Radio-Frequency Convolutional Neural Networks

# Radio-Frequency Convolutional Neural Networks
Source: [https://arxiv.org/html/2609.19279](https://arxiv.org/html/2609.19279)
Affiliation:Department of Electrical and Computer Engineering, Duke University, Durham, NC 27708, USAAffiliation:Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA

## Abstract

Running artificial intelligence \(AI\) models directly on edge devices such as smartphones, wearables, and drones offers low latency, pervasive scalability, and data privacy, but these devices rarely carry the computing capability that modern neural networks demand\. Edge accelerators have been developed in response, yet each adds computing hardware to devices already constrained in size, weight, power, and cost \(SWaP\-C\)\. An alternative lies in what these devices already carry: the frequency mixer in every wireless radio multiplies signals in time, natively performing convolution in the frequency domain\. Here we introduce radio\-frequency convolutional neural networks \(RF\-CNNs\), which repurpose existing communication hardware for CNN inference\. Multi\-channel convolutions are mapped onto frequency tones for a passive mixer to execute in a single pass\. We experimentally demonstrate thatRF\-CNNruns deep CNNs up to 26\.4 million parameters and nine layers from classification of wireless signals and images to controllable image generation, close to full\-precision performance\. Because the weights arrive over the air and the analog hardware is shared with communication, the edge device spends energy only on data preparation and readout–down to 0\.72 femtojoules per multiply\-accumulate, two orders of magnitude less than it would cost on an added digital processor\. These results suggest that deployed wireless infrastructure can bring efficient, state\-of\-the\-art AI inference to the billions of devices it already connects\.

## Introduction

Artificial intelligence \(AI\) is advancing through scaling: machine learning \(ML\) models grow, and the computation they demand grows faster than hardware efficiency improves[Sevilla et al\. \(2022\)](https://arxiv.org/html/2609.19279#bib.bib53);[Thompson et al\. \(2020\)](https://arxiv.org/html/2609.19279#bib.bib16)\. Much of this intelligence is needed at the network edge, where a data center cannot serve it for reasons such as latency, privacy, or bandwidth, so inference must run where the data is produced[Satyanarayanan \(2017\)](https://arxiv.org/html/2609.19279#bib.bib54);[Zhou et al\. \(2019\)](https://arxiv.org/html/2609.19279#bib.bib15)\. The devices at the edge—smartphones, wearables, drones—operate on fixed budgets of battery, heat, size, and cost[Xu et al\. \(2018\)](https://arxiv.org/html/2609.19279#bib.bib55)\. The prevailing response has been to equip them with dedicated accelerators; yet every addition spends from the same budgets that define the edge, and the obstacle it faces is physical rather than algorithmic: within these budgets, the hardware the computation requires cannot be carried where the computation must occur\.

The hardware to resolve this constraint is, however, already carried by every edge device that communicates\. Every wireless edge device contains radio frequency \(RF\) front ends including frequency mixers and local oscillators \(LOs\), data converters, and antennas[Razavi \(2012\)](https://arxiv.org/html/2609.19279#bib.bib56);[Weinreich and Murmann \(2022\)](https://arxiv.org/html/2609.19279#bib.bib57)\(Fig\.[1](https://arxiv.org/html/2609.19279#Sx8.F1)a\)\. The frequency mixer multiplies two signals in the time domain and, by the convolution theorem, this multiplication is equivalent to a correlation in the frequency domain[Oppenheim \(1999\)](https://arxiv.org/html/2609.19279#bib.bib2)—the same operation communication systems use to translate signals between frequency bands\. It is also the operation that a convolutional layer performs, namely the cross\-correlation of an input with a kernel\. The operation that a frequency mixer executes is therefore the operation on which convolutional inference depends\. A wireless front end is, without modification, a convolutional processor\.

Yet this correspondence has not been fully exploited\. The dedicated accelerators that have driven efficient edge inference—digital processors[Jouppi et al\. \(2017\)](https://arxiv.org/html/2609.19279#bib.bib17);[Sze et al\. \(2017\)](https://arxiv.org/html/2609.19279#bib.bib18);[Khwa et al\. \(2025\)](https://arxiv.org/html/2609.19279#bib.bib58), analog in\-memory computing[Yao et al\. \(2020\)](https://arxiv.org/html/2609.19279#bib.bib19);[Ambrogio et al\. \(2023\)](https://arxiv.org/html/2609.19279#bib.bib20);[Wan et al\. \(2022\)](https://arxiv.org/html/2609.19279#bib.bib21);[Zhang et al\. \(2023\)](https://arxiv.org/html/2609.19279#bib.bib22);[Jung et al\. \(2022\)](https://arxiv.org/html/2609.19279#bib.bib48);[Song et al\. \(2024\)](https://arxiv.org/html/2609.19279#bib.bib59), and photonic processors[Shen et al\. \(2017\)](https://arxiv.org/html/2609.19279#bib.bib23);[Feldmann et al\. \(2021\)](https://arxiv.org/html/2609.19279#bib.bib24);[Xu et al\. \(2021\)](https://arxiv.org/html/2609.19279#bib.bib25);[Chen et al\. \(2023\)](https://arxiv.org/html/2609.19279#bib.bib50);[Ahmed et al\. \(2025\)](https://arxiv.org/html/2609.19279#bib.bib46);[Hua et al\. \(2025\)](https://arxiv.org/html/2609.19279#bib.bib47)—tailor the hardware to the model, each adding computing hardware alongside the device’s existing components\. A complementary body of work computes through an innate physical mechanism[Grollier et al\. \(2020\)](https://arxiv.org/html/2609.19279#bib.bib28);[Wright et al\. \(2022\)](https://arxiv.org/html/2609.19279#bib.bib26);[Zolfagharinejad et al\. \(2025\)](https://arxiv.org/html/2609.19279#bib.bib60);[Wu et al\. \(2026\)](https://arxiv.org/html/2609.19279#bib.bib61);[Ma et al\. \(2026\)](https://arxiv.org/html/2609.19279#bib.bib62), where the computation is intrinsic to the physics; here the model is instead tailored to the mechanism, which need not match the operations a given ML architecture requires\.

Computation has also been sought within wireless communication itself: in the channel between devices, and in the front ends inside them\. One route engineers the propagation environment so that the channel emulates a layer, using the multiple\-access channel as an analog adder[Nazer and Gastpar \(2007\)](https://arxiv.org/html/2609.19279#bib.bib33);[Zhu et al\. \(2021\)](https://arxiv.org/html/2609.19279#bib.bib34), configuring intelligent surfaces to emulate convolutional and fully\-connected layers[Liu et al\. \(2022a\)](https://arxiv.org/html/2609.19279#bib.bib51);[Garcia Sanchez et al\. \(2023\)](https://arxiv.org/html/2609.19279#bib.bib35);[Reus\-Muns et al\. \(2023\)](https://arxiv.org/html/2609.19279#bib.bib36);[Hua et al\. \(2026\)](https://arxiv.org/html/2609.19279#bib.bib66);[Stylianopoulos et al\. \(2026\)](https://arxiv.org/html/2609.19279#bib.bib67), or exploiting rich\-scattering environments and time\-modulated coding metasurfaces to compute in the wave domain[del Hougne and Lerosey \(2018\)](https://arxiv.org/html/2609.19279#bib.bib65);[Chen et al\. \(2026\)](https://arxiv.org/html/2609.19279#bib.bib64)\. The other computes with the front\-end circuitry itself, through reconfigurable RF or microwave circuits for analog linear algebra, including tunable matrix multipliers[Zhu et al\. \(2024\)](https://arxiv.org/html/2609.19279#bib.bib29);[Keshavarz et al\. \(2025\)](https://arxiv.org/html/2609.19279#bib.bib63), microwave linear analog computers[Nerini and Clerckx \(2025\)](https://arxiv.org/html/2609.19279#bib.bib32), and oscillator\-based microwave networks[Govind et al\. \(2025\)](https://arxiv.org/html/2609.19279#bib.bib30)\. Most directly within this route, the passive mixer evaluates the matrix\-vector multiplication of a fully\-connected layer, with weights broadcast from a central radio[Gao et al\. \(2026\)](https://arxiv.org/html/2609.19279#bib.bib1)or analyzed in simulation[Yu and Wong \(2026\)](https://arxiv.org/html/2609.19279#bib.bib31)\. Across both routes, the computation is either emulated by engineering the radio environment, demonstrated only for single layers or shallow networks, or cast as a generic matrix\-vector multiplication that discards the weight sharing and locality of convolution\. The mixer’s own correlation is not used to compute a convolutional layer directly\.

Here we introduceRadio\-FrequencyConvolutionalNeuralNetworks \(RF\-CNNs\), which repurpose wireless communication hardware for edge inference of convolutional neural networks \(CNNs\) that underpin modern vision and signal processing[LeCun et al\. \(2002\)](https://arxiv.org/html/2609.19279#bib.bib8);[Simonyan and Zisserman \(2015\)](https://arxiv.org/html/2609.19279#bib.bib45);[LeCun et al\. \(2015\)](https://arxiv.org/html/2609.19279#bib.bib43);[Liu et al\. \(2022b\)](https://arxiv.org/html/2609.19279#bib.bib68)\(Fig\.[1](https://arxiv.org/html/2609.19279#Sx8.F1)b\)\. A passive frequency mixer, driven by a local input against weights received over the air from a central radio, performs a convolutional layer directly in the analog domain, using the mixer’s native operation rather than an engineered emulation\. The converters, oscillators, and mixers that already serve communication are thereby reused as CNN accelerators, without dedicated hardware \(Figs\.[1](https://arxiv.org/html/2609.19279#Sx8.F1)c–d\)\.

Because the mapping is native rather than approximate, it stays accurate as networks deepen: on a wireless testbed,RF\-CNNruns wide CNNs with up to 1,024 channels, as well as deep CNNs with up to nine layers at test accuracy close to full\-precision digital baselines, spanning classification and generative inference over both one\- and two\-dimensional inputs: classification of complex\-valued wireless signals and of natural images such as CIFAR\-10, and controllable image generation such as human faces\. Because the computation is carried by reuse, this scheme avoids the two energy costs that dominate edge inference: weights are received over the air rather than stored and fetched on\-device[Horowitz \(2014\)](https://arxiv.org/html/2609.19279#bib.bib37);[Sze et al\. \(2017\)](https://arxiv.org/html/2609.19279#bib.bib18), and the front end is already powered for communication[Weinreich and Murmann \(2022\)](https://arxiv.org/html/2609.19279#bib.bib57), soRF\-CNNspends energy only on encoding/decoding and transmission/reception of the local input/output\. What remains is a sub\-femtojoule\-per\-multiply\-accumulate \(fJ/MAC\) budget, down to 0\.72 fJ/MAC on the edge side—more than two orders of magnitude below what the same inference would cost on an extra digital processor added to the device[Reuther et al\. \(2025\)](https://arxiv.org/html/2609.19279#bib.bib69);[Anderson et al\. \(2023\)](https://arxiv.org/html/2609.19279#bib.bib42)\. These results show that deployed wireless infrastructure can be reused as a substrate for complete, deep convolutional inference at the edge, a step toward a unified communication and computing architecture for future networks[Letaief et al\. \(2019\)](https://arxiv.org/html/2609.19279#bib.bib39);[Saad et al\. \(2020\)](https://arxiv.org/html/2609.19279#bib.bib38);[Feng et al\. \(2024\)](https://arxiv.org/html/2609.19279#bib.bib52)\.

## Convolution by frequency mixing

The architecture ofRF\-CNNis shown in Fig\.[1](https://arxiv.org/html/2609.19279#Sx8.F1)b\. A central radio \(e\.g\., a 5G base station or Wi\-Fi access point\) broadcasts the ML weights𝐖\\mathbf\{W\}to the edge devices in its range; each device generates its ML input𝐗\\mathbf\{X\}locally and obtains the output𝐘\\mathbf\{Y\}through its own communication front end\. The computation itself takes place in a passive frequency mixer, hereafter the computing mixer, into which the remotely received𝐖\\mathbf\{W\}and the locally transmitted𝐗\\mathbf\{X\}are fed \(Fig\.[2](https://arxiv.org/html/2609.19279#Sx8.F2)a\)\. The operation the mixer performs is elementary\. It multiplies its two incident waveforms in the time domain, and time\-domain multiplication is frequency\-domain correlation: encode complex\-valued𝐗\\mathbf\{X\}and𝐖\\mathbf\{W\}as amplitudes and phases on two series of frequency tones, and the spectrum of the mixer’s output contains, tone by tone, the cross\-correlation of the two series \(Supplementary Text Section[B](https://arxiv.org/html/2609.19279#Sx11.SS2)\)\. A convolutional layer is a cross\-correlation of the input with the kernel, so the mixer computes the layer by its physics; what remains is to arrange the high\-dimensional tensors on the one\-dimensional tone series\.

This arrangement is the tone\-mapping algorithm \(Figs\.[2](https://arxiv.org/html/2609.19279#Sx8.F2)b–c\), which casts a complex\-valued, multi\-channel, one\- or two\-dimensional convolution of arbitrary dimensions into a single one\-dimensional cross\-correlation\. For brevity, we consider theconv2dlayer with square kernels \(K×KK\\times K\) and images \(I×II\\times I\), and the equal number of input and output channels asCC\(see full derivation in Supplementary Text Section[A](https://arxiv.org/html/2609.19279#Sx11.SS1)\)\. InRF\-CNN, each input channel is unrolled row by row ontoI2I^\{2\}tones within one frequency sub\-band, with zero\-padded guard tones between rows so that correlations across row boundaries contribute nothing \(Fig\.[2](https://arxiv.org/html/2609.19279#Sx8.F2)c\)\. TheCCinput channels of𝐗\\mathbf\{X\}, aligned with the corresponding channels of𝐖\\mathbf\{W\}, then occupyCCadjacent sub\-bands withC⋅I2C\\cdot I^\{2\}tones across the bandwidth \(Fig\.[2](https://arxiv.org/html/2609.19279#Sx8.F2)b\)\. The channel dimension is where the mapping pays off: every aligned sub\-band pair deposits its correlation products onto the same output tones, so the sum over input channels is carried out by the mixing itself, and𝐘\\mathbf\{Y\}emerges complete on a single output sub\-band\. One mixing operation thus executes an entire multi\-channel convolution, accumulatingK2⋅CK^\{2\}\\cdot Cproducts per output element; repeating it once per output channel,CCtimes in total with the same𝐗\\mathbf\{X\}against different𝐖\\mathbf\{W\}, completes the layer\. Finally, the time domain waveforms can be transformed from/to these frequency\-domain tone series by the \(inverse\) fast Fourier transform \(IFFT/FFT\)\.

## Energy consumption: avoided costs and remaining costs

RunningRF\-CNNon an edge device involves three energy contributions: delivering the ML weights, operating the analog RF hardware, and the digital processing at the analog\-digital boundary\. We argue that only the third should be charged to the edge device, and it is under this accounting thatRF\-CNNoffers its energy benefit\.

First, the ML weights\.𝐖\\mathbf\{W\}is received over the air from the central radio rather than stored and fetched on\-device, so the energy of weight storage and data movement, a dominant cost of digital accelerators[Horowitz \(2014\)](https://arxiv.org/html/2609.19279#bib.bib37);[Sze et al\. \(2017\)](https://arxiv.org/html/2609.19279#bib.bib18);[Anderson et al\. \(2023\)](https://arxiv.org/html/2609.19279#bib.bib42), is not spent by the edge device \(see channel and phase calibration in Supplementary Text Sections[C](https://arxiv.org/html/2609.19279#Sx11.SS3)and[D](https://arxiv.org/html/2609.19279#Sx11.SS4)\)\. Second, the analog RF hardware\.RF\-CNNreuses the existing components of the communication transceiver, including the local oscillator and amplifiers, whose energy consumption is already expended for communication, thus incurring no additional cost for computing\. Third, the digital boundary\. Encoding \(the IFFT forming the time\-domain waveform of𝐗\\mathbf\{X\},eence\_\{\\textrm\{enc\}\}\), the transmitting digital\-to\-analog converter \(DAC\) \(edace\_\{\\textrm\{dac\}\}\), the receiving analog\-to\-digital converter \(ADC\) \(eadce\_\{\\textrm\{adc\}\}\), and decoding \(edece\_\{\\textrm\{dec\}\}\) operate specifically for computation \(Fig\.[2](https://arxiv.org/html/2609.19279#Sx8.F2)e\); they define the energy per MAC reported in this work:

e=eenc\+edac\+eadc\+edec\.\\displaystyle e=e\_\{\\textrm\{enc\}\}\+e\_\{\\textrm\{dac\}\}\+e\_\{\\textrm\{adc\}\}\+e\_\{\\textrm\{dec\}\}\.\(1\)This is an edge\-side accounting: the first two contributions are displaced to the infrastructure or absorbed by communication\.

Over\-the\-air weight delivery also provides additional flexibility to further reduce the counted terms: because the central radio can transmit𝐖\\mathbf\{W\}at high power, the power of𝐗\\mathbf\{X\}required for a given computing accuracy is reduced, saving energy on the edge device\. Fig\.[2](https://arxiv.org/html/2609.19279#Sx8.F2)d quantifies this trade, mapping accuracy, expressed as effective number of bits \(ENOB\), over the power spectral densities \(PSDs\) of𝐗\\mathbf\{X\}and𝐖\\mathbf\{W\}\(see full measurements in Supplementary Text Section[H](https://arxiv.org/html/2609.19279#Sx12.SS8)\): raising the PSD of𝐖\\mathbf\{W\}reduces the PSD required of𝐗\\mathbf\{X\}, so the input PSD serves as a single operating parameter through which energy is exchanged for accuracy, a property the end\-to\-end experiments below make use of at runtime\.

Each counted term is further shared across the MACs a single mixing operation contains \(Fig\.[2](https://arxiv.org/html/2609.19279#Sx8.F2)f\)\. All four scale with1/K21/K^\{2\}, because one pass of the mixer’s correlation covers every kernel position;eence\_\{\\textrm\{enc\}\},eadce\_\{\\textrm\{adc\}\}, andedece\_\{\\textrm\{dec\}\}decrease further withCC, as the channel accumulation performed by the mixing spreads their cost; andedace\_\{\\textrm\{dac\}\}decreases with the reduced input PSDpxp\_\{x\}\. The orthogonal frequency\-division multiplexing \(OFDM\)\-based waveforms support the tone counts this structure calls for, sustaining accumulation scales ofK2⋅CK^\{2\}\\cdot Cin the magnitude of10410^\{4\}per operation\.

## Single\-layer accuracy and energy at scale

We test this expectation on a software\-defined radio testbed in a wireless setting, alongside numerical simulation \(Supplementary Text Sections[F](https://arxiv.org/html/2609.19279#Sx12.SS6)and[G](https://arxiv.org/html/2609.19279#Sx12.SS7)\), asking two questions for a single RF convolutional layer: how little energy a given accuracy requires, and how far the accuracy holds as the layer widens\.

The expected scaling is what the measurements show\. Fig\.[2](https://arxiv.org/html/2609.19279#Sx8.F2)g plots the minimum energy per MAC for 4\-bit ENOB against channel count: it follows𝒪⁡\(1K2⋅log⁡CC\)\\mathcal\{O\}\\\!\\left\(\\frac\{1\}\{K^\{2\}\}\\cdot\\frac\{\\log C\}\{C\}\\right\)for both layer types, withconv2dholding its kernel\-area advantage overconv1dthroughout\. AtC=256C=256, aconv2dlayer computes at 0\.551 fJ/MAC for 4\-bit ENOB, more than two orders of magnitude below what the same multiply\-accumulate would cost on a dedicated digital accelerator added to the device[Reuther et al\. \(2025\)](https://arxiv.org/html/2609.19279#bib.bib69);[Anderson et al\. \(2023\)](https://arxiv.org/html/2609.19279#bib.bib42)\. This trend can be scaled up toC=1,024C=\{1\{,\}024\}without significant loss of computing accuracy: an accumulation of thousands of products per output element in one analog pass, the regime of the widest layers in contemporary CNN backbones, and a scale no single operation can reach when the mixer is cast as a generic matrix\-vector multiplier\.

Accuracy follows the PSD dependence established above\. For 256\-channel layers, ENOB rises monotonically with operating energy and saturates near 5 bits \(Figs\.[2](https://arxiv.org/html/2609.19279#Sx8.F2)h,[2](https://arxiv.org/html/2609.19279#Sx8.F2)j\), with channel calibration and peak\-to\-average power ratio \(PAPR\) mitigation \(Supplementary Text Sections[C](https://arxiv.org/html/2609.19279#Sx11.SS3)and[E](https://arxiv.org/html/2609.19279#Sx11.SS5)\) applied throughout to operate the hardware at its best achievable performance\. At an operating PSD of−\-112 dBm/Hz, the outputs match full\-precision ground truth in both real and imaginary components \(Figs\.[2](https://arxiv.org/html/2609.19279#Sx8.F2)i,[2](https://arxiv.org/html/2609.19279#Sx8.F2)k\): 5\.1 bits at 1\.64 fJ/MAC forconv1dand 5\.0 bits at 0\.72 fJ/MAC forconv2d\. This 5\-bit precision meets the demands of modern quantized inference[Bondarenko et al\. \(2021\)](https://arxiv.org/html/2609.19279#bib.bib40)\. More measurements can be found in Supplementary Text Section[I](https://arxiv.org/html/2609.19279#Sx12.SS9)\.

## Deep CNN inference on time\-series signals and vision tasks

The single\-layer characterization establishes that one convolutional layer computes accurately; what practical inference requires is that many such layers chain, the analog output of each re\-encoded as the input of the next layer, without errors accumulating into failure\. We therefore runcomplete, deepCNNs of the scale used in modern edge applications: a network of fiveconv1dlayers and one fully\-connected \(FC\) layer \(2\.18 million parameters, 368\.1 million MACs\) for modulation classification of complex\-valued wireless signals on DeepSig[O’Shea et al\. \(2018\)](https://arxiv.org/html/2609.19279#bib.bib14), and a network of eightconv2dlayers and one FC layer \(9\.22 million parameters, 611\.1 million MACs\) for image classification on SVHN and CIFAR\-10 \(Figs\.[3](https://arxiv.org/html/2609.19279#Sx8.F3)a–b, and more details in Supplementary Text Section[J](https://arxiv.org/html/2609.19279#Sx12.SS10)\)\. The two architectures span the input domains a wireless edge device encounters: the one\-dimensional, complex\-valued signals native to the radio itself, for whichRF\-CNNperforms inference on exactly the kind of waveform its hardware was built to receive, and the two\-dimensional natural images of standard computer vision\.

Across all three tasks,RF\-CNNtracks its full\-precision digital counterpart closely\. On DeepSig,RF\-CNNdistinguishes ten modulation formats at 92\.9% accuracy against a full\-precision baseline of 97\.2%, operating at 3\.34 fJ/MAC \(Fig\.[3](https://arxiv.org/html/2609.19279#Sx8.F3)c\)\. On SVHN it reaches 92\.3% against 94\.7% at 0\.72 fJ/MAC, and on CIFAR\-10, 88\.7% against 90\.9% at 0\.76 fJ/MAC \(Figs\.[3](https://arxiv.org/html/2609.19279#Sx8.F3)d–e\)\. More measurements can be found in Supplementary Text Section[L](https://arxiv.org/html/2609.19279#Sx12.SS12)\. The small gap to full precision is sustained through as many as eight successive analog convolutional layers: the finite precision of each layer does not compound into degradation with depth\. Notably, the sub/few\-fJ/MAC operating points at which these accuracies are obtained are not a separate low\-power mode but the same regime characterized in Fig\.[2](https://arxiv.org/html/2609.19279#Sx8.F2); accurate deep inference and the two\-orders\-of\-magnitude energy advantage are achieved simultaneously, not traded against each other\.

The confusion matrices show thatRF\-CNN’s residual errors are not an artifact of analog computation but the ordinary errors of the underlying model \(Figs\.[3](https://arxiv.org/html/2609.19279#Sx8.F3)f–h\)\. On DeepSig, misclassification concentrates almost entirely among QPSK, PSK, and QAM, modulation formats with closely related constellations, while structurally distinct formats such as FM, GMSK, and AM\-SSB are classified nearly perfectly\. The analog hardware, in other words, inherits the semantic error structure of the digital network rather than superimposing failure modes of its own\.

Beyond validating the simulations, the accuracy\-energy curves \(Figs\.[3](https://arxiv.org/html/2609.19279#Sx8.F3)c–e\) expose an additional degree of freedom: the trade between accuracy and energy is set by a single parameter—the transmit PSD of the input—and can therefore be tuned at runtime without retraining or reconfiguration\.

