AirGC-CD: Gaussian-Circulant Precoding for Exactly Debiasable PAPR Reduction in Over-the-Air Federated Learning

Over-the-air federated learning lets edge devices transmit their local updates simultaneously, reducing the communication overhead. The resulting waveform, however, has a peak-to-average power ratio (PAPR) that grows with the model dimension, and keeping the amplifier in its linear range leaves two remedies: clipping the peaks or backing off the transmit power. Neither remedy is without cost: i) the clipping distortion appears at the receiver as a bias that cannot be removed, and ii) back-off keeps the signal intact but degrades the average signal-to-noise ratio (SNR). Independent of this trade-off, the transmission remains uncompressed, spending one channel use per model parameter, which keeps large-model training out of reach. To address these challenges, we propose AirGC-CD, an over-the-air scheme that precodes each local update with a partial Gaussian circulant matrix before clipping. In AirGC-CD, the precoder's output is exactly Gaussian regardless of the update's sparsity, so the clipping function is designed for a known distribution instead of inheriting it from the data. This enables the clipping to be inverted on average by a single scalar Bussgang gain in closed form, and we prove that the resulting aggregate is exactly unbiased, with clipping adding only variance. The clipping ratio is then the only free parameter left, trading the variance of the clipping against the SNR loss from back-off, and we derive its near-optimum in closed form. Since the precoder is linear, it also acts as a compressor, reducing the transmission from the model dimension d to the sketch dimension m at a cost of only O(dlog d) via two fast Fourier transforms, whereas a Gaussian sketch costs O(md). Experiments on five image datasets show that AirGC-CD outperforms baseline over-the-air FL schemes in most settings, particularly at low SNR, while using fewer channel uses per round.