recurrent neural network. At each round:

• The new scalar measurement (real and imaginary parts) updates the LSTM’s hidden and cell states via standard gate mechanisms.
• These states parameterize small fully connected DNN “heads” which output the transmit and receive pilot vectors for the next round, enforced to have unit $\mathbf{G} \in \mathbb{C}^{N_t \times N_r}$8 norm.

After $\mathbf{G} \in \mathbb{C}^{N_t \times N_r}$9 rounds, the final LSTM cell states summarize the entire local history of observations. Two further DNNs ($$\mathbf{G} = \sum_{\ell=1}^L \alpha_\ell\,\mathbf{a}_t(\phi^{\rm t}_\ell)\mathbf{a}_r(\phi^{\rm r}_\ell)^H,$$0, $$\mathbf{G} = \sum_{\ell=1}^L \alpha_\ell\,\mathbf{a}_t(\phi^{\rm t}_\ell)\mathbf{a}_r(\phi^{\rm r}_\ell)^H,$$1) translate these cell states into final data phase beamformers ($$\mathbf{G} = \sum_{\ell=1}^L \alpha_\ell\,\mathbf{a}_t(\phi^{\rm t}_\ell)\mathbf{a}_r(\phi^{\rm r}_\ell)^H,$$2, $$\mathbf{G} = \sum_{\ell=1}^L \alpha_\ell\,\mathbf{a}_t(\phi^{\rm t}_\ell)\mathbf{a}_r(\phi^{\rm r}_\ell)^H,$$3). If an RIS is present, a third DNN ($$\mathbf{G} = \sum_{\ell=1}^L \alpha_\ell\,\mathbf{a}_t(\phi^{\rm t}_\ell)\mathbf{a}_r(\phi^{\rm r}_\ell)^H,$$4) produces the final RIS phase vector ($$\mathbf{G} = \sum_{\ell=1}^L \alpha_\ell\,\mathbf{a}_t(\phi^{\rm t}_\ell)\mathbf{a}_r(\phi^{\rm r}_\ell)^H,$$5), applying element-wise normalization to enforce the unit-modulus constraint.

All learnable parameters (LSTM, DNN heads) are trained end-to-end to maximize the expectation of the beamforming gain over random instantiations of $$\mathbf{G} = \sum_{\ell=1}^L \alpha_\ell\,\mathbf{a}_t(\phi^{\rm t}_\ell)\mathbf{a}_r(\phi^{\rm r}_\ell)^H,$$6 and noisy measurements: $$\mathbf{G} = \sum_{\ell=1}^L \alpha_\ell\,\mathbf{a}_t(\phi^{\rm t}_\ell)\mathbf{a}_r(\phi^{\rm r}_\ell)^H,$$7 Training is performed through stochastic gradient methods (SGD/Adam) over large Monte Carlo batches.

4. Beamformer Optimization and RIS Control

The ping-pong protocol can be directly compared to closed-form solutions available under perfect CSI. For the pure MIMO case, the benchmark is the SVD-based solution ($$\mathbf{G} = \sum_{\ell=1}^L \alpha_\ell\,\mathbf{a}_t(\phi^{\rm t}_\ell)\mathbf{a}_r(\phi^{\rm r}_\ell)^H,$$8). With RIS present, alternating optimization is employed: given RIS phases, apply SVD to $$\mathbf{G} = \sum_{\ell=1}^L \alpha_\ell\,\mathbf{a}_t(\phi^{\rm t}_\ell)\mathbf{a}_r(\phi^{\rm r}_\ell)^H,$$9; with fixed beamformer pair, update each RIS phase to maximize alignment.

In the learned (active) scheme, all vectors are generated by the corresponding DNNs fed with the LSTM histories. In the RIS setting, the DNN for RIS control ingests the concatenated final LSTM cell states from both Tx and Rx.

5. Empirical Performance and Interpretability

The ping-pong sampling protocol, implemented via the deep active sensing architecture, demonstrates:

• Pilot Overhead Reduction: With only $\alpha_\ell \sim \mathcal{CN}(0,1)$0 ping-pong rounds ($\alpha_\ell \sim \mathcal{CN}(0,1)$1 pilots), the approach surpasses classical compressive sensing plus SVD, and fixed/random DNN-based schemes using 20 pilots, at SNR = 0 dB.
• SNR Robustness: Performance remains strong at SNR $\alpha_\ell \sim \mathcal{CN}(0,1)$2 dB, outperforming standard channel estimation protocols measured at SNR = 0 dB.
• Path Generalization: Networks trained on 3-path channels retain high performance with varying numbers of paths (1–5).
• Interpretability: In early rounds, the learned sensing beams are broad, approximating uniform coverage; in later rounds, they narrow to concentrate on promising directions. Final learned data beams coincide with the principal singular vectors approximately 63% (top), 26% (second), etc., of the time for $\alpha_\ell \sim \mathcal{CN}(0,1)$3 rounds.
• RIS Adaptation: Joint design of RIS phases in the active sensing process delivers an additional 1–2 dB gain compared to static or random RIS phase codes.
• Computational Complexity: At inference, each round involves an LSTM update and a few DNN forward passes, with total cost $\alpha_\ell \sim \mathcal{CN}(0,1)$4.

6. Applications and Extensions

Ping-pong sampling is immediately relevant for low-feedback, high-efficiency beam alignment in next-generation mmWave wireless systems, particularly where channel sparsity, rapid environmental dynamics, and limited RF chains render classical schemes inefficient. The approach’s adaptability and compatibility with RIS control broaden its utility to programmable and intelligent radio environments, supporting both directly connected and reflection-assisted links. A plausible implication is more robust operation under stringent pilot and SNR constraints, with direct impact on practical 5G/6G deployments (Jiang et al., 2023).