## Generative inference with latent\-code control

Generative models pose a more stringent test of analog computing precision than classification, where quantization error is largely absorbed in the final class decision[Bondarenko et al\. \(2021\)](https://arxiv.org/html/2609.19279#bib.bib40);[Ma et al\. \(2025\)](https://arxiv.org/html/2609.19279#bib.bib27): the network’s output is the image itself, produced pixel by pixel by the analog hardware, and any structured computational error is directly visible in the result\. Beyond fidelity, generation on the edge demands controllability: a device that synthesizes content is useful only if the content can be specified\. We therefore implement generative models based on InfoGAN[Chen et al\. \(2016\)](https://arxiv.org/html/2609.19279#bib.bib7), whose generator input separates into three parts with distinct roles \(Figs\.[4](https://arxiv.org/html/2609.19279#Sx8.F4)a–b\): a 62\-dimensional random seed providing sample diversity, a 10\-dimensional class label selecting*what*is generated, and a 2\-dimensional latent code steering*how*: a semantically interpretable attribute of the output\. Whether this three\-way separation survives execution on RF hardware is the question this section answers\.

We evaluate two model scales on three datasets: a four\-layer network \(one FC and threeconv2dlayers; 1\.35 million parameters, 350\.5 million MACs\) generating grayscale handwritten digits \(MNIST\) and fashion products \(FMNIST\), and a seven\-layer network \(one FC and sixconv2dlayers; 26\.4 million parameters, 5\.195 billion MACs\) generating RGB human faces \(CelebA\), executing over five billion analog MACs for every face produced\. More model details can be found in Supplementary Text Section[J](https://arxiv.org/html/2609.19279#Sx12.SS10)\.

The generated samples are well\-formed across all three tasks \(Figs\.[4](https://arxiv.org/html/2609.19279#Sx8.F4)f–h\)\. Measured over 100 generated images,RF\-CNNreaches a Fréchet inception distance \(FID\) of 90\.86 on MNIST at 1\.23 fJ/MAC, 181\.17 on FMNIST at the same energy, and 189\.10 on CelebA at 1\.56 fJ/MAC, approaching the full\-precision baselines, with measured FID following the simulated energy\-quality curve throughout the operating range \(Figs\.[4](https://arxiv.org/html/2609.19279#Sx8.F4)c–e\)\. More results can be found in Supplementary Text Section[M](https://arxiv.org/html/2609.19279#Sx12.SS13)\. Along that curve, image quality degrades gracefully: at reduced energy the samples acquire noise but remain recognizable as digits, garments, and faces \(insets of Figs\.[4](https://arxiv.org/html/2609.19279#Sx8.F4)c–e\), extending the runtime energy\-quality trade established for classification to generative workloads\.

The three\-way separation of the generator input survives intact\. The class label retains exclusive command of category: conditioning reliably produces the requested digit “0” through “9”, the requested garment, and the requested gender of face\. The latent code retains its independent, continuous handle on a single attribute: sweeping it from minimum to maximum while the label is held fixed tilts the slant of a digit, raises the height of a garment, and shifts the framing of a face, monotonically and without disturbing the class identity \(Figs\.[4](https://arxiv.org/html/2609.19279#Sx8.F4)i–k\)\. Five billion analog MACs at approximately 5\-bit precision thus preserve not only the appearance of individual outputs but the disentangled structure the generator learned in digital training, the property that practical edge generation needs, from personalized content to adaptive human\-machine interfaces and semantic scene synthesis\.

## Discussion and conclusion

We have shown that the frequency mixer, the element at the core of almost every wireless edge device, is, by its physics, a convolution engine, and that this operation alone suffices to run complete CNNs end to end in hardware on the communication front end already present at the edge\. On a wireless testbed,RF\-CNNperforms networks of up to nine layers and up to five billion analog MACs per inference—spanning modulation classification of complex\-valued wireless signals, image classification, and image generation with latent\-code control—at accuracy close to full\-precision digital baselines and at sub\-femtojoule\-per\-MAC energies, using no computing hardware beyond the reused front end\.

The present implementation is a proof of concept, and its throughput, approximately 0\.9 GOPS, is far below that of contemporary digital processors\. This reflects the testbed rather than the paradigm: the demonstration uses a single RF chain and a single frequency\-mixing pathway over a limited bandwidth\. Throughput scales directly with the wireless bandwidth and with the number of parallel mixing pathways, both of which are modest here and are routinely large in deployed radios[Govind et al\. \(2024\)](https://arxiv.org/html/2609.19279#bib.bib49)\. Massive MIMO front ends, integrated mixer arrays, and wide bandwidth available at higher frequencies \(millimeter\-wave or sub\-THz bands\) offer a path to raising throughput by orders of magnitude without altering the underlying mapping\.

Several boundaries of the approach should be stated plainly\. The reported energies are edge\-side: they exclude the power the central radio spends broadcasting weights, which is justified when that transmission is amortized across many receiving devices and drawn from grid\-connected infrastructure, but which makes the advantage one of disaggregation rather than of total system energy\. The computing accuracy saturates near five ENOB, sufficient for the tasks demonstrated but not for those demanding higher precision\. And the scheme presumes access to a radio that supplies weights over the air, so it suits devices operating within communication range rather than in isolation, and it addresses inference rather than training\.

These constraints delimit rather than diminish the opportunity\. Delivering weights over the air raises questions of confidentiality that the same wireless\-physics toolbox is well placed to answer, including physical\-layer and quantum\-secured schemes[Yin et al\. \(2020\)](https://arxiv.org/html/2609.19279#bib.bib44)\. Integrating the mixer array on\-chip would compound the energy and throughput gains while preserving compatibility with the communication stack\. And because the mapping rests on a physical operation rather than a bespoke device, it invites extension beyond convolution toward the other structured linear operations that dominate modern models\.

More broadly,RF\-CNNreframes the relationship between communication and computation in which neither is built for the other\. Correspondences between what hardware already does and what a computation requires are found rather than engineered, and the hardware need not be changed, only used for a second purpose alongside\. Much of the hardware already deployed may hold computational capability of its own\. Realized across the wireless infrastructure already deployed, this points toward an edge where communication and computation are not competing demands on separate hardware but two functions of a single physical substrate\.

![Refer to caption](https://arxiv.org/html/2609.19279v1/figure_architecture.png)Figure 1:RF\-CNN: repurposing wireless communication hardware for edge AI computing\.a, In a conventional wireless transceiver on the edge devices, radio frequency \(RF\) front end components, including frequency mixers, digital\-to\-analog converters \(DACs\), and analog\-to\-digital converters \(ADCs\), are dedicated to uplink and downlink communications\.b, InRF\-CNN, a remote radio infrastructure \(e\.g\., a 5G base station or Wi\-Fi access point\) can dynamically support either conventional wireless communication or distributed AI computation for nearby edge devices\.c, The same RF hardware is repurposed to execute convolutional neural network \(CNN\) inference directly in the analog/RF domain: the wirelessly received signals are replaced by machine learning \(ML\) weights \(𝐖\\mathbf\{W\}\), and the inputs \(𝐗\\mathbf\{X\}\) and outputs \(𝐘\\mathbf\{Y\}\) signals are generated/received locally, enabling efficient classification tasks\.d, TheRF\-CNNframework is also applicable to generative models, where RF hardware performs CNN\-based feature transformations throughout the generation pipeline\.![Refer to caption](https://arxiv.org/html/2609.19279v1/figure_scalability.png)Figure 2:The tone mapping algorithm and the characterization of a single convolutional layer\.a, On an edge device withRF\-CNN, the ML input𝐗\\mathbf\{X\}transmitted by a radio TX and the ML weight𝐖\\mathbf\{W\}received through an antenna are mixed in a computing mixer to perform RF\-domain correlation, and the output tensor𝐘\\mathbf\{Y\}is received by a radio RX\.b, Example operation of a singleconv2dlayer, where an RGB image “1” serving as𝐗\\mathbf\{X\}is mapped onto three frequency bands aligned with the three input channels of𝐖\\mathbf\{W\}, and the output tensor𝐘\\mathbf\{Y\}is extracted from the 8\.33 MHz output band after frequency\-domain correlation\.c, Illustration of the tone mapping algorithm within a single input channel, where the frequency\-domain tones of𝐗\\mathbf\{X\}and𝐖\\mathbf\{W\}are arranged such that their correlation outputs include𝐘\\mathbf\{Y\}\.d, Measured computing accuracy, expressed as effective number of bits \(ENOB\), over different power spectral density \(PSD\) combinations of𝐗\\mathbf\{X\}and𝐖\\mathbf\{W\}\.e, The energy consumptioneeis broken down into four modules:eence\_\{\\textrm\{enc\}\}for encoding,edace\_\{\\textrm\{dac\}\}for DAC,eadce\_\{\\textrm\{adc\}\}for ADC, andedece\_\{\\textrm\{dec\}\}for decoding, while other modules are either passive or shared with other functionalities\.f, The energy per multiply\-accumulate \(MAC\) scaling of aconv2dlayer, whereCCis the number of input/output channels,KKis the kernel size, andIIis the image size;pxp\_\{x\}is the PSD of𝐗\\mathbf\{X\}that determines the SNR and thus the computing accuracy\.g, Minimum energy per MAC operation required to achieve 4\-bit ENOB for a singleconv1dorconv2dlayer, showing that the energy per MAC decreases as the number of input/output channels increases\.h–j, Measured ENOB as a function of energy per MAC forconv1d\(h\) andconv2d\(j\) layers with 256 input/output channels\.i–k, Comparison between the full\-precision digital outputs and the outputs byRF\-CNNforconv1d\(i\) andconv2d\(k\), respectively, showing both real and imaginary components at an operating PSD of−\-112 dBm/Hz\.![Refer to caption](https://arxiv.org/html/2609.19279v1/figure_classifier.png)Figure 3:RF\-CNN inference for wireless communication and computer vision tasks\.a, Model architecture for wireless signal modulation classification on the DeepSig dataset, consisting of fiveconv1dlayers and one FC layer with 2\.18 million parameters and 368\.1 million MACs\.b, Model architecture for image classification on the SVHN and CIFAR\-10 datasets, consisting of eightconv2dlayers and one FC layer with 9\.22 million parameters and 611\.1 million MACs\.c–e, Classification accuracy as a function of PSD or energy efficiency for DeepSig \(c\), SVHN \(d\), and CIFAR\-10 \(e\), including the experimental measurements and simulation results\.f–h, Confusion matrices ofRF\-CNNinference results for DeepSig \(f\), SVHN \(g\), and CIFAR\-10 \(h\), respectively\.![Refer to caption](https://arxiv.org/html/2609.19279v1/figure_gan.png)Figure 4:RF\-CNN inference for generative tasks\.a, The model architecture of the InfoGAN\-based generative AI for either digit \(MNIST\) or fashion products \(FMNIST\), including one FC layer and threeconv2dlayers with 1\.35 million parameters and 350\.5 million MACs\.b, The model architecture of the InfoGAN\-based generative AI for high\-resolution RGB human faces \(CelebA\), including one FC layer and sixconv2dlayers with 26\.4 million parameters and 5,195 million MACs\.c–e, Fréchet inception distance \(FID\) as a function of energy per MAC for MNIST \(c\), FMNIST \(d\), and CelebA \(e\), including both simulations and experimental results\.f–h, Representative generated image examples from theRF\-CNNfor MNIST \(f\), FMNIST \(g\), and CelebA \(h\) at the selected high quality points\.i/j, Generated MNIST and FMNIST samples with fixed class labels and varying latent codes, demonstrating smooth latent\-space interpolation and semantic continuity\.k, Conditional human face generation with varying latent codes for female and male conditions, showing continuous and controllable changes in facial appearance\.
## Methods

### Tone Mapping Algorithm for Convolutional Layers

The frequency encoding algorithm ofRF\-CNNleverages the convolutional layers \(see definition in Supplementary Text Section[A](https://arxiv.org/html/2609.19279#Sx11.SS1)\) by cross\-correlation between two series of frequency tones\. This cross\-correlation in the frequency domain is equivalent to the time\-domain multiplication, which is inherently performed in the physics of the computing mixer\. We analyze the frequency encoding algorithm tailored for complex\-valued 2\-dimensional convolutional layers, abbreviated asconv2dlayers, which can also be generalized to other dimensions\.

Notations of input, weight, and output in conv2d layers\.For the specificconv2dlayer under consideration, let the kernel size beKh×KwK\_\{h\}\\times K\_\{w\}\(bothKhK\_\{h\}andKwK\_\{w\}are odd numbers\); by padding zeros around the input image ofPh=\(Kh−1\)/2P\_\{h\}=\(K\_\{h\}\-1\)/2andPw=\(Kw−1\)/2P\_\{w\}=\(K\_\{w\}\-1\)/2, we have the same input and output image dimensions, both of which areIh×IwI\_\{h\}\\times I\_\{w\}\. Additionally, letCinC\_\{\\textrm\{in\}\}represent the number of input channels andCoutC\_\{\\textrm\{out\}\}the number of output channels\. Accordingly, the input, weight, and output tensors of thisconv2dlayer are represented as𝐗∈ℂCin×Ih×Iw\\mathbf\{X\}\\in\\mathbb\{C\}^\{C\_\{\\textrm\{in\}\}\\times I\_\{h\}\\times I\_\{w\}\}withx⁡\[cin,ih,iw\]x\[c\_\{\\textrm\{in\}\},i\_\{h\},i\_\{w\}\]as the\(cin,ih,iw\)\(c\_\{\\textrm\{in\}\},i\_\{h\},i\_\{w\}\)\-th element,𝐖∈ℂCout×Cin×Kh×Kw\\mathbf\{W\}\\in\\mathbb\{C\}^\{C\_\{\\textrm\{out\}\}\\times C\_\{\\textrm\{in\}\}\\times K\_\{h\}\\times K\_\{w\}\}withw⁡\[cin,cout,kh,kw\]w\[c\_\{\\textrm\{in\}\},c\_\{\\textrm\{out\}\},k\_\{h\},k\_\{w\}\]as the\(cin,cout,kh,kw\)\(c\_\{\\textrm\{in\}\},c\_\{\\textrm\{out\}\},k\_\{h\},k\_\{w\}\)\-th element, and𝐘∈ℂCout×Ih×Iw\\mathbf\{Y\}\\in\\mathbb\{C\}^\{C\_\{\\textrm\{out\}\}\\times I\_\{h\}\\times I\_\{w\}\}withy⁡\[cout,ih,iw\]y\[c\_\{\\textrm\{out\}\},i\_\{h\},i\_\{w\}\]as the\(cout,ih,iw\)\(c\_\{\\textrm\{out\}\},i\_\{h\},i\_\{w\}\)\-th element, respectively\. We further assume that the kernel size is no greater than the image size asKh≤IhK\_\{h\}\\leq I\_\{h\}andKw≤IwK\_\{w\}\\leq I\_\{w\}\.

Notations of frequency tone series\.On the other hand, the series of frequency tones representing the input𝐗\\mathbf\{X\}and weight𝐖\\mathbf\{W\}in the cross\-correlation can be formulated by two complex\-valued vectors, denoted as𝐒x∈ℂLx\\mathbf\{S\}\_\{x\}\\in\\mathbb\{C\}^\{L\_\{x\}\}and𝐒w∈ℂLw\\mathbf\{S\}\_\{w\}\\in\\mathbb\{C\}^\{L\_\{w\}\}, whose lengths areLxL\_\{x\}andLwL\_\{w\}\. After the cross\-correlation, we only examine a contiguous subset of frequency tones for the output𝐘\\mathbf\{Y\}, denoted as a complex\-valued vector𝐒y∈ℂLy\\mathbf\{S\}\_\{y\}\\in\\mathbb\{C\}^\{L\_\{y\}\}with a length ofLyL\_\{y\}, which is a sub\-vector of the full cross\-correlation output with a length ofLx\+Lw−1L\_\{x\}\+L\_\{w\}\-1\(see Supplementary Text Section[B](https://arxiv.org/html/2609.19279#Sx11.SS2)for more details\)\.

Single input\-and\-output channel case\.We start with the simple case of a single input and output channel, i\.e\.,Cin=Cout=1C\_\{\\textrm\{in\}\}=C\_\{\\textrm\{out\}\}=1, as shown in Fig\.[S1](https://arxiv.org/html/2609.19279#Sx12.F1)\(a\)\. In this case, the input, weight, and output can be simplified as matrices:𝐗∈ℂIh×Iw\\mathbf\{X\}\\in\\mathbb\{C\}^\{I\_\{h\}\\times I\_\{w\}\},𝐖∈ℂKh×Kw\\mathbf\{W\}\\in\\mathbb\{C\}^\{K\_\{h\}\\times K\_\{w\}\}, and𝐘∈ℂIh×Iw\\mathbf\{Y\}\\in\\mathbb\{C\}^\{I\_\{h\}\\times I\_\{w\}\}\. Hence, their element indexing can be simplified asxih,iwx\_\{i\_\{h\},i\_\{w\}\},wkh,kww\_\{k\_\{h\},k\_\{w\}\}, andyih,iwy\_\{i\_\{h\},i\_\{w\}\}, respectively\. For the input𝐗\\mathbf\{X\}, we map it into the frequency tone series𝐒x\\mathbf\{S\}\_\{x\}withLx=\(Ih\+Ph\)⋅\(Iw\+Pw\)L\_\{x\}=\(I\_\{h\}\+P\_\{h\}\)\\cdot\(I\_\{w\}\+P\_\{w\}\)tones\. Specifically, the first row of𝐗\\mathbf\{X\}, indexed byx0,0,x1,0,…,xIh−1,0x\_\{0,0\},x\_\{1,0\},\\dots,x\_\{I\_\{h\}\-1,0\}, are mapped to the firstIhI\_\{h\}tones indexed bySx​\[0,1,…,Ih−1\]S\_\{x\}\[0,1,\\dots,I\_\{h\}\-1\]; then, a total ofPhP\_\{h\}zeros are padded next atSx​\[Ih,…,Ih\+Ph−1\]S\_\{x\}\[I\_\{h\},\\dots,I\_\{h\}\+P\_\{h\}\-1\]on𝐒x\\mathbf\{S\}\_\{x\}\. ThesePhP\_\{h\}zeros correspond to the padded zero\-pixels to the original image, which are reused for both the left padding for the first row and the right padding for the second row\. This process repeats forIwI\_\{w\}times for all theIwI\_\{w\}rows in𝐗\\mathbf\{X\}\. The lastPw⋅\(Ih\+Ph\)P\_\{w\}\\cdot\(I\_\{h\}\+P\_\{h\}\)tones are all zeros, representing the additional zero\-row padded to the original image\. To sum up, this mapping process can be formulated as:

Sx​\[i\]\\displaystyle S\_\{x\}\[i\]=\{xih,iw,i=\(Ih\+Ph\)⋅iw\+ih,0,otherwise,\\displaystyle=\\begin\{cases\}x\_\{i\_\{h\},i\_\{w\}\},&i=\(I\_\{h\}\+P\_\{h\}\)\\cdot i\_\{w\}\+i\_\{h\},\\\\ 0,&\\text\{otherwise\},\\end\{cases\}\(2\)∀ih\\displaystyle\\forall~i\_\{h\}=0,1,…,Ih−1,and​iw=0,1,…,Iw−1\.\\displaystyle=0,1,\\dots,I\_\{h\}\-1,~\\text\{and\}~i\_\{w\}=0,1,\\dots,I\_\{w\}\-1\.\(3\)Similarly, there areLw=\(Ih\+Ph\)⋅\(Iw\+Pw\)L\_\{w\}=\(I\_\{h\}\+P\_\{h\}\)\\cdot\(I\_\{w\}\+P\_\{w\}\)tones in𝐒w\\mathbf\{S\}\_\{w\}for the weight𝐖\\mathbf\{W\}\. The first row of𝐖\\mathbf\{W\}, indexed byw0,0,w1,0,…,wKh−1,0w\_\{0,0\},w\_\{1,0\},\\dots,w\_\{K\_\{h\}\-1,0\}, are mapped to the firstKhK\_\{h\}tones indexed bySw​\[0,1,…,Kh−1\]S\_\{w\}\[0,1,\\dots,K\_\{h\}\-1\]; to align the second row in𝐖\\mathbf\{W\}with the second row in𝐗\\mathbf\{X\}, we pad\(Ph\+Ih−Kh\)\(P\_\{h\}\+I\_\{h\}\-K\_\{h\}\)zeros tones in𝐒w\\mathbf\{S\}\_\{w\}indexed asSw​\[Kh,…,Ih\+Ph−1\]S\_\{w\}\[K\_\{h\},\\dots,I\_\{h\}\+P\_\{h\}\-1\]on𝐒w\\mathbf\{S\}\_\{w\}\. This process repeats forKwK\_\{w\}times for all theKwK\_\{w\}rows in𝐖\\mathbf\{W\}, while leaving all the remaining tones in𝐒w\\mathbf\{S\}\_\{w\}as zeros\. This mapping process from𝐖\\mathbf\{W\}to𝐒w\\mathbf\{S\}\_\{w\}can be formulated as:

Sw​\[i\]\\displaystyle S\_\{w\}\[i\]=\{wkh,kw,i=\(Ih\+Ph\)⋅kw\+kh,0,otherwise,\\displaystyle=\\begin\{cases\}w\_\{k\_\{h\},k\_\{w\}\},&i=\(I\_\{h\}\+P\_\{h\}\)\\cdot k\_\{w\}\+k\_\{h\},\\\\ 0,&\\text\{otherwise\},\\end\{cases\}\(4\)∀kh\\displaystyle\\forall~k\_\{h\}=0,1,…,Kh−1,and​kw=0,1,…,Kw−1\.\\displaystyle=0,1,\\dots,K\_\{h\}\-1,~\\text\{and\}~k\_\{w\}=0,1,\\dots,K\_\{w\}\-1\.\(5\)Originally, the cross\-correlation between theLxL\_\{x\}\-point𝐒x\\mathbf\{S\}\_\{x\}and theLwL\_\{w\}\-point𝐒w\\mathbf\{S\}\_\{w\}yields a\(Lx\+Lw−1\)\(L\_\{x\}\+L\_\{w\}\-1\)\-point output vector\. We only examineLyL\_\{y\}contiguous elements on the cross\-correlation output, indexed from\{l1,l1\+1,…,l2−1\}\\left\\\{l\_\{1\},l\_\{1\}\+1,\\dots,l\_\{2\}\-1\\right\\\}, which is denoted as𝐒y∈ℂLy\\mathbf\{S\}\_\{y\}\\in\\mathbb\{C\}^\{L\_\{y\}\}\. Specifically, we set the starting and ending indices as

\{l1=Lw−1=\(Ih\+Ph\)⋅\(Iw\+Pw\)−1,l2=Lx\+Lw−1=2⋅\(Ih\+Ph\)⋅\(Iw\+Pw\)−1,⇒Ly\\displaystyle\\begin\{cases\}l\_\{1\}&=L\_\{w\}\-1=\(I\_\{h\}\+P\_\{h\}\)\\cdot\(I\_\{w\}\+P\_\{w\}\)\-1,\\\\ l\_\{2\}&=L\_\{x\}\+L\_\{w\}\-1=2\\cdot\(I\_\{h\}\+P\_\{h\}\)\\cdot\(I\_\{w\}\+P\_\{w\}\)\-1,\\end\{cases\}\\Rightarrow\\quad L\_\{y\}=l2−l1=\(Ih\+Ph\)⋅\(Iw\+Pw\)\.\\displaystyle=l\_\{2\}\-l\_\{1\}=\(I\_\{h\}\+P\_\{h\}\)\\cdot\(I\_\{w\}\+P\_\{w\}\)\.\(6\)Under this setting, we notice that the mapping between the𝐘\\mathbf\{Y\}and𝐒y\\mathbf\{S\}\_\{y\}is the reverse process of that between𝐗\\mathbf\{X\}and𝐒x\\mathbf\{S\}\_\{x\}, as illustrated in Fig\.[S1](https://arxiv.org/html/2609.19279#Sx12.F1)\(a\)\. The first row of𝐘\\mathbf\{Y\}, indexed byy0,0,y1,0,…,yIh−1,0y\_\{0,0\},y\_\{1,0\},\\dots,y\_\{I\_\{h\}\-1,0\}, corresponds to the firstIhI\_\{h\}tones indexed bySy​\[0,1,…,Ih−1\]S\_\{y\}\[0,1,\\dots,I\_\{h\}\-1\]\. The subsequentPhP\_\{h\}tones are ignored and discarded\. This procedure is repeated for allIwI\_\{w\}rows in𝐘\\mathbf\{Y\}\. With the above mapping rules, we leverage the cross\-correlation from𝐒x\\mathbf\{S\}\_\{x\}and𝐒w\\mathbf\{S\}\_\{w\}to𝐒y\\mathbf\{S\}\_\{y\}, which realizes the single\-input/output\-channelconv2dfrom𝐗\\mathbf\{X\}and𝐖\\mathbf\{W\}to𝐘\\mathbf\{Y\}\.

