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Hybrid Channel Estimation with Quantized Phase Feedback for Over-the-Air Computation

Martin Dahl, Erik G. Larsson

TL;DR

The work tackles signaling overhead in coherent Over-the-Air Computation by proposing a hybrid channel-estimation scheme that combines reciprocity-based amplitude estimation with quantized phase-feedback. It analyzes two variants: Variant A uses feedback to estimate the phase while Variant B leverages reciprocity for phase estimation and incorporates periodic recalibration to combat phase noise; Variant A yields an analytically tractable MSE expression, whereas Variant B is validated via simulations showing gains under realistic phase-noise conditions. The study demonstrates that separate control of amplitude and phase precision, together with phase-quantization tailored to noise statistics, can preserve coherent aggregation with reduced signaling overhead, and highlights regimes where reciprocity suffices versus when phase feedback is advantageous. Overall, this hybrid approach provides a practical framework for reducing CSI signaling in OAC while maintaining estimation accuracy in the presence of oscillator drift and phase noise.

Abstract

To reduce the signaling overhead of over-the-air computation, a hybrid channel estimation scheme is proposed, where reciprocity-based and feedback-based channel estimation are combined. In particular, the impact of quantized phase-feedback is studied while the amplitude is assumed estimated exactly. The scheme enables selecting the estimation precision of amplitude and phase separately, depending on the importance of each. Two variants of the scheme are proposed: As shown through simulations and theory, the second variant with reciprocity-based estimation of the channel phase, and optimal quantization of phase feedback, can outperform the first variant estimating the phase by feedback only.

Hybrid Channel Estimation with Quantized Phase Feedback for Over-the-Air Computation

TL;DR

The work tackles signaling overhead in coherent Over-the-Air Computation by proposing a hybrid channel-estimation scheme that combines reciprocity-based amplitude estimation with quantized phase-feedback. It analyzes two variants: Variant A uses feedback to estimate the phase while Variant B leverages reciprocity for phase estimation and incorporates periodic recalibration to combat phase noise; Variant A yields an analytically tractable MSE expression, whereas Variant B is validated via simulations showing gains under realistic phase-noise conditions. The study demonstrates that separate control of amplitude and phase precision, together with phase-quantization tailored to noise statistics, can preserve coherent aggregation with reduced signaling overhead, and highlights regimes where reciprocity suffices versus when phase feedback is advantageous. Overall, this hybrid approach provides a practical framework for reducing CSI signaling in OAC while maintaining estimation accuracy in the presence of oscillator drift and phase noise.

Abstract

To reduce the signaling overhead of over-the-air computation, a hybrid channel estimation scheme is proposed, where reciprocity-based and feedback-based channel estimation are combined. In particular, the impact of quantized phase-feedback is studied while the amplitude is assumed estimated exactly. The scheme enables selecting the estimation precision of amplitude and phase separately, depending on the importance of each. Two variants of the scheme are proposed: As shown through simulations and theory, the second variant with reciprocity-based estimation of the channel phase, and optimal quantization of phase feedback, can outperform the first variant estimating the phase by feedback only.
Paper Structure (13 sections, 1 theorem, 19 equations, 6 figures)

This paper contains 13 sections, 1 theorem, 19 equations, 6 figures.

Table of Contents

  1. Introduction
  2. System Objective and Model
  3. Precoding and Power Control
  4. Quantization, Calibration and phase noise
  5. Uniform and Lloyd-Max Quantization
  6. Channel Estimation with Calibrated Reciprocity
  7. Phase Drift from Wiener Phase Noise
  8. Hybrid Channel Estimation
  9. Variant A: Feedback-based phase estimation
  10. Variant B: Reciprocity-based phase estimation
  11. Analytical result for Variant A
  12. Simulation results
  13. conclusions

Key Result

Lemma 1

Let $\widehat{v}$ be given by OAC with Variant A channel estimation, where $n, g_k\sim\mathcal{CN}(0,1), \text{ i.i.d. } \forall k$, then

Figures (6)

  • Figure 1: Channel estimation corresponding to steps 2,3,4 in Variant A and B.
  • Figure 2: Variant A: As $|c_k|$ is very stable, the calibration is done only once, making Variant A independent of $t$. Steps 2,3,4 are summarized by Figure \ref{['fig:tikz_estimation']}.
  • Figure 3: Variant B: The calibration is done periodically because of oscillator drift, making $t$ important. Steps 2,3,4 are summarized by Figure \ref{['fig:tikz_estimation']}.
  • Figure 4: Lloyd-Max quantizer (LMQ) vs uniform quantizer (UQ) with 20 instances of angles sampled i.i.d. from $\mathcal{N}(0,1)$.
  • Figure 5: OAC sum estimator (\ref{['eq:OAC_estimate']}) MSE (\ref{['eq:MSE_goal']}) against number of quantization bits $N$. $N=0$ means no phase feedback. $K=10$. $v_k, g_k, n \sim\mathcal{CN}(0,1)$ i.i.d. $\forall k$, averaged over $10^5$ trials.
  • ...and 1 more figures

Theorems & Definitions (2)

  • Lemma 1: Variant A MSE
  • proof