Multiple input and single output channel case\.When there areCinC\_\{\\textrm\{in\}\}input channels in theconv2d, we need to generate the frequency tone series per input channel for both𝐗\\mathbf\{X\}and𝐖\\mathbf\{W\}\. As shown in Fig\.[S1](https://arxiv.org/html/2609.19279#Sx12.F1)\(b\), we concatenate these frequency tone series from all theCinC\_\{\\textrm\{in\}\}input channels into large frequency tone series as𝐗\\mathbf\{X\}and𝐖\\mathbf\{W\}, which can be written as

𝐒x\\displaystyle\\mathbf\{S\}\_\{x\}=\[𝐒x\|cin=0,𝐒x\|cin=1,…,𝐒x\|cin=Cin−1\],\\displaystyle=\\left\[\\mathbf\{S\}\_\{x\}\|\_\{c\_\{\\textrm\{in\}\}=0\},\\mathbf\{S\}\_\{x\}\|\_\{c\_\{\\textrm\{in\}\}=1\},\\dots,\\mathbf\{S\}\_\{x\}\|\_\{c\_\{\\textrm\{in\}\}=C\_\{\\textrm\{in\}\}\-1\}\\right\],\(7\)𝐒w\\displaystyle\\mathbf\{S\}\_\{w\}=\[𝐒w\|cin=0,𝐒w\|cin=1,…,𝐒w\|cin=Cin−1\],\\displaystyle=\\left\[\\mathbf\{S\}\_\{w\}\|\_\{c\_\{\\textrm\{in\}\}=0\},\\mathbf\{S\}\_\{w\}\|\_\{c\_\{\\textrm\{in\}\}=1\},\\dots,\\mathbf\{S\}\_\{w\}\|\_\{c\_\{\\textrm\{in\}\}=C\_\{\\textrm\{in\}\}\-1\}\\right\],\(8\)whose length becomesLx=Lw=Cin⋅\(Ih\+Ph\)⋅\(Iw\+Pw\)L\_\{x\}=L\_\{w\}=C\_\{\\textrm\{in\}\}\\cdot\(I\_\{h\}\+P\_\{h\}\)\\cdot\(I\_\{w\}\+P\_\{w\}\)\. When performing the cross\-correlation between𝐒x\\mathbf\{S\}\_\{x\}and𝐒w\\mathbf\{S\}\_\{w\}, we only examine the contiguous subset ofLyL\_\{y\}tones indexed from\{l1,l1\+1,…,l2−1\}\\left\\\{l\_\{1\},l\_\{1\}\+1,\\dots,l\_\{2\}\-1\\right\\\}as the output𝐒y∈ℂLy\\mathbf\{S\}\_\{y\}\\in\\mathbb\{C\}^\{L\_\{y\}\}, where

\{l1=Lw−1=Cin⋅\(Ih\+Ph\)⋅\(Iw\+Pw\)−1,l2=Lx\+\(Ih\+Ph\)⋅\(Iw\+Pw\)−1=\(Cin\+1\)⋅\(Ih\+Ph\)⋅\(Iw\+Pw\)−1,\\displaystyle\\begin\{cases\}l\_\{1\}&=L\_\{w\}\-1=C\_\{\\textrm\{in\}\}\\cdot\(I\_\{h\}\+P\_\{h\}\)\\cdot\(I\_\{w\}\+P\_\{w\}\)\-1,\\\\ l\_\{2\}&=L\_\{x\}\+\(I\_\{h\}\+P\_\{h\}\)\\cdot\(I\_\{w\}\+P\_\{w\}\)\-1=\(C\_\{\\textrm\{in\}\}\+1\)\\cdot\(I\_\{h\}\+P\_\{h\}\)\\cdot\(I\_\{w\}\+P\_\{w\}\)\-1,\\end\{cases\}\(9\)⇒Ly=l2−l1=\(Ih\+Ph\)⋅\(Iw\+Pw\)\.\\displaystyle\\Rightarrow\\quad L\_\{y\}=l\_\{2\}\-l\_\{1\}=\(I\_\{h\}\+P\_\{h\}\)\\cdot\(I\_\{w\}\+P\_\{w\}\)\.\(10\)The mapping from𝐒y\\mathbf\{S\}\_\{y\}to𝐘\\mathbf\{Y\}remains the same as that in the single input/output channel case\. In practice, we add Zadoff\-Chu phases on each input channel on𝐒x\\mathbf\{S\}\_\{x\}and inverse phases on𝐒w\\mathbf\{S\}\_\{w\}to alleviate the peak\-to\-average power ratio \(PAPR\) of the waveforms while maintaining the calculation results \(see Supplementary Text Sections[E](https://arxiv.org/html/2609.19279#Sx11.SS5)and[K](https://arxiv.org/html/2609.19279#Sx12.SS11)\)\.

Multiple input\-and\-output channel case\.When there are multiple output channels \(Cout\>1C\_\{\\textrm\{out\}\}\>1\), we need to perform the above cross\-correlation forCoutC\_\{\\textrm\{out\}\}times in serial, each of which corresponds to one output channel\. Specifically, the input frequency tone series𝐒x\\mathbf\{S\}\_\{x\}remains the same, while the weight frequency tone series𝐒w\\mathbf\{S\}\_\{w\}corresponds to thecoutc\_\{\\textrm\{out\}\}\-th output channel for thecoutc\_\{\\textrm\{out\}\}\-th cross\-correlation; the output of thecoutc\_\{\\textrm\{out\}\}\-th cross\-correlation corresponds to thecoutc\_\{\\textrm\{out\}\}\-th output channel in𝐘\\mathbf\{Y\}\.

### Bandwidth and Computation Throughput Analysis

Assuming an accessible bandwidth ofBBof the wireless channel, we first analyze the wired/wireless bandwidths of the𝐒x\\mathbf\{S\}\_\{x\},𝐒w\\mathbf\{S\}\_\{w\}, and𝐒y\\mathbf\{S\}\_\{y\}, and their waveform timeTTunder these bandwidths\. Then, we analyze the computation throughput, i\.e\., the number of MACs per time, denoted asΛ\\Lambda\.

To simplify, we further assumeIh≫PhI\_\{h\}\\gg P\_\{h\}andIw≫PwI\_\{w\}\\gg P\_\{w\}, and have the approximation thatLx=Lw≈Cin⋅Ih​IwL\_\{x\}=L\_\{w\}\\approx C\_\{\\textrm\{in\}\}\\cdot I\_\{h\}I\_\{w\}, and thatLy≈Ih​IwL\_\{y\}\\approx I\_\{h\}I\_\{w\}\.

Bandwidth analysis\.InRF\-CNN, the weight𝐒w\\mathbf\{S\}\_\{w\}is transmitted over the wireless channel, meaningBW=BB\_\{W\}=B\. Then, the tone spacing for𝐒w\\mathbf\{S\}\_\{w\}can be given by

Δ​f=BLw≈BCin⋅Ih​Iw\.\\displaystyle\\Delta f=\\frac\{B\}\{L\_\{w\}\}\\approx\\frac\{B\}\{C\_\{\\textrm\{in\}\}\\cdot I\_\{h\}I\_\{w\}\}\.\(11\)Note that this tone spacingΔ​f\\Delta fis the same for𝐒x\\mathbf\{S\}\_\{x\}and𝐒y\\mathbf\{S\}\_\{y\}\(see Supplementary Text Section[B](https://arxiv.org/html/2609.19279#Sx11.SS2)\), so their bandwidths, denoted asBxB\_\{x\}andByB\_\{y\}, can be given by

Bx=Δ​f⋅Lx=B,\\displaystyle B\_\{x\}=\\Delta f\\cdot L\_\{x\}=B,\(12\)and

By=Δ​f⋅Ly=BCin\.\\displaystyle B\_\{y\}=\\Delta f\\cdot L\_\{y\}=\\frac\{B\}\{C\_\{\\textrm\{in\}\}\}\.\(13\)
Computation throughput analysis\.The waveform time of a single output channel can be given by1/Δ​f1/\\Delta f, which is to be repeated forCoutC\_\{\\textrm\{out\}\}times\. Therefore, the total waveform timeTTcan be given by

T=Cout⋅1Δ​f≈Cin⋅Cout⋅Ih​IwB\.\\displaystyle T=C\_\{\\textrm\{out\}\}\\cdot\\frac\{1\}\{\\Delta f\}\\approx\\frac\{C\_\{\\textrm\{in\}\}\\cdot C\_\{\\textrm\{out\}\}\\cdot I\_\{h\}I\_\{w\}\}\{B\}\.\(14\)DuringTT, a total of4​Cin⋅Cout⋅Kh​Kw⋅Ih​Iw4C\_\{\\textrm\{in\}\}\\cdot C\_\{\\textrm\{out\}\}\\cdot K\_\{h\}K\_\{w\}\\cdot I\_\{h\}I\_\{w\}real\-valued MACs are computed\. This corresponds to the computation throughputΛ\\Lambdaof

Λ=4​Cin⋅Cout⋅Kh​Kw⋅Ih​IwT≈4​Kh​Kw⋅B\.\\displaystyle\\Lambda=\\frac\{4C\_\{\\textrm\{in\}\}\\cdot C\_\{\\textrm\{out\}\}\\cdot K\_\{h\}K\_\{w\}\\cdot I\_\{h\}I\_\{w\}\}\{T\}\\approx 4K\_\{h\}K\_\{w\}\\cdot B\.\(15\)This suggests that a faster computation throughput can be achieved by a larger wireless bandwidthBB\. Also, the computation throughput ofRF\-CNNis scalable as the kernel sizeKh×KwK\_\{h\}\\times K\_\{w\}increases\.

### Energy Efficiency Analysis

Then, we analyze the energy efficiency ofRF\-CNNon the edge client\. In the following, we discuss the energy consumption of each module in detail\.

Throughout the analysis, we denote the energy consumption per real\-valued MAC of digital computing asϵmac\\epsilon\_\{\\textrm\{mac\}\}\(i\.e\.,4​ϵmac4\\epsilon\_\{\\textrm\{mac\}\}for a complex\-valued MAC\), and the energy consumption per real\-valued sample in ADC asϵadc\\epsilon\_\{\\textrm\{adc\}\}\(i\.e\.,2​ϵadc2\\epsilon\_\{\\textrm\{adc\}\}for an I/Q sample\)\. Throughout the analysis, we setϵadc=100​fJ/sample\\epsilon\_\{\\textrm\{adc\}\}=\{100\}\\thinspace\{\\textrm\{fJ/sample\}\}[Murmann \(\)](https://arxiv.org/html/2609.19279#bib.bib70)andϵmac=100​fJ/MAC\\epsilon\_\{\\textrm\{mac\}\}=\{100\}\\thinspace\{\\textrm\{fJ/MAC\}\}[Choquette \(2023\)](https://arxiv.org/html/2609.19279#bib.bib41)\.

The encoding module\.The encoding module performs the mapping from the input tensor𝐗\\mathbf\{X\}to the frequency tone series𝐒x\\mathbf\{S\}\_\{x\}, and converts the frequency\-domain𝐒x\\mathbf\{S\}\_\{x\}into the time\-domain, where aLxL\_\{x\}\-point IFFT is conducted in digital computing\. Therefore, the energy consumption of the encoding module per layer, denoted asEencE\_\{\\textrm\{enc\}\}, is

Eenc=2​Lx​log2​Lx⋅ϵmac≈2​Cin⋅Ih​Iw​log2⁡\(Cin⋅Ih​Iw\)⋅ϵmac\.\\displaystyle E\_\{\\textrm\{enc\}\}=2L\_\{x\}\\log\_\{2\}L\_\{x\}\\cdot\\epsilon\_\{\\textrm\{mac\}\}\\approx 2C\_\{\\textrm\{in\}\}\\cdot I\_\{h\}I\_\{w\}\\log\_\{2\}\(C\_\{\\textrm\{in\}\}\\cdot I\_\{h\}I\_\{w\}\)\\cdot\\epsilon\_\{\\textrm\{mac\}\}\.\(16\)Given the number of real\-valued MACs per layer as4​Cin⋅Cout⋅Kh​Kw⋅Ih​Iw4C\_\{\\textrm\{in\}\}\\cdot C\_\{\\textrm\{out\}\}\\cdot K\_\{h\}K\_\{w\}\\cdot I\_\{h\}I\_\{w\}, the energy consumption per MAC in the encoding module, denoted aseence\_\{\\textrm\{enc\}\}, is

eenc\\displaystyle e\_\{\\textrm\{enc\}\}=Eenc4⋅Cin⋅Cout⋅Kh​Kw⋅Ih​Iw=12⋅log2⁡\(Cin⋅Ih​Iw\)Cout⋅Kh​Kw⋅ϵmac\.\\displaystyle=\\frac\{E\_\{\\textrm\{enc\}\}\}\{4\\cdot C\_\{\\textrm\{in\}\}\\cdot C\_\{\\textrm\{out\}\}\\cdot K\_\{h\}K\_\{w\}\\cdot I\_\{h\}I\_\{w\}\}=\\frac\{1\}\{2\}\\cdot\\frac\{\\log\_\{2\}\(C\_\{\\textrm\{in\}\}\\cdot I\_\{h\}I\_\{w\}\)\}\{C\_\{\\textrm\{out\}\}\\cdot K\_\{h\}K\_\{w\}\}\\cdot\\epsilon\_\{\\textrm\{mac\}\}\.\(17\)
The digital\-to\-analog converter \(DAC\) module\.On the edge client, the DAC module generates the analog waveform carrying the input𝐗\\mathbf\{X\}, whose energy consumption, denoted asEdacE\_\{\\textrm\{dac\}\}, can be expressed as the product of the transmitted power \(denoted asPxP\_\{x\}\) and the waveform timeTTderived in equation \([14](https://arxiv.org/html/2609.19279#Sx9.E14)\)\. Thereby, we have the energy consumption of the DAC module per MVM as

Edac=Px⋅T≈Px⋅Cin⋅Cout⋅Ih​IwB=Cin⋅Cout⋅Ih​Iw⋅px,\\displaystyle E\_\{\\textrm\{dac\}\}=P\_\{x\}\\cdot T\\approx P\_\{x\}\\cdot\\frac\{C\_\{\\textrm\{in\}\}\\cdot C\_\{\\textrm\{out\}\}\\cdot I\_\{h\}I\_\{w\}\}\{B\}=C\_\{\\textrm\{in\}\}\\cdot C\_\{\\textrm\{out\}\}\\cdot I\_\{h\}I\_\{w\}\\cdot p\_\{x\},\(18\)where we definepx=Px/Bp\_\{x\}=P\_\{x\}/Bas the power spectral density \(PSD\) of the DACs\. By normalizing it with the number of real\-valued MACs per layer, the energy consumption per MAC in the DAC module, denoted asedace\_\{\\textrm\{dac\}\}, is

edac\\displaystyle e\_\{\\textrm\{dac\}\}=Edac4⋅Cin⋅Cout⋅Kh​Kw⋅Ih​Iw≈14⋅1Kh​Kw⋅px\.\\displaystyle=\\frac\{E\_\{\\textrm\{dac\}\}\}\{4\\cdot C\_\{\\textrm\{in\}\}\\cdot C\_\{\\textrm\{out\}\}\\cdot K\_\{h\}K\_\{w\}\\cdot I\_\{h\}I\_\{w\}\}\\approx\\frac\{1\}\{4\}\\cdot\\frac\{1\}\{K\_\{h\}K\_\{w\}\}\\cdot p\_\{x\}\.\(19\)
The analog\-to\-digital converter \(ADC\) module\.The ADC module receives the output frequency tone series𝐒y\\mathbf\{S\}\_\{y\}from the computing mixer\. Given the smaller bandwidthByB\_\{y\}of𝐒y\\mathbf\{S\}\_\{y\}in equation \([13](https://arxiv.org/html/2609.19279#Sx9.E13)\), the ADC can also operate at a smaller sampling rate ofByB\_\{y\}to fully capture𝐒y\\mathbf\{S\}\_\{y\}\. Throughout the total waveform timeTT, the total energy consumed by ADC, denoted asEadcE\_\{\\textrm\{adc\}\}is given by

Eadc=T⋅By⋅2​ϵadc≈Cin⋅Cout⋅Ih​IwB⋅BCin⋅2​ϵadc=2​Cout⋅Ih​Iw⋅ϵadc\.\\displaystyle E\_\{\\textrm\{adc\}\}=T\\cdot B\_\{y\}\\cdot 2\\epsilon\_\{\\textrm\{adc\}\}\\approx\\frac\{C\_\{\\textrm\{in\}\}\\cdot C\_\{\\textrm\{out\}\}\\cdot I\_\{h\}I\_\{w\}\}\{B\}\\cdot\\frac\{B\}\{C\_\{\\textrm\{in\}\}\}\\cdot 2\\epsilon\_\{\\textrm\{adc\}\}=2C\_\{\\textrm\{out\}\}\\cdot I\_\{h\}I\_\{w\}\\cdot\\epsilon\_\{\\textrm\{adc\}\}\.\(20\)Normalized by the number of MACs computed, we have the energy per MAC term for the ADC module as

eadc=Eadc4⋅Cin⋅Cout⋅Kh​Kw⋅Ih​Iw≈12⋅1Cin⋅Kh​Kw⋅ϵadc\.\\displaystyle e\_\{\\textrm\{adc\}\}=\\frac\{E\_\{\\textrm\{adc\}\}\}\{4\\cdot C\_\{\\textrm\{in\}\}\\cdot C\_\{\\textrm\{out\}\}\\cdot K\_\{h\}K\_\{w\}\\cdot I\_\{h\}I\_\{w\}\}\\approx\\frac\{1\}\{2\}\\cdot\\frac\{1\}\{C\_\{\\textrm\{in\}\}\\cdot K\_\{h\}K\_\{w\}\}\\cdot\\epsilon\_\{\\textrm\{adc\}\}\.\(21\)
The decoding module\.The decoding module performs FFT in digital to extract the computing output𝐘\\mathbf\{Y\}from the frequency domain𝐒y\\mathbf\{S\}\_\{y\}\. Specifically, aLyL\_\{y\}\-point FFT is performed per output channelcoutc\_\{\\textrm\{out\}\}, involving the MAC number of2​Ly​log2​Ly2L\_\{y\}\\log\_\{2\}L\_\{y\}; the totalCoutC\_\{\\textrm\{out\}\}repeats yield the total energy consumption of this decoding module as

Edec=Cout⋅2​Ly​log2​Ly⋅ϵmac≈2​Cout⋅Ih​Iw​log2⁡\(Ih​Iw\)⋅ϵmac\.\\displaystyle E\_\{\\textrm\{dec\}\}=C\_\{\\textrm\{out\}\}\\cdot 2L\_\{y\}\\log\_\{2\}L\_\{y\}\\cdot\\epsilon\_\{\\textrm\{mac\}\}\\approx 2C\_\{\\textrm\{out\}\}\\cdot I\_\{h\}I\_\{w\}\\log\_\{2\}\(I\_\{h\}I\_\{w\}\)\\cdot\\epsilon\_\{\\textrm\{mac\}\}\.\(22\)Correspondingly, the energy per MAC is given by

edec=Edec4⋅Cin⋅Cout⋅Kh​Kw⋅Ih​Iw≈12⋅log2⁡\(Ih​Iw\)Cin⋅Kh​Kw⋅ϵmac\.\\displaystyle e\_\{\\textrm\{dec\}\}=\\frac\{E\_\{\\textrm\{dec\}\}\}\{4\\cdot C\_\{\\textrm\{in\}\}\\cdot C\_\{\\textrm\{out\}\}\\cdot K\_\{h\}K\_\{w\}\\cdot I\_\{h\}I\_\{w\}\}\\approx\\frac\{1\}\{2\}\\cdot\\frac\{\\log\_\{2\}\(I\_\{h\}I\_\{w\}\)\}\{C\_\{\\textrm\{in\}\}\\cdot K\_\{h\}K\_\{w\}\}\\cdot\\epsilon\_\{\\textrm\{mac\}\}\.\(23\)
Total energy efficiency\.Putting equations \([16](https://arxiv.org/html/2609.19279#Sx9.E16)\), \([18](https://arxiv.org/html/2609.19279#Sx9.E18)\), \([20](https://arxiv.org/html/2609.19279#Sx9.E20)\), and \([22](https://arxiv.org/html/2609.19279#Sx9.E22)\) together, we have the total energy consumption perconv2dlayer as

E\\displaystyle E≈2​Cin⋅Ih​Iw​log2⁡\(Cin⋅Ih​Iw\)⋅ϵmac⏟Eenc\+Cin⋅Cout⋅Ih​Iw⋅px⏟Edac\\displaystyle\\approx\\underbrace\{2C\_\{\\textrm\{in\}\}\\cdot I\_\{h\}I\_\{w\}\\log\_\{2\}\(C\_\{\\textrm\{in\}\}\\cdot I\_\{h\}I\_\{w\}\)\\cdot\\epsilon\_\{\\textrm\{mac\}\}\}\_\{E\_\{\\textrm\{enc\}\}\}\+\\underbrace\{C\_\{\\textrm\{in\}\}\\cdot C\_\{\\textrm\{out\}\}\\cdot I\_\{h\}I\_\{w\}\\cdot p\_\{x\}\}\_\{E\_\{\\textrm\{dac\}\}\}\(24\)\+2​Cout⋅Ih​Iw⋅ϵadc⏟Eadc\+2​Cout⋅Ih​Iw​log2⁡\(Ih​Iw\)⋅ϵmac⏟Edec\.\\displaystyle\\quad\+\\underbrace\{2C\_\{\\textrm\{out\}\}\\cdot I\_\{h\}I\_\{w\}\\cdot\\epsilon\_\{\\textrm\{adc\}\}\}\_\{E\_\{\\textrm\{adc\}\}\}\+\\underbrace\{2C\_\{\\textrm\{out\}\}\\cdot I\_\{h\}I\_\{w\}\\log\_\{2\}\(I\_\{h\}I\_\{w\}\)\\cdot\\epsilon\_\{\\textrm\{mac\}\}\}\_\{E\_\{\\textrm\{dec\}\}\}\.\(25\)Similarly, summing up equations \([17](https://arxiv.org/html/2609.19279#Sx9.E17)\), \([19](https://arxiv.org/html/2609.19279#Sx9.E19)\), \([21](https://arxiv.org/html/2609.19279#Sx9.E21)\) and \([23](https://arxiv.org/html/2609.19279#Sx9.E23)\) yields the energy consumption per MAC as

e\\displaystyle e≈12⋅log2⁡\(Cin⋅Ih​Iw\)Cout⋅Kh​Kw⋅ϵmac⏟eenc\+14⋅1Kh​Kw⋅px⏟edac\+12⋅1Cin⋅Kh​Kw⋅ϵadc⏟eadc\+12⋅log2⁡\(Ih​Iw\)Cin⋅Kh​Kw⋅ϵmac⏟edec\\displaystyle\\approx\\underbrace\{\\frac\{1\}\{2\}\\cdot\\frac\{\\log\_\{2\}\(C\_\{\\textrm\{in\}\}\\cdot I\_\{h\}I\_\{w\}\)\}\{C\_\{\\textrm\{out\}\}\\cdot K\_\{h\}K\_\{w\}\}\\cdot\\epsilon\_\{\\textrm\{mac\}\}\}\_\{e\_\{\\textrm\{enc\}\}\}\+\\underbrace\{\\frac\{1\}\{4\}\\cdot\\frac\{1\}\{K\_\{h\}K\_\{w\}\}\\cdot p\_\{x\}\}\_\{e\_\{\\textrm\{dac\}\}\}\+\\underbrace\{\\frac\{1\}\{2\}\\cdot\\frac\{1\}\{C\_\{\\textrm\{in\}\}\\cdot K\_\{h\}K\_\{w\}\}\\cdot\\epsilon\_\{\\textrm\{adc\}\}\}\_\{e\_\{\\textrm\{adc\}\}\}\+\\underbrace\{\\frac\{1\}\{2\}\\cdot\\frac\{\\log\_\{2\}\(I\_\{h\}I\_\{w\}\)\}\{C\_\{\\textrm\{in\}\}\\cdot K\_\{h\}K\_\{w\}\}\\cdot\\epsilon\_\{\\textrm\{mac\}\}\}\_\{e\_\{\\textrm\{dec\}\}\}\(26\)=12⋅1Kh​Kw⋅\(12⋅px\+1Cin⋅ϵadc\+\[log2⁡\(Cin⋅Ih​Iw\)Cout\+log2⁡\(Ih​Iw\)Cin\]⋅ϵmac\)\.\\displaystyle=\\frac\{1\}\{2\}\\cdot\\frac\{1\}\{K\_\{h\}K\_\{w\}\}\\cdot\\left\(\\frac\{1\}\{2\}\\cdot p\_\{x\}\+\\frac\{1\}\{C\_\{\\textrm\{in\}\}\}\\cdot\\epsilon\_\{\\textrm\{adc\}\}\+\\left\[\\frac\{\\log\_\{2\}\(C\_\{\\textrm\{in\}\}\\cdot I\_\{h\}I\_\{w\}\)\}\{C\_\{\\textrm\{out\}\}\}\+\\frac\{\\log\_\{2\}\(I\_\{h\}I\_\{w\}\)\}\{C\_\{\\textrm\{in\}\}\}\\right\]\\cdot\\epsilon\_\{\\textrm\{mac\}\}\\right\)\.\(27\)We notice that the energy efficiency ofRF\-CNNis also scalable for the kernel sizeKh×KwK\_\{h\}\\times K\_\{w\}\. In addition, the energy per MAC decreases as the computing scale,CinC\_\{\\textrm\{in\}\}andCoutC\_\{\\textrm\{out\}\}increase, and converges to the thermal dynamic limit \(TDL\),eTDLe\_\{\\textrm\{TDL\}\}, as

eTDL=limCin,Cout→∞e=14⋅1Kh​Kw⋅px\.\\displaystyle e\_\{\\textrm\{TDL\}\}=\\lim\_\{C\_\{\\textrm\{in\}\},C\_\{\\textrm\{out\}\}\\rightarrow\\infty\}e=\\frac\{1\}\{4\}\\cdot\\frac\{1\}\{K\_\{h\}K\_\{w\}\}\\cdot p\_\{x\}\.\(28\)

## Acknowledgments

Funding\.Z\.G\., Y\.C\., and T\.C\. acknowledge partial support from the NSF Athena AI Institute for Edge Computing under award \#2112562\. Z\.G\., D\.E\., and T\.C\. acknowledge partial support from ARO under grant W911NF\-25\-1\-0241\. D\.E\. acknowledges support from AFRL under Cooperative Agreement No\. FA8750\-25\-2\-0500\. S\.\-Y\.M\. acknowledges support from the DARPA INSPIRED program \(HR001123S0052\) and from the NSF Engineering Research Center for Quantum Networks \(CQN\) under award \#1941583\.

Author Contributions\.Z\.G\., D\.E\., and T\.C\. developed the original concept and system architecture\. Z\.G\., S\.\-Y\.M\., and T\.C\. designed the experimental setup and evaluation plan\. Z\.G\. conducted the experiments and analyzed the data\. Z\.G\., S\.\-Y\.M\., and T\.C\. prepared the manuscript with the inputs from Y\.C\. and D\.E\.\.

Correspondence\.Requests for information should be directed to Tingjun Chen \([tingjun\.chen@duke\.edu](mailto:[email protected])\)\.

Supplementary Information: Radio\-Frequency Convolutional Neural Networks

Zhihui Gao1, Shi\-Yuan Ma2, Yiran Chen1, Dirk Englund2, and Tingjun Chen1

1Department of Electrical and Computer Engineering, Duke University, Durham, NC 27708, USA

2Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA

###### Contents

1. [AConvolutional Layer Basics and Formulation](https://arxiv.org/html/2609.19279#Sx11.SS1)
2. [BUsing Frequency Mixer for Convolution and Cross\-Correlation](https://arxiv.org/html/2609.19279#Sx11.SS2)
3. [CPhysical Channel Calibration](https://arxiv.org/html/2609.19279#Sx11.SS3)
4. [DPhase Calibration with Preambles and Postambles](https://arxiv.org/html/2609.19279#Sx11.SS4)
5. [EWaveform PAPR Alleviation](https://arxiv.org/html/2609.19279#Sx11.SS5)
6. [FAdditive White Gaussian Noise Channel based Simulations](https://arxiv.org/html/2609.19279#Sx12.SS6)
7. [GSoftware\-Defined Radio Testbed for Experiments](https://arxiv.org/html/2609.19279#Sx12.SS7)
8. [HFrequency Mixer Computing Accuracy Benchmark](https://arxiv.org/html/2609.19279#Sx12.SS8)
9. [IGeneral\-Purposeconv2dBenchmarks over Scalability](https://arxiv.org/html/2609.19279#Sx12.SS9)
10. [JComplex\-Valued CNN\-based Classification and Generative Models](https://arxiv.org/html/2609.19279#Sx12.SS10)
11. [KWaveform Amplitude Selection against PAPR](https://arxiv.org/html/2609.19279#Sx12.SS11)
12. [LPower Spectral Density Trade\-off on the CNN Models for Classification](https://arxiv.org/html/2609.19279#Sx12.SS12)
13. [MPower Spectral Density Trade\-off on the CNN Models for Generation](https://arxiv.org/html/2609.19279#Sx12.SS13)
14. [References](https://arxiv.org/html/2609.19279#bib)

## Supplementary Information: Theory

### AConvolutional Layer Basics and Formulation

RF\-CNNis designed to accelerate general convolutional layers of any dimensions\. Without loss of generality, we start with the basics of the \(complex\-valued\) 2D convolutional layer \(abbreviated asconv2d\) that is widely used in modern DL models \(e\.g\., object detection, image classification, semantic segmentation, and generative models\)\. Then, we present the extension to convolutional layers with higher dimensions\.

2D convolutional layer modeling\.We define the input to aconv2dlayer as a \(complex\-valued\) three\-dimensional tensor,𝐗∈ℂCin×Ih×Iw\\mathbf\{X\}\\in\\mathbb\{C\}^\{C\_\{\\textrm\{in\}\}\\times I\_\{h\}\\times I\_\{w\}\}, whereCinC\_\{\\textrm\{in\}\}is the number of channels \(e\.g\.,Cin=3C\_\{\\textrm\{in\}\}=3for an RGB\-colored image\), andIh×IwI\_\{h\}\\times I\_\{w\}is the dimension of the input’s height and width\. For thecinc\_\{\\textrm\{in\}\}\-th channel, the element in theihi\_\{h\}\-th row andiwi\_\{w\}\-th column is denoted asx⁡\[cin,ih,iw\]∈ℂx\[c\_\{\\textrm\{in\}\},i\_\{h\},i\_\{w\}\]\\in\\mathbb\{C\}\. The trainable weights in thisconv2dlayer are defined as a four\-dimensional tensor,𝐖∈ℂCin×Cout×Kh×Kw\\mathbf\{W\}\\in\\mathbb\{C\}^\{C\_\{\\textrm\{in\}\}\\times C\_\{\\textrm\{out\}\}\\times K\_\{h\}\\times K\_\{w\}\}, whereCoutC\_\{\\textrm\{out\}\}is the number of output channels, andKh×KwK\_\{h\}\\times K\_\{w\}is the dimension of the 2D kernel along the height and width dimensions, respectively\. For thecinc\_\{\\textrm\{in\}\}\-th input channel to thecoutc\_\{\\textrm\{out\}\}\-th output channel, the kernel element in thekhk\_\{h\}\-th row andkwk\_\{w\}\-th column is denoted asw⁡\[cin,cout,kh,kw\]∈ℂw\[c\_\{\\textrm\{in\}\},c\_\{\\textrm\{out\}\},k\_\{h\},k\_\{w\}\]\\in\\mathbb\{C\}\. Without loss of generality, we assume that the kernel dimensions,KhK\_\{h\}andKwK\_\{w\}, are both odd numbers, and we further assume a zero\-padding sizePh=\(Kh−1\)/2P\_\{h\}=\(K\_\{h\}\-1\)/2andPw=\(Kw−1\)/2P\_\{w\}=\(K\_\{w\}\-1\)/2along the two dimensions\. Under this definition, the height and width dimensions of the output are identical to those of the input, while the number of channels changes fromCinC\_\{\\textrm\{in\}\}toCoutC\_\{\\textrm\{out\}\}\. Therefore, the output of theconv2dlayer can be defined as𝐘∈ℂCout×Ih×Iw\\mathbf\{Y\}\\in\\mathbb\{C\}^\{C\_\{\\textrm\{out\}\}\\times I\_\{h\}\\times I\_\{w\}\}, whosecoutc\_\{\\textrm\{out\}\}\-th channel’s element in theihi\_\{h\}\-th row andiwi\_\{w\}\-th column isy⁡\[cout,ih,iw\]∈ℂy\[c\_\{\\textrm\{out\}\},i\_\{h\},i\_\{w\}\]\\in\\mathbb\{C\}\. Using such notations, theconv2doperation, denoted asconv​\(⋅\)\\textsf\{conv\}\\left\(\\cdot\\right\), can be formulated as

𝐘=conv​\(𝐖,𝐗\),where\\displaystyle\\mathbf\{Y\}=\\textsf\{conv\}\\left\(\\mathbf\{W\},\\mathbf\{X\}\\right\),\\ \\text\{where\}y⁡\[cout,ih,iw\]=∑cin=1Cin∑kh=1Kh∑kw=1Kww⁡\[cin,cout,kh,kw\]⋅x⁡\[cin,ih−Ph\+kh,iw−Pw\+kw\]\.\\displaystyle y\[c\_\{\\textrm\{out\}\},i\_\{h\},i\_\{w\}\]=\\sum\_\{c\_\{\\textrm\{in\}\}=1\}^\{C\_\{\\textrm\{in\}\}\}\\sum\_\{k\_\{h\}=1\}^\{K\_\{h\}\}\\sum\_\{k\_\{w\}=1\}^\{K\_\{w\}\}w\[c\_\{\\textrm\{in\}\},c\_\{\\textrm\{out\}\},k\_\{h\},k\_\{w\}\]\\cdot x\[c\_\{\\textrm\{in\}\},i\_\{h\}\-P\_\{h\}\+k\_\{h\},i\_\{w\}\-P\_\{w\}\+k\_\{w\}\]\.\(S1\)Typically, there are a total number of\(Cin​Cout⋅Kh​Kw\)\(C\_\{\\textrm\{in\}\}C\_\{\\textrm\{out\}\}\\cdot K\_\{h\}K\_\{w\}\)complex\-valued trainable parameters in𝐖\\mathbf\{W\}in such aconv2dlayer, corresponding to a total number of\(Cin​Cout⋅Kh​Kw⋅Ih​Iw\)\(C\_\{\\textrm\{in\}\}C\_\{\\textrm\{out\}\}\\cdot K\_\{h\}K\_\{w\}\\cdot I\_\{h\}I\_\{w\}\)complex\-valued MACs, or\(4⋅Cin​Cout⋅Kh​Kw⋅Ih​Iw\)\(4\\cdot C\_\{\\textrm\{in\}\}C\_\{\\textrm\{out\}\}\\cdot K\_\{h\}K\_\{w\}\\cdot I\_\{h\}I\_\{w\}\)real\-valued MACs\.

Arbitrary\-dimension convolutional layer\.We further extend the definition for aconv2dlayer to a generalDD\-dimensional convolutional layer, where the input is a\(D\+1\)\(D\+1\)\-dimensional tensor𝐗∈ℂCin×I1×⋯×ID\\mathbf\{X\}\\in\\mathbb\{C\}^\{C\_\{\\textrm\{in\}\}\\times I\_\{1\}\\times\\cdots\\times I\_\{D\}\}, the weight is a\(D\+2\)\(D\+2\)\-dimensional tensor𝐖∈ℂCin×Cout×K1×⋯×KD\\mathbf\{W\}\\in\\mathbb\{C\}^\{C\_\{\\textrm\{in\}\}\\times C\_\{\\textrm\{out\}\}\\times K\_\{1\}\\times\\cdots\\times K\_\{D\}\}, and the output is a\(D\+1\)\(D\+1\)\-dimensional tensor𝐘∈ℂCout×I1×⋯×ID\\mathbf\{Y\}\\in\\mathbb\{C\}^\{C\_\{\\textrm\{out\}\}\\times I\_\{1\}\\times\\cdots\\times I\_\{D\}\}\. Following equation \([S1](https://arxiv.org/html/2609.19279#Sx11.E1)\), theDD\-dimensional convolution operation can be written as

𝐘=conv​\(𝐖,𝐗\),where\\displaystyle\\mathbf\{Y\}=\\textsf\{conv\}\\left\(\\mathbf\{W\},\\mathbf\{X\}\\right\),\\ \\text\{where\}y⁡\[cout,i1,…,iD\]=∑cin=1Cin∑k1=1K1…​∑kD=1KDw⁡\[cin,cout,k1,…,kD\]⋅x⁡\[cin,i1−P1\+k1,…,iD−PD\+kD\]\.\\displaystyle y\[c\_\{\\textrm\{out\}\},i\_\{1\},\\dots,i\_\{D\}\]=\\sum\_\{c\_\{\\textrm\{in\}\}=1\}^\{C\_\{\\textrm\{in\}\}\}\\sum\_\{k\_\{1\}=1\}^\{K\_\{1\}\}\\dots\\sum\_\{k\_\{D\}=1\}^\{K\_\{D\}\}w\[c\_\{\\textrm\{in\}\},c\_\{\\textrm\{out\}\},k\_\{1\},\\dots,k\_\{D\}\]\\cdot x\[c\_\{\\textrm\{in\}\},i\_\{1\}\-P\_\{1\}\+k\_\{1\},\\dots,i\_\{D\}\-P\_\{D\}\+k\_\{D\}\]\.\(S2\)
Fully\-connected \(FC\) layer as a special case of convolutional layer\.A fully\-connected \(FC\) layer with an input size ofHinH\_\{\\textrm\{in\}\}and an output size ofHoutH\_\{\\textrm\{out\}\}can be formulated as a matrix\-vector multiplication that𝐘=𝐖⋅𝐗\\mathbf\{Y\}=\\mathbf\{W\}\\cdot\\mathbf\{X\}, where𝐗∈ℂHin\\mathbf\{X\}\\in\\mathbb\{C\}^\{H\_\{\\textrm\{in\}\}\},𝐖∈ℂHin\\mathbf\{W\}\\in\\mathbb\{C\}^\{H\_\{\\textrm\{in\}\}\}, and𝐘∈ℂHout\\mathbf\{Y\}\\in\\mathbb\{C\}^\{H\_\{\\textrm\{out\}\}\}\. There are a total number ofHin​HoutH\_\{\\textrm\{in\}\}H\_\{\\textrm\{out\}\}complex\-valued trainable parameters in𝐖\\mathbf\{W\}, corresponding to a total number ofHin​HoutH\_\{\\textrm\{in\}\}H\_\{\\textrm\{out\}\}complex\-valued MACs, or4​Hin​Hout4H\_\{\\textrm\{in\}\}H\_\{\\textrm\{out\}\}real\-valued MACs\. This FC layer can be viewed as a special case of aconv2dlayer withCin=HinC\_\{\\textrm\{in\}\}=H\_\{\\textrm\{in\}\}andCout=HoutC\_\{\\textrm\{out\}\}=H\_\{\\textrm\{out\}\}withKh=Kw=Ih=Iw=1K\_\{h\}=K\_\{w\}=I\_\{h\}=I\_\{w\}=1\. Therefore, inferring an FC layer can either be accelerated byRF\-CNNor the existing work[Gao et al\. \(2026\)](https://arxiv.org/html/2609.19279#bib.bib1)\.

### BUsing Frequency Mixer for Convolution and Cross\-Correlation

We conduct analog computing for complex\-valued convolution using a \(passive\) frequency mixer, which is the core computing operation to drive the convolution layers inRF\-CNN\. According to the convolution theorem[Oppenheim \(1999\)](https://arxiv.org/html/2609.19279#bib.bib2);[Gao et al\. \(2026\)](https://arxiv.org/html/2609.19279#bib.bib1), element\-wise multiplication in the time domain is equivalent to the convolution in the frequency domain; when one of the operands is conjugated in the time domain, the convolution in the frequency domain becomes cross\-correlation\.

Convolution and cross\-correlation formulation\.We consider the convolution between two complex\-valued vectors with lengthsL1L\_\{1\}andL2L\_\{2\}, respectively, denoted as𝐒1∈ℂL1\\mathbf\{S\}\_\{1\}\\in\\mathbb\{C\}^\{L\_\{1\}\}and𝐒2∈ℂL2\\mathbf\{S\}\_\{2\}\\in\\mathbb\{C\}^\{L\_\{2\}\}\. The output of this convolution, denoted as𝐒o∈ℂLo\\mathbf\{S\}\_\{o\}\\in\\mathbb\{C\}^\{L\_\{o\}\}, whereLo=L1\+L2−1L\_\{o\}=L\_\{1\}\+L\_\{2\}\-1, can be formulated as

𝐒o=𝐒1∗𝐒2,whereSo\[l\]=∑l′=max⁡\{0,l−L2\+1\}min⁡\{L1−1,l\}S1\[l′\]⋅S2\[l−l′\],∀l=0,1,…,L1\+L2−2\.\\displaystyle\\mathbf\{S\}\_\{o\}=\\mathbf\{S\}\_\{1\}\\ast\\mathbf\{S\}\_\{2\},~\\text\{where\}~S\_\{o\}\[l\]=\\sum\_\{l^\{\\prime\}=\\max\\\{0,l\-L\_\{2\}\+1\\\}\}^\{\\min\\\{L\_\{1\}\-1,l\\\}\}S\_\{1\}\[l^\{\\prime\}\]\\cdot S\_\{2\}\[l\-l^\{\\prime\}\],\\ \\forall l=0,1,\\dots,L\_\{1\}\+L\_\{2\}\-2\.\(S3\)This convolution contains a total number ofL1​L2L\_\{1\}L\_\{2\}complex\-valued MACs, or4​L1​L24L\_\{1\}L\_\{2\}real\-valued MACs\. Similarly, one can formulate the cross\-correlation operation, whose only difference from the convolution is that𝐒o\\mathbf\{S\}\_\{o\}is not flipped, given by

𝐒o′=𝐒1⋆𝐒2,whereSo′\[l\]=∑l′=max⁡\{0,l−L2\+1\}min⁡\{L1−1,l\}S1\[l′\]⋅S2\[l′\+L2−l−1\],∀l=0,1,…,L1\+L2−2\.\\displaystyle\\mathbf\{S\}\_\{o\}^\{\\prime\}=\\mathbf\{S\}\_\{1\}\\star\\mathbf\{S\}\_\{2\},~\\text\{where\}~S\_\{o\}^\{\\prime\}\[l\]=\\sum\_\{l^\{\\prime\}=\\max\\\{0,l\-L\_\{2\}\+1\\\}\}^\{\\min\\\{L\_\{1\}\-1,l\\\}\}S\_\{1\}\[l^\{\\prime\}\]\\cdot S\_\{2\}\[l^\{\\prime\}\+L\_\{2\}\-l\-1\],\\ \\forall l=0,1,\\dots,L\_\{1\}\+L\_\{2\}\-2\.\(S4\)
Waveform generation using IFFT\.The two vectors,𝐒1\\mathbf\{S\}\_\{1\}and𝐒2\\mathbf\{S\}\_\{2\}, are mapped to two series of frequency tones in the frequency domain\. We assume the analog computing process is completed within a time duration ofTT, which corresponds to a tone spacing ofΔ​f=1/T\\Delta f=1/Tfor each frequency tone series\. For𝐒1\\mathbf\{S\}\_\{1\}, we map theL1L\_\{1\}elements toL1L\_\{1\}tones, where thell\-th element,S1​\[l\]S\_\{1\}\[l\], is assigned on thell\-th tone at the frequency ofl⋅Δ​fl\\cdot\\Delta f\. Its corresponding time domain I/Q sample sequence, or*I/Q waveform*, is anL1L\_\{1\}\-point vector,𝐬1∈ℂL1\\mathbf\{s\}\_\{1\}\\in\\mathbb\{C\}^\{L\_\{1\}\}, obtained by conducting anL1L\_\{1\}\-point IFFT, i\.e\.,

𝐬1=iFFT\{𝐒1\},wheres1\[l\]=1L1∑l′=0L1−1S1\[l′\]⋅ej​2​π​l′​lL1,∀l=0,1,…,L1−1\.\\displaystyle\\mathbf\{s\}\_\{1\}=\\textsf\{iFFT\}\\left\\\{\\mathbf\{S\}\_\{1\}\\right\\\},\\ \\text\{where\}~s\_\{1\}\[l\]=\\frac\{1\}\{L\_\{1\}\}\\sum\_\{l^\{\\prime\}=0\}^\{L\_\{1\}\-1\}S\_\{1\}\[l^\{\\prime\}\]\\cdot\{e\}^\{\{j\}2\\pi\\frac\{l^\{\\prime\}l\}\{L\_\{1\}\}\},~\\forall l=0,1,\\dots,L\_\{1\}\-1\.\(S5\)ThisL1L\_\{1\}\-point IFFT operation contains a total number ofL12⋅log2⁡L1\\frac\{L\_\{1\}\}\{2\}\\cdot\\log\_\{2\}L\_\{1\}complex\-valued MACs, or2​L1⋅log2⁡L12L\_\{1\}\\cdot\\log\_\{2\}L\_\{1\}real\-valued MACs\. Then, this I/Q waveform𝐬1\\mathbf\{s\}\_\{1\}, which carries the information of𝐒1\\mathbf\{S\}\_\{1\}, is streamed to a DAC at the sampling rate ofL1⋅Δ​fL\_\{1\}\\cdot\\Delta f, generating a continuous\-time waveforms1​\(t\)s\_\{1\}\(t\)lasting for a time duration ofTT\. This waveform occupies a bandwidth ofL1⋅Δ​fL\_\{1\}\\cdot\\Delta fin the frequency domain\. Similarly, this process can be applied to𝐒2\\mathbf\{S\}\_\{2\}\. First, anL2L\_\{2\}\-point IFFT with2​L2⋅log2⁡L22L\_\{2\}\\cdot\\log\_\{2\}L\_\{2\}MACs provides the time\-domain I/Q waveform𝐬2∈ℂL2\\mathbf\{s\}\_\{2\}\\in\\mathbb\{C\}^\{L\_\{2\}\}\. Then, a DAC operating at a sampling rate ofL2⋅Δ​fL\_\{2\}\\cdot\\Delta fgenerates the analog waveform ofs2​\(t\)s\_\{2\}\(t\)with a time duration ofTT, occupying a bandwidth ofL2⋅Δ​fL\_\{2\}\\cdot\\Delta f\. This waveform generation process is illustrated in the left half of Fig\.[S2](https://arxiv.org/html/2609.19279#Sx12.F2)\(a\)\.

Waveform analysis for the convolution output\.The two generated analog waveforms are fed into a frequency mixer, or, namely, a computing mixer inRF\-CNN\. When performing up\-conversion, this computing mixer yields the output waveform as a time\-domain multiplication of the two input waveforms[Gao et al\. \(2026\)](https://arxiv.org/html/2609.19279#bib.bib1)as

so​\(t\)=s1​\(t\)⋅s2​\(t\),∀t∈\[0,T\)\.\\displaystyle s\_\{o\}\(t\)=s\_\{1\}\(t\)\\cdot s\_\{2\}\(t\),~\\forall t\\in\[0,T\)\.\(S6\)Based on the convolution theorem, the frequency domain representation ofso​\(t\)s\_\{o\}\(t\)is the convolution between𝐒1\\mathbf\{S\}\_\{1\}and𝐒2\\mathbf\{S\}\_\{2\}, which contains a total number ofLoL\_\{o\}tones spaced atΔ​f\\Delta fand spanning over a bandwidth ofLo⋅Δ​fL\_\{o\}\\cdot\\Delta f\. Therefore, to fully receive this signal without frequency aliasing, an ADC operating at a sampling rate ofLo⋅Δ​fL\_\{o\}\\cdot\\Delta fis required to sample this output waveform, which yields anLoL\_\{o\}\-point I/Q sample sequence in digital as𝐬o∈ℂLo\\mathbf\{s\}\_\{o\}\\in\\mathbb\{C\}^\{L\_\{o\}\}\. The convolution results,𝐒o\\mathbf\{S\}\_\{o\}, between𝐒1\\mathbf\{S\}\_\{1\}and𝐒2\\mathbf\{S\}\_\{2\}, can be extracted from𝐬o\\mathbf\{s\}\_\{o\}by performing anLoL\_\{o\}\-point digital FFT, i\.e\.,

𝐒o=FFT\{𝐬o\},whereSo\[l\]=∑l′=0Lo−1so\[l′\]⋅e−j​2​π​l′​lLo−1,∀l=0,1,…,Lo−1\.\\displaystyle\\mathbf\{S\}\_\{o\}=\\textsf\{FFT\}\\left\\\{\\mathbf\{s\}\_\{o\}\\right\\\},\\ \\text\{where\}~S\_\{o\}\[l\]=\\sum\_\{l^\{\\prime\}=0\}^\{L\_\{o\}\-1\}s\_\{o\}\[l^\{\\prime\}\]\\cdot\{e\}^\{\-\{j\}2\\pi\\frac\{l^\{\\prime\}l\}\{L\_\{o\}\-1\}\},~\\forall l=0,1,\\dots,L\_\{o\}\-1\.\(S7\)ThisLoL\_\{o\}\-point FFT contains a total number of\(Lo2⋅log2⁡Lo\)\(\\frac\{L\_\{o\}\}\{2\}\\cdot\\log\_\{2\}L\_\{o\}\)complex\-valued MACs, or\(2​Lo⋅log2⁡Lo\)\(2L\_\{o\}\\cdot\\log\_\{2\}L\_\{o\}\)real\-valued MACs\. The waveform analysis diagram for𝐒o\\mathbf\{S\}\_\{o\}is shown in the right half of Fig\.[S2](https://arxiv.org/html/2609.19279#Sx12.F2)\(a\)\.

Waveform analysis for partial convolution output\.In practice, only a subset of elements in the output is needed, which are usually contiguous\. For example, we consider the desired output vector𝐒↓∈ℂL↓\\mathbf\{S\}\_\{\\downarrow\}\\in\\mathbb\{C\}^\{L\_\{\\downarrow\}\}elements indexed at\{l1,l1\+1,…,l2−1\}\\left\\\{l\_\{1\},l\_\{1\}\+1,\\dots,l\_\{2\}\-1\\right\\\}on the full output vector𝐒o\\mathbf\{S\}\_\{o\}, where0≤l1<l2≤Lo0\\leq l\_\{1\}<l\_\{2\}\\leq L\_\{o\}andL↓=l2−l1L\_\{\\downarrow\}=l\_\{2\}\-l\_\{1\}\. Hereby, we only need to examine the middleL↓L\_\{\\downarrow\}tones indexed at\{l1,l1\+1,…,l2−1\}\\left\\\{l\_\{1\},l\_\{1\}\+1,\\dots,l\_\{2\}\-1\\right\\\}in the frequency domain\. To achieve this, we first apply an additional frequency mixer, which down\-convertsso​\(t\)s\_\{o\}\(t\)byl1⋅Δ​fl\_\{1\}\\cdot\\Delta f\. This shifts the desiredL↓L\_\{\\downarrow\}tones to the baseband at the frequencies of\{0,Δ​f,…,L↓⋅Δ​f\}\\\{0,\\Delta f,\\dots,L\_\{\\downarrow\}\\cdot\\Delta f\\\}\. Then, a low\-pass filter \(LPF\) with a stop band atL↓⋅Δ​fL\_\{\\downarrow\}\\cdot\\Delta fis applied, which only passes the desiredL↓L\_\{\\downarrow\}tones\. We denote this filtered waveform ass↓​\(t\)s\_\{\\downarrow\}\(t\), which still lasts for a time duration ofTTbut occupies a reduced bandwidth ofL↓⋅Δ​fL\_\{\\downarrow\}\\cdot\\Delta f\. Given its smaller bandwidth, a pair of ADCs operating at a sampling rate ofL↓⋅Δ​fL\_\{\\downarrow\}\\cdot\\Delta fis sufficient to sample this down\-converted waveforms↓​\(t\)s\_\{\\downarrow\}\(t\)using I/Q demodulation, which yields anL↓L\_\{\\downarrow\}\-point I/Q sample sequence in digital, denoted by𝐬↓∈ℂL↓\\mathbf\{s\}\_\{\\downarrow\}\\in\\mathbb\{C\}^\{L\_\{\\downarrow\}\}\. Then, anL↓L\_\{\\downarrow\}\-point FFT is applied on𝐬↓\\mathbf\{s\}\_\{\\downarrow\}to extract the desired output vector𝐒↓\\mathbf\{S\}\_\{\\downarrow\}as

𝐒↓=FFT\{𝐬↓\},whereS↓\[l\]=∑l′=0L↓−1s↓\[l′\]⋅e−j​2​π​l′​lL↓−1,∀l=0,1,…,L↓−1,\\displaystyle\\mathbf\{S\}\_\{\\downarrow\}=\\textsf\{FFT\}\\left\\\{\\mathbf\{s\}\_\{\\downarrow\}\\right\\\},\\ \\text\{where\}~S\_\{\\downarrow\}\[l\]=\\sum\_\{l^\{\\prime\}=0\}^\{L\_\{\\downarrow\}\-1\}s\_\{\\downarrow\}\[l^\{\\prime\}\]\\cdot\{e\}^\{\-\{j\}2\\pi\\frac\{l^\{\\prime\}l\}\{L\_\{\\downarrow\}\-1\}\},~\\forall l=0,1,\\dots,L\_\{\\downarrow\}\-1,\(S8\)which requires a total number of\(2​L↓⋅log2⁡L↓\)\(2L\_\{\\downarrow\}\\cdot\\log\_\{2\}L\_\{\\downarrow\}\)real\-valued MACs\. The whole waveform analysis diagram for𝐒↓\\mathbf\{S\}\_\{\\downarrow\}is illustrated in Fig\.[S2](https://arxiv.org/html/2609.19279#Sx12.F2)\(b\)\.

Cross\-correlation by down\-conversion\.When the computing mixer is configured and used for down\-conversion, the output waveform becomes the time\-domain multiplication ofs1​\(t\)s\_\{1\}\(t\)and the conjugate ofs2​\(t\)s\_\{2\}\(t\)[Gao et al\. \(2026\)](https://arxiv.org/html/2609.19279#bib.bib1)\. In this case, equation \([S6](https://arxiv.org/html/2609.19279#Sx11.E6)\) can be written as

so′​\(t\)=s1​\(t\)⋅s2¯​\(t\),∀t∈\[0,T\)\.\\displaystyle s\_\{o\}^\{\\prime\}\(t\)=s\_\{1\}\(t\)\\cdot\\overline\{s\_\{2\}\}\(t\),~\\forall t\\in\[0,T\)\.\(S9\)This conjugate in the time domain reflects vector\-flipping in the frequency domain for𝐒w\\mathbf\{S\}\_\{w\}, and therefore the convolution becomes the cross\-correlation defined in equation \([S4](https://arxiv.org/html/2609.19279#Sx11.E4)\)\. This means that the output of equation \([S7](https://arxiv.org/html/2609.19279#Sx11.E7)\) becomes𝐒o′=𝐒1⋆𝐒2\\mathbf\{S\}\_\{o\}^\{\\prime\}=\\mathbf\{S\}\_\{1\}\\star\\mathbf\{S\}\_\{2\}, and the output of equation \([S8](https://arxiv.org/html/2609.19279#Sx11.E8)\) becomes𝐒↓′\\mathbf\{S\}\_\{\\downarrow\}^\{\\prime\}, which is a subset of𝐒o′\\mathbf\{S\}\_\{o\}^\{\\prime\}\.

### CPhysical Channel Calibration

The transmission of waveforms,s1​\(t\)s\_\{1\}\(t\),s2​\(t\)s\_\{2\}\(t\), andso​\(t\)s\_\{o\}\(t\), or their frequency\-domain tone series,𝐒1\\mathbf\{S\}\_\{1\},𝐒2\\mathbf\{S\}\_\{2\}, and𝐒o\\mathbf\{S\}\_\{o\}, through a physical channel introduces distortions that must be calibrated before convolution computing\. This distortion comes from the unflattened frequency response introduced by the embedded anti\-aliasing filters for the wired channel, or the multipath effect or timing delay for the wireless channel\.

Physical channel modeling\.We model the physical channel in the frequency domain, which element\-wisely multiplies to the corresponding tone series\. We formulate this process as a function as𝐂o=ℋ⁡\(𝐂1,𝐂2\)\\mathbf\{C\}\_\{o\}=\\mathcal\{H\}\\left\(\\mathbf\{C\}\_\{1\},\\mathbf\{C\}\_\{2\}\\right\), where𝐇1∈ℂL1\\mathbf\{H\}\_\{1\}\\in\\mathbb\{C\}^\{L\_\{1\}\}and𝐇2∈ℂL2\\mathbf\{H\}\_\{2\}\\in\\mathbb\{C\}^\{L\_\{2\}\}are the input tone series, and𝐂o∈ℂLo\\mathbf\{C\}\_\{o\}\\in\\mathbb\{C\}^\{L\_\{o\}\}is the output tone series after the physical channel\. Specifically, we define𝐇1∈ℂL1\\mathbf\{H\}\_\{1\}\\in\\mathbb\{C\}^\{L\_\{1\}\},𝐇2∈ℂL2\\mathbf\{H\}\_\{2\}\\in\\mathbb\{C\}^\{L\_\{2\}\}, and𝐇o∈ℂLo\\mathbf\{H\}\_\{o\}\\in\\mathbb\{C\}^\{L\_\{o\}\}as the channel frequency response for tone series𝐂1\\mathbf\{C\}\_\{1\},𝐂2\\mathbf\{C\}\_\{2\}and𝐂o\\mathbf\{C\}\_\{o\}, respectively\. In addition, we denote the element\-wise multiplication \(or Hadamard multiplication\) between two equal dimension vectors by⊙\\odot, as well as the element\-wise division \(or Hadamard division\) by⊘\\oslash\. Generally, these channel responses are complex\-valued, and their amplitudes and phases are continuous and smooth over the frequency\. Then, the distorted version of equation \([S3](https://arxiv.org/html/2609.19279#Sx11.E3)\) including the physical channel distortion can be rewritten as

𝐂o\\displaystyle\\mathbf\{C\}\_\{o\}=ℋ⁡\(𝐂1,𝐂2\)=𝐇o⊙\(\(𝐇1⊙𝐂1\)∗\(𝐇2⊙𝐂2\)\)\\displaystyle=\\mathcal\{H\}\\left\(\\mathbf\{C\}\_\{1\},\\mathbf\{C\}\_\{2\}\\right\)=\\mathbf\{H\}\_\{o\}\\odot\\left\(\\left\(\\mathbf\{H\}\_\{1\}\\odot\\mathbf\{C\}\_\{1\}\\right\)\\ast\\left\(\\mathbf\{H\}\_\{2\}\\odot\\mathbf\{C\}\_\{2\}\\right\)\\right\)\(S10\)where​Co​\[l\]\\displaystyle\\text\{where\}~C\_\{o\}\[l\]=Ho​\[l\]⋅∑l′=max⁡\{0,l−L2\+1\}min⁡\{L1−1,l\}H1​\[l′\]⋅C1​\[l′\]⋅H2​\[l−l′\]⋅C2​\[l−l′\],\\displaystyle=H\_\{o\}\[l\]\\cdot\\sum\_\{l^\{\\prime\}=\\max\\\{0,l\-L\_\{2\}\+1\\\}\}^\{\\min\\\{L\_\{1\}\-1,l\\\}\}H\_\{1\}\[l^\{\\prime\}\]\\cdot C\_\{1\}\[l^\{\\prime\}\]\\cdot H\_\{2\}\[l\-l^\{\\prime\}\]\\cdot C\_\{2\}\[l\-l^\{\\prime\}\],\(S11\)∀l\\displaystyle\\forall l=0,1,…,L1\+L2−2\.\\displaystyle=0,1,\\dots,L\_\{1\}\+L\_\{2\}\-2\.\(S12\)
Pre\- and post\-channel calibration\.To calculate the convolution𝐒o=𝐒1∗𝐒2\\mathbf\{S\}\_\{o\}=\\mathbf\{S\}\_\{1\}\\ast\\mathbf\{S\}\_\{2\}, we pre\-compensate the physical channel distortion by pre\-calibrating the input tone series𝐂1\\mathbf\{C\}\_\{1\}and𝐂2\\mathbf\{C\}\_\{2\}before the transmission as

\{𝐂1=𝐒1⊘𝐇1,whereC1\[l\]=S1\[l\]/H1\[l\],∀l=0,1,…,L1−1𝐂2=𝐒2⊘𝐇2,whereC2\[l\]=S2\[l\]/H2\[l\],∀l=0,1,…,L2−1,\\displaystyle\\begin\{cases\}\\mathbf\{C\}\_\{1\}=\\mathbf\{S\}\_\{1\}\\oslash\\mathbf\{H\}\_\{1\},\\quad\\text\{where\}~C\_\{1\}\[l\]=S\_\{1\}\[l\]/H\_\{1\}\[l\],\\quad\\forall l=0,1,\\dots,L\_\{1\}\-1\\\\ \\mathbf\{C\}\_\{2\}=\\mathbf\{S\}\_\{2\}\\oslash\\mathbf\{H\}\_\{2\},\\quad\\text\{where\}~C\_\{2\}\[l\]=S\_\{2\}\[l\]/H\_\{2\}\[l\],\\quad\\forall l=0,1,\\dots,L\_\{2\}\-1,\\end\{cases\}\(S13\)as well as post\-calibrating the output tone series𝐂o\\mathbf\{C\}\_\{o\}after the reception by

𝐒o=𝐂o⊘𝐇o,whereSo\[l\]=Co\[l\]/Ho\[l\],∀l=0,1,…,Lo−1\.\\displaystyle\\mathbf\{S\}\_\{o\}=\\mathbf\{C\}\_\{o\}\\oslash\\mathbf\{H\}\_\{o\},\\quad\\text\{where\}~S\_\{o\}\[l\]=C\_\{o\}\[l\]/H\_\{o\}\[l\],\\quad\\forall l=0,1,\\dots,L\_\{o\}\-1\.\(S14\)This process is illustrated in Fig\.[S2](https://arxiv.org/html/2609.19279#Sx12.F2)\(c\)\. This indicates that the key to the physical channel calibration is to estimate the channel response vectors,𝐇1\\mathbf\{H\}\_\{1\},𝐇2\\mathbf\{H\}\_\{2\}, and𝐇o\\mathbf\{H\}\_\{o\}\. In practice, these pre\- and post\-calibration can be fused with the IFFT and FFT operations in equations \([S5](https://arxiv.org/html/2609.19279#Sx11.E5)\) and \([S7](https://arxiv.org/html/2609.19279#Sx11.E7)\) without introducing additional digital computing costs\.

Physical channel estimation using pre\-defined training sequences\.The physical channels,𝐇1\\mathbf\{H\}\_\{1\},𝐇2\\mathbf\{H\}\_\{2\}, and𝐇o\\mathbf\{H\}\_\{o\}can be estimated using pre\-defined training sequences sent through them and the minimum mean squared error \(MMSE\) method\. Specifically, we randomize a set of𝐇1\(i\)\\mathbf\{H\}\_\{1\}^\{\(i\)\},𝐇2\(i\)\\mathbf\{H\}\_\{2\}^\{\(i\)\}, and conduct the transmission process to obtain the corresponding output tone series𝐂o^\(i\)\\widehat\{\\mathbf\{C\}\_\{o\}\}^\{\(i\)\}\. The channel estimation can be formulated as solving the following optimization problem:

\{𝐇1⋆,𝐇2⋆,𝐇o⋆\}\\displaystyle\\left\\\{\\mathbf\{H\}\_\{1\}^\{\\star\},\\mathbf\{H\}\_\{2\}^\{\\star\},\\mathbf\{H\}\_\{o\}^\{\\star\}\\right\\\}=arg⁡min⁡∑i𝐇1,𝐇2,𝐇o⁡\|𝐂o^\(i\)−ℋ⁡\(𝐂1\(i\),𝐂2\(i\)\)\|2\\displaystyle=\\arg\\min\_\{\\mathbf\{H\}\_\{1\},\\mathbf\{H\}\_\{2\},\\mathbf\{H\}\_\{o\}\}\\sum\_\{i\}\\left\|\{\\widehat\{\\mathbf\{C\}\_\{o\}\}^\{\(i\)\}\-\\mathcal\{H\}\\left\(\\mathbf\{C\}\_\{1\}^\{\(i\)\},\\mathbf\{C\}\_\{2\}^\{\(i\)\}\\right\)\}\\right\|^\{2\}\(S15\)=arg⁡min⁡∑i𝐇1,𝐇2,𝐇o⁡\|𝐂o^\(i\)−𝐇o⊙\(\(𝐇1⊙𝐂1\(i\)\)∗\(𝐇2⊙𝐂2\(i\)\)\)\|2\.\\displaystyle=\\arg\\min\_\{\\mathbf\{H\}\_\{1\},\\mathbf\{H\}\_\{2\},\\mathbf\{H\}\_\{o\}\}\\sum\_\{i\}\\left\|\{\\widehat\{\\mathbf\{C\}\_\{o\}\}^\{\(i\)\}\-\\mathbf\{H\}\_\{o\}\\odot\\left\(\\left\(\\mathbf\{H\}\_\{1\}\\odot\\mathbf\{C\}\_\{1\}^\{\(i\)\}\\right\)\\ast\\left\(\\mathbf\{H\}\_\{2\}\\odot\\mathbf\{C\}\_\{2\}^\{\(i\)\}\\right\)\\right\)\}\\right\|^\{2\}\.\(S16\)We adopt the Adam optimizer[Kingma and Ba \(2015\)](https://arxiv.org/html/2609.19279#bib.bib11)with a learning rate of10−310^\{\-3\}, and run iterations until the MMSE converges\. Generally, smaller values ofL1L\_\{1\},L2L\_\{2\}, andLoL\_\{o\}may not be sufficient to accurately describe the physical channel, while larger values may introduce too many parameters and therefore lead to over\-fitting\. We empirically selectL1=L2=1,000L\_\{1\}=L\_\{2\}=1,000andLo=1,999L\_\{o\}=1,999to optimize𝐇1⋆\\mathbf\{H\}\_\{1\}^\{\\star\},𝐇2⋆\\mathbf\{H\}\_\{2\}^\{\\star\}, and𝐇o⋆\\mathbf\{H\}\_\{o\}^\{\\star\}once, and use linear interpolation respectively for amplitudes and phases to obtain the physical channels at other lengths\. Under this setting, we randomize 100 such training sequences for the optimization\.

### DPhase Calibration with Preambles and Postambles

The input and weight waveforms ofRF\-CNNare structured into \(wired\) input and \(wireless\) weight packets\. These two packets are expected to be aligned in time and fed into the computing mixer for computing the target convolution operations, as shown in Fig\.[S3](https://arxiv.org/html/2609.19279#Sx12.F3)\(a\)\. Typically, the two transmitters \(s1​\(t\)s\_\{1\}\(t\)ands2​\(t\)s\_\{2\}\(t\)\) and the receiver \(s↓​\(t\)s\_\{\\downarrow\}\(t\)\) are not synchronized in time and frequency, requiring a detection process at the receiver\. This process identifies the starting point of the transmitted waveforms in time and aligns the carrier frequency at the receiver\. Specifically, we follow the detection process of the low\-sampling\-rate Wi\-Fi monitoring system[Gao et al\. \(2023\)](https://arxiv.org/html/2609.19279#bib.bib4)in the implementation ofRF\-CNN\. In this section, we mainly focus on the output packet and its phase calibration\.

Packet formulation and time\-frequency representation\.As shown in Fig\.[S3](https://arxiv.org/html/2609.19279#Sx12.F3)\(a\), a packet consists of\(K\+2\)\(K\+2\)symbols concatenated over time:KKpayload symbols sandwiched between two pre\-defined preamble and postamble symbols\. TheseKKpayload symbols are used to performKKequal\-dimension convolution operations \(denoting its output length asL↓L\_\{\\downarrow\}, and tone spacing asΔ​F\\Delta F\), while the preamble and postamble symbols are used for phase estimation and calibration\. The waveforms of theseK\+2K\+2symbols in the time domain can be converted into the frequency domain usingL↓L\_\{\\downarrow\}\-point FFT; by concatenating these frequency\-domain representations, we obtain the time\-frequency representation of the output packet, denoted as𝒮∈ℂ\(K\+2\)×L↓\\mathcal\{S\}\\in\\mathbb\{C\}^\{\(K\+2\)\\times L\_\{\\downarrow\}\}, as shown in Fig\.[S3](https://arxiv.org/html/2609.19279#Sx12.F3)\(b\)\. Specifically,𝒮\\mathcal\{S\}can be further divided into𝒮\(payload\)∈ℂK×L↓\\mathcal\{S\}^\{\\textrm\{\(payload\)\}\}\\in\\mathbb\{C\}^\{K\\times L\_\{\\downarrow\}\}for the payload, as well as𝒮\(pre\)∈ℂL↓\\mathcal\{S\}^\{\\textrm\{\(pre\)\}\}\\in\\mathbb\{C\}^\{L\_\{\\downarrow\}\}and𝒮\(post\)∈ℂL↓\\mathcal\{S\}^\{\\textrm\{\(post\)\}\}\\in\\mathbb\{C\}^\{L\_\{\\downarrow\}\}for the preamble and postamble symbols\.

Time\-frequency phase offset formulation\.In practice, the phases of the experimentally observed𝒮′\\mathcal\{S\}^\{\\prime\}are offset from the expected one𝒮\\mathcal\{S\}, and this offset varies linearly over time \(due to the carrier frequency offset of the receiver\), and over frequency \(due to the packet detection timing offset on the receiver\)\. Denoting the carrier frequency offset asΔ​F\\Delta Fand the timing offset asΔ​τ\\Delta\\tau, the experimental𝒮′\\mathcal\{S\}^\{\\prime\}with the phase offset can be formulated as

S′\[k,l\]=S\[k,l\]⋅ej​2​π⋅\(k⋅Δ​FΔ​f\+l⋅Δ​τ​Δ​f\),∀k=0,1,…,K\+1,∀l=0,1,…,L↓−1,\\displaystyle S^\{\\prime\}\[k,l\]=S\[k,l\]\\cdot\{e\}^\{\{j\}2\\pi\\cdot\\left\(k\\cdot\\frac\{\\Delta F\}\{\\Delta f\}\+l\\cdot\\Delta\\tau\\Delta f\\right\)\},\\ \\forall k=0,1,\\dots,K\+1,\\ \\forall l=0,1,\\dots,L\_\{\\downarrow\}\-1,\(S17\)whereΔ​f\\Delta fis the tone spacing \(see definition in Supplementary Text Section[B](https://arxiv.org/html/2609.19279#Sx11.SS2)\)\. This is demonstrated in Fig\.[S3](https://arxiv.org/html/2609.19279#Sx12.F3)\(c\)\. Breaking it down, the experimental preamble and postamble symbols \(𝒮\(pre\)′\\mathcal\{S\}^\{\\textrm\{\(pre\)\}\\prime\}and𝒮\(post\)′\\mathcal\{S\}^\{\\textrm\{\(post\)\}\\prime\}\) are with phase offsets given by

S\(pre\)′\[l\]\\displaystyle S^\{\\textrm\{\(pre\)\}\\prime\}\[l\]=S\(pre\)\[l\]⋅ej​2​π⋅\(0⋅Δ​FΔ​f\+l⋅Δ​τ​Δ​f\),∀l=0,1,…,L↓−1;\\displaystyle=S^\{\\textrm\{\(pre\)\}\}\[l\]\\cdot\{e\}^\{\{j\}2\\pi\\cdot\\left\(0\\cdot\\frac\{\\Delta F\}\{\\Delta f\}\+l\\cdot\\Delta\\tau\\Delta f\\right\)\},\\ \\forall l=0,1,\\dots,L\_\{\\downarrow\}\-1;\(S18\)S\(post\)′\[l\]\\displaystyle S^\{\\textrm\{\(post\)\}\\prime\}\[l\]=S\(post\)\[l\]⋅ej​2​π⋅\(\(K\+1\)⋅Δ​FΔ​f\+l⋅Δ​τ​Δ​f\),∀l=0,1,…,L↓−1\.\\displaystyle=S^\{\\textrm\{\(post\)\}\}\[l\]\\cdot\{e\}^\{\{j\}2\\pi\\cdot\\left\(\(K\+1\)\\cdot\\frac\{\\Delta F\}\{\\Delta f\}\+l\\cdot\\Delta\\tau\\Delta f\\right\)\},\\ \\forall l=0,1,\\dots,L\_\{\\downarrow\}\-1\.\(S19\)Similarly, the experimental payload symbols with the phase offset can be expressed as

S\(payload\)′\[k,l\]=S\(payload\)\[k,l\]⋅ej​2​π⋅\(\(k\+1\)⋅Δ​FΔ​f\+l⋅Δ​τ​Δ​f\),∀k=0,1,…,K−1,∀l=0,1,…,L↓−1\.\\displaystyle S^\{\\textrm\{\(payload\)\}\\prime\}\[k,l\]=S^\{\\textrm\{\(payload\)\}\}\[k,l\]\\cdot\{e\}^\{\{j\}2\\pi\\cdot\\left\(\(k\+1\)\\cdot\\frac\{\\Delta F\}\{\\Delta f\}\+l\\cdot\\Delta\\tau\\Delta f\\right\)\},\\ \\forall k=0,1,\\dots,K\-1,\\ \\forall l=0,1,\\dots,L\_\{\\downarrow\}\-1\.\(S20\)Therefore, to correct this phase offset, we need to estimate the two unknown variables,Δ​F\\Delta FandΔ​τ\\Delta\\tau\. In contrast to the relatively stable channel calibration discussed in Supplementary Text Section[C](https://arxiv.org/html/2609.19279#Sx11.SS3), the values ofΔ​F\\Delta FandΔ​τ\\Delta\\taufluctuate unpredictably due to variations in packet detection and synchronization and, as a result, this estimation and calibration must be conducted for every packet\.

Packet preamble and postamble design\.We pre\-define the frequency\-domain preambles on the input and weight packets as vectors with uniform amplitudes𝒰⁡\(0,1\)\\mathcal\{U\}\(0,1\)and random phases following the independent uniform distribution𝒰⁡\(−π,π\)\\mathcal\{U\}\(\-\\pi,\\pi\)\. In this way, the transmitted time\-domain preamble waveforms have a low PAPR to alleviate the saturation issue \(see Supplementary Text Section[E](https://arxiv.org/html/2609.19279#Sx11.SS5)\)\. In addition, this yields a relatively flat power distribution in the frequency domain after the convolution operation, ensuring good SNR values of𝒮\(pre\)\\mathcal\{S\}^\{\\textrm\{\(pre\)\}\}for any choices ofl1l\_\{1\}andl2l\_\{2\}\(see definition in Supplementary Text Section[B](https://arxiv.org/html/2609.19279#Sx11.SS2)\)\. The postamble𝒮\(post\)\\mathcal\{S\}^\{\\textrm\{\(post\)\}\}is the duplication of the preamble𝒮\(pre\)\\mathcal\{S\}^\{\\textrm\{\(pre\)\}\}\.

Carrier frequency offset estimation\.The carrier frequency offsetΔ​F\\Delta Fis estimated by comparing the phase offset between the preamble and postamble symbols\. Specifically, we have

ΔF=Δ​f2​π​\(K\+1\)⋅∠\[∑l=0L↓−1S\(pre\)′\[l\]⋅S\(post\)′¯\[l\]\]\.\\displaystyle\\Delta F=\\frac\{\\Delta f\}\{2\\pi\(K\+1\)\}\\cdot\\angle\{\}\\left\[\\sum\_\{l=0\}^\{L\_\{\\downarrow\}\-1\}S^\{\\textrm\{\(pre\)\}\\prime\}\[l\]\\cdot\\overline\{S^\{\\textrm\{\(post\)\}\\prime\}\}\[l\]\\right\]\.\(S21\)where∠⁡\(⋅\)\\angle\{\(\}\\cdot\)and\(⋅\)¯\\overline\{\(\\cdot\)\}denote the angle and conjugate of a complex number\.

Timing offset estimation\.The timing offsetΔ​τ\\Delta\\tauis estimated by independently by examining the phase offset over frequency on both the preamble or postamble symbol, and then averaging the two results to improve the accuracy\. This process can be written as

Δ​τ\\displaystyle\\Delta\\tau=12​π​Δ​f⋅\(∑l=0L↓−2∠\[S\(pre\)′\[l\]⋅S\(pre\)′¯\[l\+1\]\]L↓−1\+∑l=0L↓−2∠\[S\(post\)′\[l\]⋅S\(post\)′¯\[l\+1\]\]L↓−1\)\.\\displaystyle=\\frac\{1\}\{2\\pi\\Delta f\}\\cdot\\left\(\\frac\{\\sum\_\{l=0\}^\{L\_\{\\downarrow\}\-2\}\\angle\{\}\\left\[S^\{\\textrm\{\(pre\)\}\\prime\}\[l\]\\cdot\\overline\{S^\{\\textrm\{\(pre\)\}\\prime\}\}\[l\+1\]\\right\]\}\{L\_\{\\downarrow\}\-1\}\+\\frac\{\\sum\_\{l=0\}^\{L\_\{\\downarrow\}\-2\}\\angle\{\}\\left\[S^\{\\textrm\{\(post\)\}\\prime\}\[l\]\\cdot\\overline\{S^\{\\textrm\{\(post\)\}\\prime\}\}\[l\+1\]\\right\]\}\{L\_\{\\downarrow\}\-1\}\\right\)\.\(S22\)

### EWaveform PAPR Alleviation

Consider a DAC generating a waveform ofs⁡\(t\)s\(t\)from the time\-domain I/Q sample sequence𝐬\\mathbf\{s\}\.

Saturation and quantization error of a DAC\.In practice, a DAC has a maximum output waveform range fors⁡\(t\)s\(t\), beyond which the generated analog waveforms in analog might be saturated and will be truncated to the maximum output voltage\. This maximum voltage corresponds to a maximum amplitude of the input I/Q waveform for𝐬\\mathbf\{s\}, which can be formulated as a normalized constraint given by

maxl⁡\|s⁡\[l\]\|≤1\.\\displaystyle\\max\_\{l\}\\left\|\{s\[l\]\}\\right\|\\leq 1\.\(S23\)At the same time, a DAC has a finite resolution for the input I/Q samples𝐬\\mathbf\{s\}, which introduces quantization errors when the input sample amplitudesls\_\{l\}is too small\. Specifically, abb\-bit DAC has a constant quantization step of2−\(b−1\)2^\{\-\(b\-1\)\}withs⁡\[l\]∈\[−1,\+1\]s\[l\]\\in\\left\[\-1,\+1\\right\], corresponding to a mean squared quantization error of\(3⋅4b\)−1\\left\(3\\cdot 4^\{b\}\\right\)^\{\-1\}[Lathi and Ding \(2019\)](https://arxiv.org/html/2609.19279#bib.bib5)\. The average power of the output waveforms⁡\(t\)s\(t\)is proportional to the mean squared value of𝐬\\mathbf\{s\}, i\.e\.,1L​∑l=0L−1\|s⁡\[l\]\|2\\frac\{1\}\{L\}\\sum\_\{l=0\}^\{L\-1\}\\left\|\{s\[l\]\}\\right\|^\{2\}\. A smaller average power of𝐬\\mathbf\{s\}leads to a smaller signal\-to\-quantization\-noise ratio \(SQNR\) of the output waveforms⁡\(t\)s\(t\), given by

SQNR​\{𝐬\}=10​log10⁡\(1L​∑l=0L−1\|s⁡\[l\]\|2\(3⋅4b\)−1\)=10​log10⁡\(1L​∑l=0L−1\|s⁡\[l\]\|2\)\+4\.77\+6\.02​b\.\\displaystyle\\textsf\{SQNR\}\\left\\\{\\mathbf\{s\}\\right\\\}=10\\log\_\{10\}\\left\(\\frac\{\\frac\{1\}\{L\}\\sum\_\{l=0\}^\{L\-1\}\\left\|\{s\[l\]\}\\right\|^\{2\}\}\{\\left\(3\\cdot 4^\{b\}\\right\)^\{\-1\}\}\\right\)=10\\log\_\{10\}\\left\(\\frac\{1\}\{L\}\\sum\_\{l=0\}^\{L\-1\}\\left\|\{s\[l\]\}\\right\|^\{2\}\\right\)\+4\.77\+6\.02b\.\(S24\)Therefore, we notice a trade\-off between the peak power and average power of the waveform on the amplitude of𝐬\\mathbf\{s\}: a smaller\|s⁡\[l\]\|\\left\|\{s\[l\]\}\\right\|is less likely to be saturated, while yielding a smaller SQNR\.

Peak\-to\-average power ratio \(PAPR\) and waveform backoff\.The PAPR of the I/Q sample sequence𝐬\\mathbf\{s\}can be calculated as

PAPR​\{𝐬\}=10​log10⁡\(maxl⁡\|s⁡\[l\]\|21L​∑l=0L−1\|s⁡\[l\]\|2\)\.\\displaystyle\\textsf\{PAPR\}\\left\\\{\\mathbf\{s\}\\right\\\}=10\\log\_\{10\}\\left\(\\frac\{\\max\_\{l\}\\left\|\{s\[l\]\}\\right\|^\{2\}\}\{\\frac\{1\}\{L\}\\sum\_\{l=0\}^\{L\-1\}\\left\|\{s\[l\]\}\\right\|^\{2\}\}\\right\)\.\(S25\)Putting equations \([S23](https://arxiv.org/html/2609.19279#Sx11.E23)\), \([S24](https://arxiv.org/html/2609.19279#Sx11.E24)\), and \([S25](https://arxiv.org/html/2609.19279#Sx11.E25)\) together, we have

SQNR​\{𝐬\}≤−PAPR​\{𝐬\}\+4\.77\+6\.02​b\.\\displaystyle\\textsf\{SQNR\}\\left\\\{\\mathbf\{s\}\\right\\\}\\leq\-\\textsf\{PAPR\}\\left\\\{\\mathbf\{s\}\\right\\\}\+4\.77\+6\.02b\.\(S26\)This indicates that a smaller PAPR of𝐬\\mathbf\{s\}leads to a higher upper bound of the corresponding SQNR, which is beneficial for alleviating the quantization error introduced by the finite\-resolution DAC\. In practice, we define a backoff coefficientα\\alphathat scales the amplitude of the transmitted I/Q waveform𝐬\\mathbf\{s\}so that its mean square becomes1/α1/\\alpha\. Under this definition, the scaled waveform𝐬′\\mathbf\{s\}^\{\\prime\}is given by:

𝐬′=1α⋅1L∑l′=0L−1\|s\[l′\]\|2⋅𝐬,wheres′\[l\]=1α⋅1L∑l′=0L−1\|s\[l′\]\|2⋅s\[l\],∀l=0,1,…,L−1\.\\displaystyle\\mathbf\{s\}^\{\\prime\}=\\sqrt\{\\frac\{1\}\{\\alpha\\cdot\\frac\{1\}\{L\}\\sum\_\{l^\{\\prime\}=0\}^\{L\-1\}\\left\|\{s\[l^\{\\prime\}\]\}\\right\|^\{2\}\}\}\\cdot\\mathbf\{s\},\\ \\text\{where\}~s^\{\\prime\}\[l\]=\\sqrt\{\\frac\{1\}\{\\alpha\\cdot\\frac\{1\}\{L\}\\sum\_\{l^\{\\prime\}=0\}^\{L\-1\}\\left\|\{s\[l^\{\\prime\}\]\}\\right\|^\{2\}\}\}\\cdot s\[l\],\\ \\forall l=0,1,\\dots,L\-1\.\(S27\)In this way, the scaled I/Q waveform𝐬′\\mathbf\{s\}^\{\\prime\}with a backoff coefficient ofα\\alphacan tolerate a maximum PAPR of10​log10​α10\\log\_\{10\}\\alpha\(dB\)\. However, plugging in equation \([S24](https://arxiv.org/html/2609.19279#Sx11.E24)\), its SQNR is reduced to

SQNR​\{𝐬′\}=−10​log10​α\+4\.77\+6\.02​b\.\\displaystyle\\textsf\{SQNR\}\\left\\\{\\mathbf\{s\}^\{\\prime\}\\right\\\}=\-10\\log\_\{10\}\\alpha\+4\.77\+6\.02b\.\(S28\)To summarize, a higher PAPR of𝐬\\mathbf\{s\}requires a larger backoff coefficientα\\alphato avoid saturation, which leads to a smaller SQNR\. In addition, a larger backoff coefficientα\\alpharequires a higher amplification gain to compensate for the reduced amplitude of the transmitted I/Q waveform𝐬′\\mathbf\{s\}^\{\\prime\}, which is limited by the hardware constraints of the employed radios \(see Supplementary Text Section[G](https://arxiv.org/html/2609.19279#Sx12.SS7)\)\. Therefore, mitigating the PAPR of𝐬\\mathbf\{s\}is critical to the computing accuracy ofRF\-CNN’s analog computing\.

Mitigating the impact of PAPR using Zadoff\-Chu phases over input channels\.In today’s DL workloads, the input tensor𝐗\\mathbf\{X\}is usually nonnegative for two main reasons: \(*i*\) for the firstconv2dlayer, the input image RGB pixels are non\-negative within the range of\[0,255\]\[0,255\]for the three input channels; and \(*ii*\) for the followingconv2dor FC layers, the input tensor goes through ReLU, which yields nonnegative values\. Such a nonnegative input tensor𝐗\\mathbf\{X\}results in a strong direct current \(DC\) component in the time domain I/Q sample sequence after IFFT, which leads to a high PAPR\. To mitigate this issue, we employ aCinC\_\{\\textrm\{in\}\}\-point Zadoff\-Chu sequence, denoted as𝚽\(ZC\)=\[ϕcin\(ZC\)\]∈\[−π,\+π\]Cin\\mathbf\{\\Phi\}^\{\(\\textrm\{ZC\}\)\}=\\left\[\\phi^\{\(\\textrm\{ZC\}\)\}\_\{c\_\{\\textrm\{in\}\}\}\\right\]\\in\\left\[\-\\pi,\+\\pi\\right\]^\{C\_\{\\textrm\{in\}\}\}, where

ϕ\(ZC\)cin=e−j​π​cin​\(cin\+1\)Cin,∀cin=0,1,…,Cin−1\.\\displaystyle\\phi^\{\(\\textrm\{ZC\}\)\}\_\{c\_\{\\textrm\{in\}\}\}=\{e\}^\{\-\{j\}\\pi\\frac\{c\_\{\\textrm\{in\}\}\(c\_\{\\textrm\{in\}\}\+1\)\}\{C\_\{\\textrm\{in\}\}\}\},\\ \\forall c\_\{\\textrm\{in\}\}=0,1,\\dots,C\_\{\\textrm\{in\}\}\-1\.\(S29\)Specifically, we respectively multiply this Zadoff\-Chu sequence to all theCinC\_\{\\textrm\{in\}\}input channels on the input tensor𝐗\\mathbf\{X\}before mapping to the frequency\-domain tone series; also, we multiply the element\-wise inverse Zadoff\-Chu sequence to all theCinC\_\{\\textrm\{in\}\}weight channels on the weight tensor𝐖\\mathbf\{W\}\. It can be easily verified that this operation does not change the final convolution output𝐘\\mathbf\{Y\}\. On the other hand, this operation effectively spreads the energy of the input tensor𝐗\\mathbf\{X\}over the time domain I/Q sample sequence after IFFT, thereby mitigating the PAPR issue\.

## Supplementary Information: Experiment

### FAdditive White Gaussian Noise Channel based Simulations

We simulateRF\-CNN’s performance on the ML tasks for both classification models and the generative models\.

In the simulation, we employ the additive white Gaussian noise \(AWGN\) channel with no channel distortions, where the computing accuracy is impacted by the Gaussian noise\. Also, we consider an ideal analog multiplication on the computing mixer\. Specifically, given the two waveformss1​\(t\)s\_\{1\}\(t\)ands2​\(t\)s\_\{2\}\(t\), the simulated process of the computing mixer in equation \([S6](https://arxiv.org/html/2609.19279#Sx11.E6)\) can be rewritten as

so​\(t\)=s1​\(t\)⋅s2​\(t\)\+n⁡\(t\),∀t∈\[0,T\)\.\\displaystyle s\_\{o\}\(t\)=s\_\{1\}\(t\)\\cdot s\_\{2\}\(t\)\+n\(t\),~\\forall t\\in\[0,T\)\.\(S30\)wheren⁡\(t\)n\(t\)is a complex\-valued Gaussian noise\. Theoretically, the PSD of the Gaussian noise is given by \-174 dBm/Hz\. However, this PSD is not practical, and is usually much higher than this number due to the noise figure of the radio RX, insertion loss of the frequency mixer, etc\. To align the simulation results with experiments, we define the equivalent noise PSD, denoted aspnp\_\{n\}, to include the factors mentioned above\. This equivalent noise PSD may vary over different convolutional layers and, thus, different ML tasks\.

### GSoftware\-Defined Radio Testbed for Experiments

We implement and evaluateRF\-CNNon a software\-defined radio \(SDR\) testbed, which consists of a central radio that transmits/broadcasts the model weights, and one edge client that performs the analog computing locally, as shown in Fig\.[S4](https://arxiv.org/html/2609.19279#Sx12.F4)\(a\)\. There are two main setups for our experiments: \(*i*\) a wired setup, where the model weights are delivered via a cable, and \(*ii*\) a wireless setup, where the model weights are delivered over a wireless channel\. Specifically in the wireless setup, the central radio is implemented by a radio transmitter \(𝐖\\mathbf\{W\}\) and a directional antenna; the edge client is comprised of one radio transmitter \(𝐗\\mathbf\{X\}\), one radio receiver \(𝐘\\mathbf\{Y\}\), a frequency mixer for analog computing, and a directional antenna\. In the SDR testbed, we utilize the USRP X310, a high\-performance SDR with a UBX\-160 daughterboard, as the transmitter and receiver in our SDR testbed; the Mini\-Circuits ZEM\-4300\+[Mini\-Circuits \(\)](https://arxiv.org/html/2609.19279#bib.bib6)as the computing frequency mixer; and the Tupavco TP514 Yagi directional antenna for wireless transmission\.

USRP X310 as the radio transmitter\.As a radio transmitter, the USRP X310 can be summarized into three main components: \(*i*\) a pair of I/Q DACs, which supports a sampling rate from 0\.196 MHz to 200 MHz with a resolution of 16 bits; \(*ii*\) an internal frequency mixer for up\-converting the baseband I/Q waveforms to the RF band at the carrier frequency from 10 MHz to 6 GHz; and \(*iii*\) a pair of power amplifiers \(PAs\) with a tunable gain at 0–31\.5 dB, as shown in Fig\.[S4](https://arxiv.org/html/2609.19279#Sx12.F4)\(b\)\. The maximum transmit power of the USRP X310 is approximately \+23\.5 dBm with a 0 dB\-PAPR\-waveform, and is reduced to \+23\.5\-α\\alphadBm given the backoff coefficient ofα\\alpha, as discussed in Supplementary Text Section[E](https://arxiv.org/html/2609.19279#Sx11.SS5)\.

USRP X310 as the radio receiver\.As shown in Fig\.[S4](https://arxiv.org/html/2609.19279#Sx12.F4)\(b\), there are three components when the USRP X310 serves as a radio receiver: \(*i*\) an internal frequency mixer that down\-converts the RF signals with a carrier frequency from 10 MHz to 6 GHz; \(*ii*\) low\-pass filters \(LPF\) with a cutoff frequency of half of the set sampling rate to avoid frequency aliasing; and \(*iii*\) a pair of I/Q ADCs, which supports a sampling rate from 0\.196 MHz to 200 MHz with a resolution of 14 bits\.

ZEM\-4300\+ frequency mixer as the computing mixer\.The Mini\-Circuits ZEM\-4300\+[Mini\-Circuits \(\)](https://arxiv.org/html/2609.19279#bib.bib6)is a passive double\-balanced diode mixer for either frequency down\-conversion or up\-conversion\. InRF\-CNN, we consider the down\-conversion case, where its LO and RF ports are configured as inputs and the IF port as output, as shown in Fig\.[S4](https://arxiv.org/html/2609.19279#Sx12.F4)\(c\)\. For the conventional usage in a communication system, the input power to the LO port is typically around \+7 dBm to drive the diodes much higher than its voltage threshold, where the frequency mixer behaves as an on\-off switch that modulates the RF waveform,sRF​\(t\)s\_\{\\textrm\{RF\}\}\(t\)with a switching pattern determined by the \(conjugated\) LO waveformsLO​\(t\)s\_\{\\textrm\{LO\}\}\(t\)as

sIF​\(t\)∝sgn​\(s¯LO​\(t\)\)⋅sRF​\(t\),\\displaystyle s\_\{\\textrm\{IF\}\}\(t\)\\propto\\textsf\{sgn\}\(\\overline\{s\}\_\{\\textrm\{LO\}\}\(t\)\)\\cdot s\_\{\\textrm\{RF\}\}\(t\),\(S31\)wheresgn​\(⋅\)\\textsf\{sgn\}\(\\cdot\)is the sign function\. InRF\-CNN’s computing usage, the input power to the LO port is reduced to approximately \-3 dBm \(see measurements in Supplementary Text Section[H](https://arxiv.org/html/2609.19279#Sx12.SS8)\), which drives the diodes around the threshold voltage, where its behavior is similar to the analog multiplier that

sIF​\(t\)∝s¯LO​\(t\)⋅sRF​\(t\)\.\\displaystyle s\_\{\\textrm\{IF\}\}\(t\)\\propto\\overline\{s\}\_\{\\textrm\{LO\}\}\(t\)\\cdot s\_\{\\textrm\{RF\}\}\(t\)\.\(S32\)On the other hand, ZEM\-4300\+ supports the input carrier frequency range of 0\.3–4\.3 GHz for the LO and RF ports, and a carrier frequency range of 0–1\.0 GHz for the IF port, which conformsRF\-CNN’s computing usage \(see Supplementary Text Section[H](https://arxiv.org/html/2609.19279#Sx12.SS8)\)\.

Tupavco TP514 Yagi directional antenna\.For the wireless setup, we use a pair of Tupavco TP514 Yagi directional antenna as the TX/RX antenna to establish the wireless link between the central radio and the edge client\. This Yagi antenna is designed for dual frequency bands at 0\.80–0\.96 GHz and 1\.7–2\.5 GHz, covering the ISM band at 0\.915 GHz utilized in our experiments\. In addition, it provides an antenna gain of 9 dBi in the designated direction for both TX and RX\. Given the backoff coefficientα=0\.04\\alpha=0\.04\(tolerating PAPR up to 14 dB without saturation\), and the maximum transmission power of \+23\.5 dBm by USRP X310, the wireless link distance is recommended to be around 1 m to maintain a receiving power of \-3 dBm on the edge client\.

### HFrequency Mixer Computing Accuracy Benchmark

We first benchmark and optimize the computing accuracy of the employed frequency mixer, ZEM\-4300\+[Mini\-Circuits \(\)](https://arxiv.org/html/2609.19279#bib.bib6), over two varying factors: \(*i*\) the carrier frequency to the RF port, and \(*ii*\) the input power levels to the RF \(for𝐗\\mathbf\{X\}\) and LO \(for𝐖\\mathbf\{W\}\) ports\. Note that these benchmarks are performed in the wired setup to avoid the impact of wireless channel impairments\.

General\-purposeconv2doperation setup\.We employ a set of default general\-purposeconv2doperations to benchmark the computing accuracy\. Specifically, we consider aconv2dlayer \(see definition in Supplementary Text Section[A](https://arxiv.org/html/2609.19279#Sx11.SS1)\) with the number of input channelsCin∈\{64,128,256\}C\_\{\\textrm\{in\}\}\\in\\left\\\{64,128,256\\right\\\}, and the number of output channelsCout=1C\_\{\\textrm\{out\}\}=1, and a kernel dimension ofKh=Kw=3K\_\{h\}=K\_\{w\}=3; the input image size is set toIh=Iw=16I\_\{h\}=I\_\{w\}=16\. Note that we vary the number of input channelsCinC\_\{\\textrm\{in\}\}for their different downsampling ratios \(see[Methods](https://arxiv.org/html/2609.19279#Sx9)section\)\. We randomize𝐗\\mathbf\{X\}and𝐖\\mathbf\{W\}with i\.i\.d\. Gaussian distributed amplitudes from𝒩⁡\(0,1\)\\mathcal\{N\}\(0,1\), and uniformly distributed phases from𝒰⁡\(−π,\+π\)\\mathcal\{U\}\(\-\\pi,\+\\pi\)\. This randomization is repeated for 100 times to obtain statistically meaningful results\. In addition, limited by the unlicensed ISM band[\(2020\)](https://arxiv.org/html/2609.19279#bib.bib13), the wireless transmission of𝐖\\mathbf\{W\}is always constrained to the carrier frequency of 915 MHz and a bandwidth of 25 MHz\.

Computing accuracy metrics: normalized RMSE and resolution bit\.To quantify the computing accuracy, we consider two metrics: \(*i*\) the normalized root mean square error \(RMSE\), and \(*ii*\) the equivalent resolution bit\. For the digital computed𝐘\\mathbf\{Y\}, we first normalize it so that its standard deviation equals1/31/3, i\.e\.,𝐘←𝐘/\(3⋅std​\(𝐘\)\)\\mathbf\{Y\}\\leftarrow\\mathbf\{Y\}/\(3\\cdot\\textsf\{std\}\(\\mathbf\{Y\}\)\)\. After this normalization, the real and imaginary components of𝐘\\mathbf\{Y\}mostly fall within the range of\[−1,\+1\]\[\-1,\+1\]\. Then, we normalize the analog computed𝐘^\\widehat\{\\mathbf\{Y\}\}by the same standard deviation as𝐘^←𝐘^/\(3⋅std​\(𝐘\)\)\\widehat\{\\mathbf\{Y\}\}\\leftarrow\\widehat\{\\mathbf\{Y\}\}/\(3\\cdot\\textsf\{std\}\(\\mathbf\{Y\}\)\)\. Hence, we define the normalized RMSE as:

RMSE=𝔼⁡\[1Cout⋅Ih⋅Iw​∑cout=1Cout∑ih=1Ih∑iw=1Iw\|y⁡\[cout,ih,iw\]−y^​\[cout,ih,iw\]\|2\]\.\\displaystyle\\textsf\{RMSE\}=\\sqrt\{\\mathbb\{E\}\\left\[\\frac\{1\}\{C\_\{\\textrm\{out\}\}\\cdot I\_\{h\}\\cdot I\_\{w\}\}\\sum\_\{c\_\{\\textrm\{out\}\}=1\}^\{C\_\{\\textrm\{out\}\}\}\\sum\_\{i\_\{h\}=1\}^\{I\_\{h\}\}\\sum\_\{i\_\{w\}=1\}^\{I\_\{w\}\}\\left\|\{y\[c\_\{\\textrm\{out\}\},i\_\{h\},i\_\{w\}\]\-\\widehat\{y\}\[c\_\{\\textrm\{out\}\},i\_\{h\},i\_\{w\}\]\}\\right\|^\{2\}\\right\]\}\.\(S33\)Next, we define the resolution bit as−log2⁡\(RMSE/2\)\-\\log\_\{2\}\(\\textsf\{RMSE\}/2\), where the division by22comes from𝐘\\mathbf\{Y\}’s range of\[−1,\+1\]\[\-1,\+1\]\.

Benchmark the carrier frequency to RF port\.We vary the carrier frequency of𝐗\\mathbf\{X\}to benchmark the computing accuracy, while fixing that of𝐖\\mathbf\{W\}at 0\.915 GHz of the ISM band\. In the down\-conversion, the carrier frequency of𝐘\\mathbf\{Y\}is the difference between𝐗\\mathbf\{X\}and𝐖\\mathbf\{W\}\. Specifically, the carrier frequency of𝐗\\mathbf\{X\}is varied from 1\.00 GHz to 3\.00 GHz with a step of 0\.05 GHz, while fixing the input PSDs to the RF \(for𝐗\\mathbf\{X\}\) and LO ports \(for𝐖\\mathbf\{W\}\) to \-102 dBm/Hz and \-77 dBm/Hz, respectively\. These correspond to the power levels of \-28 dBm and \-3 dBm over a bandwidth of 25 MHz\. As shown in Fig\.[S5](https://arxiv.org/html/2609.19279#Sx12.F5), the trendency is similar over the three considered numbers of input channelsCin∈\{64,128,256\}C\_\{\\textrm\{in\}\}\\in\\left\\\{64,128,256\\right\\\}\. Specifically forCin=128C\_\{\\textrm\{in\}\}=128, the normalized RMSE is around 0\.056–0\.064 \(approximately 5\-bit resolution\) when the carrier frequency of𝐗\\mathbf\{X\}is set to 1\.00–2\.50 GHz, excluding an abnomality at around 1\.80 GHz where the carrier frequencies of leaked𝐖\\mathbf\{W\}at 0\.915 GHz is overlapped with that of𝐗\\mathbf\{X\}at 0\.885 GHz\. When further increasing the carrier frequency of𝐗\\mathbf\{X\}to 3\.00 GHz, the normalized RMSE gradually increases to around 0\.098 \(approximately 4\.35\-bit resolution\)\.

Benchmark the input power levels to RF and LO ports\.We follow the above\-mentionedconv2doperation setup to benchmark the computing accuracy over varying input power levels to the RF \(𝐗\\mathbf\{X\}\) and LO \(𝐖\\mathbf\{W\}\) ports\. Specifically, we vary the PSD combination of𝐗\\mathbf\{X\}and𝐖\\mathbf\{W\}respectively from−\-116 dBm/Hz to−\-66 dBm/Hz with a step of 1 dB, corresponding to power levels of−\-42 dBm to\+\+8 dBm over the bandwidth of 25 MHz\. Similarly, the computing accuracy trend is similar for the three selected numbers of input channels ofCin∈\{64,128,256\}C\_\{\\textrm\{in\}\}\\in\\left\\\{64,128,256\\right\\\}, as shown in Fig\.[S6](https://arxiv.org/html/2609.19279#Sx12.F6)\. Specifically forCin=128C\_\{\\textrm\{in\}\}=128, an minimized normalized RMSE of around 0\.043 \(approximately 5\.54\-bit resolution\) is achieved when the input power levels to the RF and LO ports are around \-81 dBm/Hz and \-81 dBm/Hz \(corresponding to \-7\.02 dBm power\), respectively\. This indicates the optimal operating point for the diodes in the frequency mixer as an analog multiplier\. InRF\-CNN, the input power level to the LO port is fixed, depending on the central radio’s broadcast power, while that to the RF port is adjustable by the edge client, but is minimized to improve the energy efficiency\. For example, under a fixed LO power of \-77 dBm/Hz \(\-3\.02 dBm\), we have an normalized RMSE of 0\.056 \(resolution bit of 5\.16\) with at the RF power of \-90 dBm/Hz \(\-16\.02 dBm\); when reducing that to \-110 dBm/Hz \(\-36\.02 dBm\), the normalized RMSE is increased to around 0\.165 \(resolution bit of 3\.60\)\.

Default settings for the frequency mixer\.In the implementation ofRF\-CNN, we empirically select the carrier frequency of𝐗\\mathbf\{X\}to be 1\.00 GHz; given the carrier frequency of𝐖\\mathbf\{W\}at 0\.915 GHz, that of𝐘\\mathbf\{Y\}is set to 0\.085 GHz after the down\-conversion\. In addition, we select the input power levels to the LO ports to be \-77 dBm/Hz \(\-3\.02 dBm\)\.

### IGeneral\-Purposeconv2dBenchmarks over Scalability

We benchmark the complex\-valuedconv1dandconv2dlayers with randomized𝐗\\mathbf\{X\}and𝐖\\mathbf\{W\}over varying numbers of input/output channels and PSDs\. Throughout this section, all the elements in𝐗\\mathbf\{X\}and𝐖\\mathbf\{W\}are i\.i\.d\. with a uniformly distributed amplitudes from𝒰⁡\(0,1\)\\mathcal\{U\}\(0,1\)and a uniformly distributed phase from𝒰⁡\(−π,\+π\)\\mathcal\{U\}\(\-\\pi,\+\\pi\)\. Similar to the Supplementary Text Section[H](https://arxiv.org/html/2609.19279#Sx12.SS8), we fix the kernel size of theconv2dlayer toKh×Kw=3×3K\_\{h\}\\times K\_\{w\}=3\\times 3, and the input image size toIh×Iw=16×16I\_\{h\}\\times I\_\{w\}=16\\times 16\. As forconv1dlayers, we configure the kernel size toK=3K=3, and the input image size toI=64I=64\.

Impact over varying number of input/output channels\.We first examine the computing accuracy over a varying number of input and output channels, with a fixed input PSD to the RF at \-77 dBm/Hz\. In particular, we adjust the number of input and output channels equally, settingCin=Cout∈\{20,21,…,210\}C\_\{\\textrm\{in\}\}=C\_\{\\textrm\{out\}\}\\in\\left\\\{2^\{0\},2^\{1\},\\dots,2^\{10\}\\right\\\}\. A detailed comparison regarding the real/imaginary components, amplitude, and phase between the𝐘\\mathbf\{Y\}by full precision and𝐘^\\widehat\{\\mathbf\{Y\}\}byRF\-CNNis shown in Fig\.[S7](https://arxiv.org/html/2609.19279#Sx12.F7)\. Specifically, Fig\.[S7](https://arxiv.org/html/2609.19279#Sx12.F7)\(a\) shows the results ofconv1d, while Fig\.[S7](https://arxiv.org/html/2609.19279#Sx12.F7)\(b\) shows the results ofconv2d\. Over a different number of input/output channels, the normalized RMSE is stable between 0\.031–0\.076 \(approximately 4\.72–6\.01\-bit resolution\) forconv1d, and 0\.048–0\.065 \(approximately 4\.94–5\.49\-bit resolution\) forconv2d\. This illustratesRF\-CNN’s scalability over the number of input/output channels, which is crucial for supporting increasingly large CNN models\.

Impact over varying input PSD\.For each number of input/output channels, we further examine the computing accuracy over varying input PSD to the RF port\. This varying PSD potentially impacts the SNR of the output waveform, and thus the computing accuracy\. Fig\.[S8](https://arxiv.org/html/2609.19279#Sx12.F8)\(a\) shows the results ofconv1d, and Fig\.[S8](https://arxiv.org/html/2609.19279#Sx12.F8)\(b\) shows the results ofconv2d\. For instance, withCin=cout=64C\_\{\\textrm\{in\}\}=c\_\{\\textrm\{out\}\}=64, the normalized RMSE is 0\.326 \(equivalent to 2\.62\-bit resolution\) at a PSD of \-140 dBm/Hz forconv2d\. The computing accuracy improves at a rate of approximately 6\.02 dB/bit as the input PSD decreases, until reaching a PSD of \-115 dBm/Hz, where Gaussian noise becomes the dominant factor affecting accuracy\. Beyond this point, increasing the PSD leads to a saturation in computing accuracy, maintaining an RMSE of approximately 0\.057–0\.063 \(corresponding to a resolution bit of 5\) within the PSD range of \-115—100 dBm/Hz, limited by the imperfections of the frequency mixer\. This trend is similar for the other numbers of input/output channels ranging from202^\{0\}to2102^\{10\}\. Comparing the results ofconv1dandconv2dlayers, theconv1dlayers fluctuate more\.

Energy efficiency with different computing accuracy requirements\.The energy efficiency per MAC ofRF\-CNNis discussed in[Methods](https://arxiv.org/html/2609.19279#Sx9)section\. Specifically, the energy per MAC is given bye=eenc\+edac\+eadc\+edece=e\_\{\\textrm\{enc\}\}\+e\_\{\\textrm\{dac\}\}\+e\_\{\\textrm\{adc\}\}\+e\_\{\\textrm\{dec\}\}, whereedace\_\{\\textrm\{dac\}\}by the DACs is determined by the PSD that impacts the computing accuracy, while the other three terms,eence\_\{\\textrm\{enc\}\}by digital encoding,eadce\_\{\\textrm\{adc\}\}by ADC, andedece\_\{\\textrm\{dec\}\}by digital decoding are constant, and can be scaled down by the computing scale,CinC\_\{\\textrm\{in\}\}andCoutC\_\{\\textrm\{out\}\}\. Fig\.[S9](https://arxiv.org/html/2609.19279#Sx12.F9)and Fig\.[S10](https://arxiv.org/html/2609.19279#Sx12.F10)respectively showcase the energy scaling ofconv1dandconv2dlayers\. Specifically forconv2d, we first show the energy termeence\_\{\\textrm\{enc\}\},eadce\_\{\\textrm\{adc\}\}andedece\_\{\\textrm\{dec\}\}over varying number of input/output channels in Fig\.[S10](https://arxiv.org/html/2609.19279#Sx12.F10)\(a\), whose energy per MAC is reduced from 44\.44 fJ/MAC, 5\.56 fJ/MAC and 44\.44 fJ/MAC to 0\.10 fJ/MAC, 0\.01 fJ/MAC and 0\.04 fJ/MAC whenCin=CoutC\_\{\\textrm\{in\}\}=C\_\{\\textrm\{out\}\}is increased from202^\{0\}to2102^\{10\}\. Then, we show the minimumedace\_\{\\textrm\{dac\}\}to achieve a computing accuracy of 3/4/5 bits and the corresponding total energy per MACeein Fig\.[S8](https://arxiv.org/html/2609.19279#Sx12.F8)\(b\)\. Overall, the termedace\_\{\\textrm\{dac\}\}increases slightly with the number of input/output channels\. Specifically forCin=Cout=210C\_\{\\textrm\{in\}\}=C\_\{\\textrm\{out\}\}=2^\{10\}, the minimumedace\_\{\\textrm\{dac\}\}to achieve a computing accuracy of 3/4/5 bits is 1\.40 aJ/MAC, 8\.83 aJ/MAC and 0\.22 fJ/MAC, respectively; these correspond to the total energy per MACeeof 0\.15 fJ/MAC, 0\.16 fJ/MAC, and 0\.37 fJ/MAC\. The total energy per MACeecan be scaled up by increasing the computing scale, and will converge to the termedace\_\{\\textrm\{dac\}\}\. To conclude, the energy efficiency ofRF\-CNNis approximately 1–3 orders of magnitude higher than the state\-of\-the\-art digital computing with 100 fJ/MAC\.

### JComplex\-Valued CNN\-based Classification and Generative Models

The CNN models, both for classification and the generation, are trained digitally, and then directly evaluated inRF\-CNN’s analog computing on the software\-defined radio testbed\. Note that all parameters in these models are complex\-valued, in order to showcaseRF\-CNN’s capability to support complex MAC operations\. A summary of the six models is presented in Fig\.[S11](https://arxiv.org/html/2609.19279#Sx12.F11)and Table[S1](https://arxiv.org/html/2609.19279#Sx12.T1)\.

Complex\-valued layers\.There are mainly two types of layers used in the CNN models inRF\-CNN: \(*i*\) linear layers, including theconv2dlayers and the fully\-connected layers \(see Supplementary Text Section[A](https://arxiv.org/html/2609.19279#Sx11.SS1)\), and \(*ii*\) non\-linear layers, such as BatchNorm, ReLU/LeakyReLU, max pooling and etc\. These non\-linear layers are implemented by extending their real\-valued definitions: two parallel layers are defined respectively for the real and imaginary components\. For example, there are two sets of mean and variance parameters in the BatchNorm layer, one for the real component and the other for the imaginary component\.

6\-layer CNN models for DeepSig classification\.DeepSig[O’Shea et al\. \(2018\)](https://arxiv.org/html/2609.19279#bib.bib14)is a wireless signal dataset for modulation classification\. Originally, there are 28 classes of modulation types, but we pick and merge them into 10 representative classes, which are

- •*OOK*: on\-off keying\.
- •*ASK*: 4/8\-amplitude shift keying\.
- •*BPSK*: binary phase shift keying\.
- •*QPSK*: quadrature phase shift keying\.
- •*PSK*: 8/16/32/64/128\-phase shift keying\.
- •*QAM*: 16/32/64/128/256\-quadrature amplitude modulation\.
- •*AM\-SSB*: amplitude modulation – single sideband, either with carrier or carrier suppressed\.
- •*AM\-DSB*: amplitude modulation – double sideband, either with carrier or carrier suppressed\.
- •*FM*: frequency modulation\.
- •*GMSK*: gaussian minimum\-shift keying\.

All of these wireless signals are represented as complex\-valued 1,024\-point vectors\. The dataset provides wireless signals with SNRs varying from \-20 dB to \+30 dB\. We train the classification model on the signals with SNRs of 10–30 dB, and test the model using the 30 dB signals only\. We consider one\-shot, three\-shot, and five\-shot classifications, which infer 1/3/5 different wireless signals of the same modulation, and vote for the final prediction\. This multi\-shot setup is based on the fact that a wireless signal can exist for a longer time than the vector length provided in the dataset\. Given the one\-dimensional input and the complex\-valued nature of the signals, we build a 6\-layer\-CNN model with fiveconv1dlayers \(namely,Conv1toConv5\) and one FC layer \(namely,FC1\), where the kernel size of allconv1dlayers is33, and the number of channels are1→256→256→512→512→5121\\rightarrow 256\\rightarrow 256\\rightarrow 512\\rightarrow 512\\rightarrow 512; eachconv1dlayer is followed by a BatchNorm layer, a ReLU layer, and a max pooling layers with a kernel size of44that shrinks the signals’ dimension by44\. The final FC layer maps the feature into a 10\-point complex\-valued vector for the 10\-classification task\. To summarize, this 6\-layer CNN model has a total of 2\.18 M parameters and 368\.1 M MACs\.

9\-layer CNN models for SVHN and CIFAR\-10 classification\.The SVHN[Netzer et al\. \(2011\)](https://arxiv.org/html/2609.19279#bib.bib3)dataset is composed of32×3232\\times 32RGB images of house numbers \(“0” to “9”\), and the CIFAR\-10 dataset is composed of32×3232\\times 32RGB images of ten classes of objects \(plane, car, bird, cat, deer, dog, frog, horse, ship, and truck\)\. Given the same dimensions of these two datasets, we tailor the standard VGG11 architecture[Simonyan and Zisserman \(2015\)](https://arxiv.org/html/2609.19279#bib.bib45)to build a 9\-layer CNN model architecture that is applied to the image classification tasks on both datasets\. Specifically, the 9\-layer CNN model consists of eightconv2dlayers \(namely,Conv1toConv8\) and one FC layer \(namely,FC1\), where the kernel size of allconv2dlayers, and the number of channels are3→64→128→256→256→512→512→512→5123\\rightarrow 64\\rightarrow 128\\rightarrow 256\\rightarrow 256\\rightarrow 512\\rightarrow 512\\rightarrow 512\\rightarrow 512; eachconv2dlayer is followed by a 2D BatchNorm layer, a ReLU layer, andConv1/2/4/6/8layers have additional 2D max pooling layers with a kernel size of2×22\\times 2and a stride of22\. Over these eightconv2dlayers, the input image with a size of3×32×323\\times 32\\times 32is converted into a feature of512×1×1512\\times 1\\times 1, which is then flattened as a one\-dimensional vector into theFC1layer that projects it into a 10\-point complex\-valued vector\. Finally, we extract the real components of the output vector as the model’s prediction of the input image being ten classes\. To summarize, this 9\-layer CNN model architecture has a total of 9\.22 M parameters and 611\.1 M MACs\.

4\-layer CNN model for the MNIST and FMNIST generation\.MNIST[LeCun et al\. \(2002\)](https://arxiv.org/html/2609.19279#bib.bib8)dataset contains28×2828\\times 28grayscale images of handwritten digits \(“0” to “9”\), and FMNIST[Xiao et al\. \(2017\)](https://arxiv.org/html/2609.19279#bib.bib9)dataset contains28×2828\\times 28grayscale images of ten classes of fashion clothes \(coat, boot, shirt, sneaker, pullover, bag, scandal, dress, trousers, and T\-shirt\)\. Similarly, we employ the same model architecture to generate fake images for both datasets following InfoGAN[Chen et al\. \(2016\)](https://arxiv.org/html/2609.19279#bib.bib7)\. The generator is a complex\-valued 4\-layer CNN model with one FC layer \(FC1\) and threeconv2dlayers \(Conv1toConv3\)\. It takes a 74\-point real\-valued vector as input, including a 62\-point random seed vector, a 10\-point one\-hot vector to represent the image’s being ten classes, and a 2\-point latent code vector continuously ranging from 0 to 1 to represent the image’s hidden features, such as the rotation and thickness of the digits or fashion products\. This input is first projected by theFC1layer into a 6,272\-point complex\-valued vector, which is then reshaped into a feature of128×7×7128\\times 7\\times 7\. Then, this feature is processed by threeconv2dlayers with a kernel size of3×33\\times 3with a channel number of128→512128\\rightarrow 512,128→256128\\rightarrow 256, and64→164\\rightarrow 1, respectively\. TheConv1andConv2layers are followed by PixelShuffle, 2D BatchNorm, and LeakyReLU with a negative slope of0\.20\.2\. Note that the PixelShuffle layers perform 2D up\-sampling by2×22\\times 2while reducing the channel number by44\. TheConv3layer is followed by one 2D BatchNorm, and performs Tanh as the output activation function on the real component only, which converts the output into the range of\[−1,\+1\]\[\-1,\+1\]\. In summary, this 4\-layer CNN generative model has a total of 1\.35 M parameters and 350\.5 M MACs\. The discriminator is a real\-valued 5\-layer CNN model with fourconv2dlayers and one FC layer, which \(*i*\) distinguishes the real images from the fake images by the generator, \(*ii*\) classifies the input images into ten classes, and \(*iii*\) estimates the latent code vector of the input images\. The fourconv2dlayers have a kernel size of3×33\\times 3and2×22\\times 2stride, and the channel numbers are1→16→32→64→1281\\rightarrow 16\\rightarrow 32\\rightarrow 64\\rightarrow 128; they are all followed by 2D BatchNorm, LeakyReLU with a negative slope of0\.20\.2, and Dropout with a probability of 25%\. The last FC layer projects the feature into a 14\-point real\-valued vector composed of a 2\-point vector indicating the input image’s being real or fake, a 10\-point vector indicating the input image’s being one of ten classes, and a 2\-point vector indicating the input image’s latent code vector\.

7\-layer CNN model for CelebA generation\.We further employ a complex\-valued 7\-layer CNN model to generate fake images of RGB human faces with128×128128\\times 128based on the CelebA[Liu et al\. \(2015\)](https://arxiv.org/html/2609.19279#bib.bib10)dataset\. The input is still a 74\-point real\-valued vector, where the 10\-point one\-hot vector for the label is randomly generated\. The firstFC1layer projects the input into a 16,384\-point complex\-valued vector, which is reshaped into a feature of1024×8×81024\\times 8\\times 8\. Then, this feature is processed by sixconv2dlayers with a kernel size of3×33\\times 3and a channel number of1024→20481024\\rightarrow 2048,512→1024512\\rightarrow 1024,256→512256\\rightarrow 512,128→256128\\rightarrow 256,64→3264\\rightarrow 32and32→332\\rightarrow 3, respectively\. TheConv1toConv5layers are followed by PixelShuffle, 2D BatchNorm and LeakyReLU with a negative slope of0\.20\.2, while the lastConv6layer is followed by one 2D BatchNorm and performs Tanh as the output activation function on the real component only\. This 7\-layer\-CNN generator has a total of 26\.4 M parameters and 5,195 M MACs\. The discriminator is a real\-valued 6\-layer CNN model with fiveconv2dlayers and one FC layer\. The fiveconv2dlayers have a kernel size of3×33\\times 3and2×22\\times 2stride, and the channel numbers are3→64→128→256→512→10243\\rightarrow 64\\rightarrow 128\\rightarrow 256\\rightarrow 512\\rightarrow 1024; they are all followed by 2D BatchNorm, LeakyReLU with a negative slope of0\.20\.2, and Dropout with probability of 25%\. The last FC layer projects the feature into a 14\-point real\-valued vector with the same composition as the discriminator for MNIST and FMNIST\.

Model training\.The two classification models for SVHN and CIFAR\-10 are trained for 200 epochs using SGD[Bottou \(2010\)](https://arxiv.org/html/2609.19279#bib.bib12)with a learning rate of10−210^\{\-2\}and a weight decay of10−210^\{\-2\}\. Over the epochs, we select the model with the best classification accuracy as the final model for evaluation inRF\-CNN’s analog computing\. The three generative models for SVHN, CIFAR\-10 and CelebA are trained with the InfoGAN loss[Chen et al\. \(2016\)](https://arxiv.org/html/2609.19279#bib.bib7)for 200 epochs using Adam[Kingma and Ba \(2015\)](https://arxiv.org/html/2609.19279#bib.bib11)with a learning rate of2×10−42\\times 10^\{\-4\}until the loss converges\. These training processes are all performed on a single NVIDIA A100 GPU\.

### KWaveform Amplitude Selection against PAPR

As discussed in Supplementary Text Section[E](https://arxiv.org/html/2609.19279#Sx11.SS5), balancing the trade\-off between PAPR and SQNR, a\.k\.a\. selecting the backoff coefficientα\\alpha, is crucial forRF\-CNN’s computing accuracy\. Hereby, we benchmark the computing accuracy over varyingα\\alphato select a properα\\alphafor eachconv1d,conv2d, and FC layer in the employed CNN models\. At the same time, we identify the key layers that are particularly sensitive to computing errors and replace them fromRF\-CNN’s analog computing to full precision digital computing if the errors exceed acceptable levels\.

Definition of the single\-layer and end\-to\-end inference RMSEs\.We define the single\-layer and end\-to\-end inference RMSEs as the metrics to benchmark the computing accuracy on these CNN models\. When examining thell\-th layer, we feed the input𝐗\\mathbf\{X\}of this layer by inferring the previousl−1l\-1layers in full precision\. Then, we infer thell\-th layer byRF\-CNN\(𝐘^\\widehat\{\\mathbf\{Y\}\}\) and full precision \(𝐘\\mathbf\{Y\}\)\. The RMSE following equation \([S33](https://arxiv.org/html/2609.19279#Sx12.E33)\) between𝐘^\\widehat\{\\mathbf\{Y\}\}and𝐘\\mathbf\{Y\}is defined as the single\-layer inference RMSE\. By comparing to the randomized single layer in Supplementary Text Section[I](https://arxiv.org/html/2609.19279#Sx12.SS9), this metric quantifies how the real𝐗\\mathbf\{X\}and𝐖\\mathbf\{W\}to thell\-th layer impact the computing accuracy\. Then, we infer the resting layers after thell\-th layer in full precision with𝐘^\\widehat\{\\mathbf\{Y\}\}and𝐘\\mathbf\{Y\}, respectively\. The RMSE between the final outputs of the whole CNN model is defined as the end\-to\-end inference RMSE\. This end\-to\-end inference RMSE provides a fair comparison between the sensitivity of different layers\. In the rest of this section, all the benchmark measurements are based on three signals/images in the dataset, and the optimizedα\\alphais then applied to the whole testing dataset\.

Benchmark the backoff coefficient for 6\-layer CNN model for the DeepSig classification\.We first benchmark the backoff coefficientα\\alphausing the end\-to\-end inference RMSE for the 6\-layer CNN model for the DeepSig classification\. Fig\.[S12](https://arxiv.org/html/2609.19279#Sx12.F12)shows the end\-to\-end RMSE as a function of the backoff coefficientα\\alphafor each layer in the 6\-layer CNN model for the DeepSig classification\. TheConv1layer encounters the highest end\-to\-end RMSE \(\>0\.30\>0\.30\) over all the choices ofα\\alpha\. This is because of the highest PAPR of its input wireless signals, and becauseCin=1C\_\{\\textrm\{in\}\}=1, where the Zadoff\-Chu phases cannot apply \(see Supplementary Text Section[E](https://arxiv.org/html/2609.19279#Sx11.SS5)\)\. Therefore, we infer theConv1layer in digital full precision\. Fortunately, thisConv1layer only occupies 0\.85% of the MACs, so its digital inference does not largely change the overall energy efficiency \(or energy per MAC\) of the whole model’s inference\. For the followingConv2layer, the RMSE first drops because of its better robustness to overcome the saturation issue, and then increases because of a too low SQNR\. In observation of this, we select theα\\alphaas 34 dB to achieve the minimum RMSE\. Then, the RMSE gradually decreases over the layers\. This is because a largerCinC\_\{\\textrm\{in\}\}enables the Zadoff\-Chu phases to reduce the PAPR more effectively, and a smallerα\\alphais good enough to avoid the saturation\.

Benchmark the backoff coefficient for 9\-layer CNN model for the SVHN/CIFAR\-10 classification\.We then benchmark the 9\-layer CNN model for the SVHN/CIFAR\-10 classification\. The end\-to\-end inference RMSE on the SVHN classification is shown in Fig\.[S13](https://arxiv.org/html/2609.19279#Sx12.F13)\. Most layers \(Conv1/2/3/4\) have decreasing end\-to\-end inference RMSE with the increasingα\\alpha, which is expected as a largerα\\alphatolerates a higher PAPR without the saturation issue\. Also, the optimized RMSE decreases over the layers\. This is because the later layers have a largerCinC\_\{\\textrm\{in\}\}, and thus the Zadoff\-Chu phases can alleviate the PAPR issue more effectively; to avoid saturation, a smallerα\\alphais required, which leads to a better SQNR and thus a higher computing accuracy\. A similar trend can be found in the model for CIFAR\-10 classification, as shown in Fig\.[S14](https://arxiv.org/html/2609.19279#Sx12.F14)\.

Benchmark the backoff coefficient for the generative models\.We further benchmark the backoff coefficientα\\alphafor the three generative models\. Similarly, most layers have a decreasing end\-to\-end inference RMSE with the increasingα\\alpha, and then increase\. Different from the three models for classifications, the generative models have a decreasing number of channels over the layers, which leads to a reverse trend of the end\-to\-end inference RMSE over the layers\. For example, in the 7\-layer CNN model for CelebA generation, the end\-to\-end inference RMSE ofFC1is 0\.054, while that of theConv6is increased to 0\.200\. To maintain a good model inference accuracy, we performConv6in digital full precision\.

The single\-layer RMSE measurements\.After picking the optimalα\\alphafor all the layers in all the models, we showcase the comparison between the full precision output andRF\-CNNoutput, as well as the single\-layer RMSE of the six CNN models in Fig\.[S18](https://arxiv.org/html/2609.19279#Sx12.F18)\(DeepSig classification\), Fig\.[S19](https://arxiv.org/html/2609.19279#Sx12.F19)\(SVHN classification\), Fig\.[S20](https://arxiv.org/html/2609.19279#Sx12.F20)\(CIFAR\-10 classification\), Fig\.[S21](https://arxiv.org/html/2609.19279#Sx12.F21)\(MNIST generation\), Fig\.[S22](https://arxiv.org/html/2609.19279#Sx12.F22)\(FMNIST generation\), and Fig\.[S23](https://arxiv.org/html/2609.19279#Sx12.F23)\(CelebA generation\)\. Note that the digital inferred layers \(Conv1in the DeepSig classification, andConv6in the CelebA generation\) are skipped\. Generally, the distribution of these DL outputs follows specific patterns, unlike that in the scalability measurement in Supplementary Text Section[I](https://arxiv.org/html/2609.19279#Sx12.SS9)that follows a Gaussian distribution\. Therefore, the corresponding single\-layer RMSE can be different\.

### LPower Spectral Density Trade\-off on the CNN Models for Classification

InRF\-CNN, the PSD of𝐗\\mathbf\{X\}determines the SNR, and therefore the computing accuracy\. This further affects the overall energy efficiency ofRF\-CNNas a trade\-off\. We provide additional results by simulations and experiments across a larger PSD or energy efficiency range for the six CNN models\.

Single/multi\-shot classification accuracy over PSD for DeepSig\.Due to the continuity of the wireless signals in the DeepSig dataset, we can infer multiple signals of the same modulation type to improve the classification accuracy\. Hereby, we consider the one\-shot, three\-shot, and five\-shot classifications, which infer 1/3/5 different wireless signals of the same modulation, and vote for the final prediction, as shown in Fig\.[S24](https://arxiv.org/html/2609.19279#Sx12.F24), Fig\.[S25](https://arxiv.org/html/2609.19279#Sx12.F25), and Fig\.[S26](https://arxiv.org/html/2609.19279#Sx12.F26), respectively\. Specifically for the one\-shot classification in Fig\.[S24](https://arxiv.org/html/2609.19279#Sx12.F24)\(a\), the simulated and experimental results byRF\-CNNexhibit a consistent trend as the PSD varies\. At very low PSD values \(e\.g\., below approximately \-125 dBm/Hz\), the classification accuracy remains below 20%, indicating thatRF\-CNNis heavily corrupted by the Gaussian noises\. As the PSD increases to \-112 dBm/Hz \(3\.34 fJ/MAC\), the experimental classification accuracy is increased to 87\.6%; The classification accuracy is further increased up to 88\.7% at the PSD of \-107 dBm/Hz \(\(4\.48 fJ/MAC\)\), only 4\.2% gap from the full\-precision accuracy of 92\.9%, as shown in Fig\.[S24](https://arxiv.org/html/2609.19279#Sx12.F24)\(b\)\. A similar trend can be observed for the three\-shot and five\-shot classifications, as shown in Fig\.[S25](https://arxiv.org/html/2609.19279#Sx12.F25)and Fig\.[S26](https://arxiv.org/html/2609.19279#Sx12.F26), respectively\. The experimental classification accuracy at \-107 dBm/Hz \(\(4\.48 fJ/MAC\)\) is improved to 92\.8% and 95\.2% by inferring more signals of the same modulation type\.

Classification accuracy over PSD for SVHN and CIFAR\-10\.We then evaluate the classification accuracy over varying PSD for the 9\-layer CNN model for the SVHN and CIFAR\-10 classification\. Fig\.[S27](https://arxiv.org/html/2609.19279#Sx12.F27)shows the results for the SVHN classification, where the experimental classification accuracy is around 93\.3% at the PSD of \-117 dBm/Hz \(0\.76 fJ/MAC\), only 1\.4% gap from the full\-precision accuracy of 94\.7%\. A similar trend can be observed for the CIFAR\-10 classification, as shown in Fig\.[S28](https://arxiv.org/html/2609.19279#Sx12.F28), where the experimental classification accuracy is around 90\.0% at the PSD of \-112 dBm/Hz \(0\.88 fJ/MAC\), only 0\.9% gap from the full\-precision accuracy of 90\.9%\.

### MPower Spectral Density Trade\-off on the CNN Models for Generation

We evaluate the trade\-off between the input PSD and the FID of the generated images by the three generative models on the MNIST, FMNIST and CelebA datasets, respectively\.

Frechet inception distance \(FID\) as the metric for generative images\.FID is a widely used metric to evaluate the quality of generated images, which maps a set of real and fake images into a feature space by a pre\-trained Inception network, and then measures the distance between the distribution of the real and fake images\. Generally, a smaller FID indicates a higher fidelity of the generated images, and thus a better performance of the generative model\.

PSD\-FID trade\-off on MNIST and FMNIST datasets\.We evaluate the trade\-off between the PSD \(or energy efficiency\) and the FID of the generated images by the four\-layer generative models on the MNIST and FMNIST datasets\. These FIDs are calculated across 100 generated images with 10 images per label\. Fig\.[S29](https://arxiv.org/html/2609.19279#Sx12.F29)\(a\) shows the trade\-off on the MNIST dataset, where the FID is around 81\.33 when PSD is \-102 dBm/Hz \(2\.89 fJ/MAC\), while the FID is increased to around 108\.29 when PSD is reduced to around \-117 dBm/Hz \(1\.11 fJ/MAC\)\. We further show examples of digit 1–9 in Fig\.[S29](https://arxiv.org/html/2609.19279#Sx12.F29)\(b\) at the PSD of \-132– \-102 dBm/Hz at the step of 5 dB, corresponding to the FIDs of 81\.33–295\.04\. Similarly, Fig\.[S30](https://arxiv.org/html/2609.19279#Sx12.F30)\(a\) shows the trade\-off on the FMNIST dataset, where the FID is around 170\.97 when PSD is \-102 dBm/Hz \(2\.89 fJ/MAC\), while the FID is increased to around 188\.15 when PSD is reduced to around \-117 dBm/Hz \(1\.11 fJ/MAC\)\. We further show examples of the ten classes of fashion clothes in Fig\.[S30](https://arxiv.org/html/2609.19279#Sx12.F30)\(b\) at the PSD of \-132 – \-102 dBm/Hz at the step of 5 dB, corresponding to the FIDs of 170\.97–346\.73\.

PSD\-FID trade\-off on CelebA dataset\.We repeat the same experiments for the 7\-layer CNN model on the CelebA dataset to generate fake human faces of 128×\\times128 with RGB\. Each reported FID is calculated across 100 generated images\. Due to the higher complexity of the model and dataset, the digital FID at full precision is 112\.55\. In comparison, the FID is around 183\.22 when PSD is \-102 dBm/Hz \(2\.77 fJ/MAC\), while the FID is increased to around 206\.83 when PSD is reduced to around \-112 dBm/Hz \(1\.17 fJ/MAC\)\. We further show examples of the generated human faces in Fig\.[S31](https://arxiv.org/html/2609.19279#Sx12.F31)\(b\) at the PSD of \-132 – \-102 dBm/Hz at the step of 5 dB, corresponding to the FIDs of 183\.22 – 393\.33\.

![Refer to caption](https://arxiv.org/html/2609.19279v1/conv_algorithm.png)Figure S1:The tone mapping algorithm of RF\-CNN to achieve conv2d layers\.a–b, For the single\-input/output\-channel case, the tone series of𝐗\\mathbf\{X\}is the flattened 2D image, and that of𝐖\\mathbf\{W\}is aligned in rows; the correlation result of each alignment generates one output element of𝐘\\mathbf\{Y\}\.c, For the multiple\-input\-channel case, the tone series of𝐗\\mathbf\{X\}and𝐖\\mathbf\{W\}are aligned for each channel, and are therefore summed up after the correlation\.d, For the multiple\-output\-channel case, the inference is performed independently for each output channel using the same tone series of𝐗\\mathbf\{X\}\.![Refer to caption](https://arxiv.org/html/2609.19279v1/supplementary_convolution_diagram.png)Figure S2:The hardware diagrams for the frequency mixer\-driven convolutions\.a, The hardware diagram for the full convolutions from the input vectors𝐒1\\mathbf\{S\}\_\{1\}and𝐒2\\mathbf\{S\}\_\{2\}to the output𝐒o\\mathbf\{S\}\_\{o\}\.b, The hardware diagram for the partial convolution, whose output𝐒↓\\mathbf\{S\}\_\{\\downarrow\}is a segment of𝐒o\\mathbf\{S\}\_\{o\}\.c, The hardware diagram with the pre\- and post\-channel calibration, which compensates the physical channel \(wired or wireless\) distortion𝐂1\\mathbf\{C\}\_\{1\},𝐂2\\mathbf\{C\}\_\{2\}, and𝐂o\\mathbf\{C\}\_\{o\}\.![Refer to caption](https://arxiv.org/html/2609.19279v1/phase_offset.png)Figure S3:During the wireless transmission, the phase offset on the output packet\.a, The packet structure ofRF\-CNNwith preamble/postamble symbols before and after the payload in time\.b, After conducting the FFT for each symbol, the time\-domain output packet becomes a time\-frequency representation\.c, The phase offset on the output packet in the time\-frequency representation is linear over time and frequency\.![Refer to caption](https://arxiv.org/html/2609.19279v1/supplementary_experiement_setup.png)Figure S4:The experiment setup on the software\-defined radio testbed\.a, InRF\-CNN’s implementation with a central radio broadcasting ML model \(𝐖\\mathbf\{W\}\) and an edge client performing local ML inference from ML input𝐗\\mathbf\{X\}to the ML output𝐘\\mathbf\{Y\}using a frequency mixer\.b, The hardware diagram of a USRP X310, which is equivalent to a DAC as a transmitter, and a low\-pass filter followed by an ADC as a receiver\.c, The employed ZEM\-4300\+ as the computing mixer, including an LO port receiving wireless𝐖\\mathbf\{W\}, an RF port receiving wired𝐗\\mathbf\{X\}, and an IF port sending𝐘\\mathbf\{Y\}\.![Refer to caption](https://arxiv.org/html/2609.19279v1/supplementary_mixer_carrier.png)Figure S5:The computing accuracy measurement over different carrier frequencies of𝐗\\mathbf\{X\}with that of𝐖\\mathbf\{W\}fixed at 0\.915 GHz, where we select 1\.0 GHz \(red star\) in RF\-CNN’s implementation\.a, The normalized RMSE with the number of input channelsCin∈\{64,128,256\}C\_\{\\textrm\{in\}\}\\in\\left\\\{64,128,256\\right\\\}\.b, The computing accuracy in ENOB with the number of input channelsCin∈\{64,128,256\}C\_\{\\textrm\{in\}\}\\in\\left\\\{64,128,256\\right\\\}\.![Refer to caption](https://arxiv.org/html/2609.19279v1/supplementary_both_power.png)Figure S6:The computing accuracy measurement over different input PSD to the LO port \(𝐖\\mathbf\{W\}\) and the RF port \(𝐗\\mathbf\{X\}\), where we fix the PSD of𝐖\\mathbf\{W\}to \-77 dBm/Hz \(red dashed line\)\.a, The normalized RMSE with the number of input channelsCin∈\{64,128,256\}C\_\{\\textrm\{in\}\}\\in\\left\\\{64,128,256\\right\\\}\.b, The computing accuracy in ENOB with the number of input channelsCin∈\{64,128,256\}C\_\{\\textrm\{in\}\}\\in\\left\\\{64,128,256\\right\\\}\.![Refer to caption](https://arxiv.org/html/2609.19279v1/supplementary_scalability_scatter.png)Figure S7:The comparison between the digital full precision output \(torch\.complex64in PyTorch\) andRF\-CNN’s output\. Specifically, we break down the complex values into real/imaginary components, or amplitudes and phases; we further vary the number of input channelsCin∈\{1,2,4,8,…,1024\}C\_\{\\textrm\{in\}\}\\in\\left\\\{1,2,4,8,\\dots,1024\\right\\\}\.![Refer to caption](https://arxiv.org/html/2609.19279v1/supplementary_scalability_energy.png)Figure S8:The computing accuracy measurement over the PSD of𝐗\\mathbf\{X\}with the number of input channelsCin∈\{1,2,4,8,16,32,64,128,256,512,1024\}C\_\{\\textrm\{in\}\}\\in\\left\\\{1,2,4,8,16,32,64,128,256,512,1024\\right\\\}ofconv1d\(a\) andconv2d\(b\)\.![Refer to caption](https://arxiv.org/html/2609.19279v1/supplementary_scalability_scale_1D.png)Figure S9:For conv1d layers, RF\-CNN’s energy efficiency per MAC over different numbers of input/output channels\.a, The scaling of the energy efficiency terms \(eence\_\{\\textrm\{enc\}\}for encoding,eadce\_\{\\textrm\{adc\}\}for ADC sampling, andedece\_\{\\textrm\{dec\}\}for decoding\) over the number of channels\.b, The minimum energy per MAC by the DACsedace\_\{\\textrm\{dac\}\}to achieve 3/4/5 ENOBs, and the corresponding total energy per MAC ofRF\-CNN\.![Refer to caption](https://arxiv.org/html/2609.19279v1/supplementary_scalability_scale.png)Figure S10:For conv2d layers, RF\-CNN’s energy efficiency per MAC over different numbers of input/output channels\.a, The scaling of the energy efficiency terms \(eence\_\{\\textrm\{enc\}\}for encoding,eadce\_\{\\textrm\{adc\}\}for ADC sampling, andedece\_\{\\textrm\{dec\}\}for decoding\) over the number of channels\.b, The minimum energy per MAC by the DACsedace\_\{\\textrm\{dac\}\}to achieve 3/4/5 ENOBs, and the corresponding total energy per MAC ofRF\-CNN\.![Refer to caption](https://arxiv.org/html/2609.19279v1/supplementary_model_architecture.png)Figure S11:The four CNN model architectures employed in RF\-CNN’s evaluations on six ML tasks\.a, the 6\-layer CNN model withconv1dlayers for the wireless signal modulation classification task on the DeepSig dataset\.b, the 9\-layer CNN model withconv2dfor both the RGB digit classification task on the SVHN dataset, and the object classification task on the CIFAR\-10 dataset\.c, the 4\-layer CNN model withconv2dtrained from InfoGAN[Chen et al\. \(2016\)](https://arxiv.org/html/2609.19279#bib.bib7)to generate gray images of digits \(MNIST\) and fashion clothes \(FMNIST\)\.d, the 7\-layer CNN model withconv2dtrained from InfoGAN to generate RGB images of human faces \(CelebA\)\.![Refer to caption](https://arxiv.org/html/2609.19279v1/supplementary_wave_amp_DeepSig.png)Figure S12:The backoff coefficientα\\alphameasurement per layer for the 6\-layer CNN model for DeepSig classification, where the red stars refer to the selectedα\\alpha\. Note that the end\-to\-end RMSE inConv1is too large, and is inferred in digital full precision\.![Refer to caption](https://arxiv.org/html/2609.19279v1/supplementary_wave_amp_SVHN.png)Figure S13:The backoff coefficientα\\alphameasurement per layer for the 9\-layer CNN model for SVHN classification, where the red stars refer to the selectedα\\alpha\.![Refer to caption](https://arxiv.org/html/2609.19279v1/supplementary_wave_amp_CIFAR-10.png)Figure S14:The backoff coefficientα\\alphameasurement per layer for the 9\-layer CNN model for CIFAR\-10 classification, where the red stars refer to the selectedα\\alpha\.![Refer to caption](https://arxiv.org/html/2609.19279v1/supplementary_wave_amp_MNIST.png)Figure S15:The backoff coefficientα\\alphameasurement per layer for the 4\-layer CNN model for MNIST generation, where the red stars refer to the selectedα\\alpha\.![Refer to caption](https://arxiv.org/html/2609.19279v1/supplementary_wave_amp_FMNIST.png)Figure S16:The backoff coefficientα\\alphameasurement per layer for the 4\-layer CNN model for FMNIST generation, where the red stars refer to the selectedα\\alpha\.![Refer to caption](https://arxiv.org/html/2609.19279v1/supplementary_wave_amp_CelebA.png)Figure S17:The backoff coefficientα\\alphameasurement per layer for the 7\-layer CNN model for CelebA generation, where the red stars refer to the selectedα\\alpha\. Note that the end\-to\-end RMSE inConv3is too large, and is inferred in digital full precision\.![Refer to caption](https://arxiv.org/html/2609.19279v1/supplementary_scatter_DL_deepsig.png)Figure S18:The per\-layer comparison between the full precision output andRF\-CNNoutput regarding real/imaginary/amplitude/angle on the 6\-layer CNN model for the DeepSig classification\.![Refer to caption](https://arxiv.org/html/2609.19279v1/supplementary_scatter_DL_SVHN.png)Figure S19:The per\-layer comparison between the full precision output andRF\-CNNoutput regarding real/imaginary/amplitude/angle on the 6\-layer CNN model for the SVHN generation\.![Refer to caption](https://arxiv.org/html/2609.19279v1/supplementary_scatter_DL_CIFAR-10.png)Figure S20:The per\-layer comparison between the full precision output andRF\-CNNoutput regarding real/imaginary/amplitude/angle on the 6\-layer CNN model for the CIFAR\-10 generation\.![Refer to caption](https://arxiv.org/html/2609.19279v1/supplementary_scatter_DL_MNIST.png)Figure S21:The per\-layer comparison between the full precision output andRF\-CNNoutput regarding real/imaginary/amplitude/angle on the 6\-layer CNN model for the MNIST generation\.![Refer to caption](https://arxiv.org/html/2609.19279v1/supplementary_scatter_DL_FMNIST.png)Figure S22:The per\-layer comparison between the full precision output andRF\-CNNoutput regarding real/imaginary/amplitude/angle on the 6\-layer CNN model for the FMNIST generation\.![Refer to caption](https://arxiv.org/html/2609.19279v1/supplementary_scatter_DL_CelebA.png)Figure S23:The per\-layer comparison between the full precision output andRF\-CNNoutput regarding real/imaginary/amplitude/angle on the 7\-layer CNN model for the CelebA generation\.![Refer to caption](https://arxiv.org/html/2609.19279v1/supplementary_classification_energy_deepsig_1shot.png)Figure S24:The evaluation results of the 6\-layer CNN model for the one\-shot DeepSig classification\.a, The classification accuracy over PSD of𝐗\\mathbf\{X\}by experiments and simulations\.b, The confusion matrices of three selected points at \-107/\-112/\-117 dBm/Hz\.![Refer to caption](https://arxiv.org/html/2609.19279v1/supplementary_classification_energy_deepsig_3shot.png)Figure S25:The evaluation results of the 6\-layer CNN model for the three\-shot DeepSig classification\.a, The classification accuracy over PSD of𝐗\\mathbf\{X\}by experiments and simulations\.b, The confusion matrices of three selected points at \-107/\-112/\-117 dBm/Hz\.![Refer to caption](https://arxiv.org/html/2609.19279v1/supplementary_classification_energy_deepsig_5shot.png)Figure S26:The evaluation results of the 6\-layer CNN model for the five\-shot DeepSig classification\.a, The classification accuracy over PSD of𝐗\\mathbf\{X\}by experiments and simulations\.b, The confusion matrices of three selected points at \-107/\-112/\-117 dBm/Hz\.![Refer to caption](https://arxiv.org/html/2609.19279v1/supplementary_classification_energy_svhn.png)Figure S27:The evaluation results of the 9\-layer CNN model for the SVHN classification\.a, The classification accuracy over PSD of𝐗\\mathbf\{X\}by experiments and simulations\.b, The confusion matrices of three selected points at \-117/\-122/\-127 dBm/Hz\.![Refer to caption](https://arxiv.org/html/2609.19279v1/supplementary_classification_energy_cifar10.png)Figure S28:The evaluation results of the 9\-layer CNN model for the CIFAR\-10 classification\.a, The classification accuracy over PSD of𝐗\\mathbf\{X\}by experiments and simulations\.b, The confusion matrices of three selected points at \-112/\-117/\-122 dBm/Hz\.![Refer to caption](https://arxiv.org/html/2609.19279v1/supplementary_generative_energy_mnist.png)Figure S29:The evaluation results of the 4\-layer CNN model for the MNIST generation\.a, The FID over PSD of𝐗\\mathbf\{X\}by experiments and simulations, which is computed over 100 generated images\.b, The generated examples of digit “0” to “9” of selected points at \-132/\-127/\-122/\-117/\-112/\-107/\-102 dBm/Hz\.![Refer to caption](https://arxiv.org/html/2609.19279v1/supplementary_generative_energy_fmnist.png)Figure S30:The evaluation results of the 4\-layer CNN model for the FMNIST generation\.a, The FID over PSD of𝐗\\mathbf\{X\}by experiments and simulations, which is computed over 100 generated images\.b, The generated examples of ten fashion products of selected points at \-132/\-127/\-122/\-117/\-112/\-107/\-102 dBm/Hz\.![Refer to caption](https://arxiv.org/html/2609.19279v1/supplementary_generative_energy_celeba.png)Figure S31:The evaluation results of the 7\-layer CNN model for the CelebA generation\.a, The FID over PSD of𝐗\\mathbf\{X\}by experiments and simulations, which is computed over 100 generated images\.b, The generated examples of human faces of selected points at \-132/\-112/\-107/\-102 dBm/Hz\.Table S1:The overview of the CNN models implemented by RF\-CNN’s evaluation\.a, The model architecture summary of the six CNN models\.b, The energy efficiency w\.r\.t\. energy per MAC of the six CNN models\.

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