End-to-end Learning for OFDM: From Neural Receivers to Pilotless Communication

Fayçal Ait Aoudia; Jakob Hoydis

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End-to-end Learning for OFDM: From Neural Receivers to Pilotless Communication

Fayçal Ait Aoudia, Jakob Hoydis

arXiv:2009.05261v3Sep 11, 2020cs.ITcs.LGeess.SP

TL;DR

The paper addresses the limited evidence for end-to-end learning over realistic time- and frequency-selective wireless channels. It evaluates neural receivers and jointly learned pilotless OFDM transmission, finding comparable BER with higher throughput when orthogonal pilots are removed.

Problem
End-to-end learning gains had not been quantified over realistic OFDM channels with time- and frequency-selective fading and imperfect channel knowledge.
Method
The study evaluates a neural receiver and jointly learned transmitter-receiver schemes using either QAM with superimposed pilots or a learned zero-mean constellation over OFDM fading channels.
Results
The pilotless schemes achieve BERs similar to the pilot-based neural-receiver baseline while removing orthogonal pilots and increasing throughput by around 7%.
Takeaways & Limitations
Jointly learned transmitters and neural receivers could remove DMRS-related control overhead in beyond-5G communication systems.

Abstract

from arXiv · show

Previous studies have demonstrated that end-to-end learning enables significant shaping gains over additive white Gaussian noise (AWGN) channels. However, its benefits have not yet been quantified over realistic wireless channel models. This work aims to fill this gap by exploring the gains of end-to-end learning over a frequency- and time-selective fading channel using orthogonal frequency division multiplexing (OFDM). With imperfect channel knowledge at the receiver, the shaping gains observed on AWGN channels vanish. Nonetheless, we identify two other sources of performance improvements. The first comes from a neural network (NN)-based receiver operating over a large number of subcarriers and OFDM symbols which allows to significantly reduce the number of orthogonal pilots without loss of bit error rate (BER). The second comes from entirely eliminating orthognal pilots by jointly learning a neural receiver together with either superimposed pilots (SIPs), linearly combined with conventional quadrature amplitude modulation (QAM), or an optimized constellation geometry. The learned geometry works for a wide range of signal-to-noise ratios (SNRs), Doppler and delay spreads, has zero mean and does hence not contain any form of superimposed pilots. Both schemes achieve the same BER as the pilot-based baseline with around 7% higher throughput. Thus, we believe that a jointly learned transmitter and receiver are a very interesting component for beyond-5G communication systems which could remove the need and associated control overhead for demodulation reference signals (DMRSs).

I. INTRODUCTION

This work evaluates end-to-end learning for OFDM over realistic time- and frequency-selective fading channels, where AWGN shaping gains vanish with imperfect channel knowledge. It instead studies neural receivers and jointly learned transmission schemes that reduce or eliminate orthogonal pilots while maintaining BER.

Motivation: Realistic wireless evaluations of end-to-end learning must account for Doppler and delay spread, which are absent from many prior AWGN, block-fading, or static-channel studies.The paper uses a controllable model based on Jakes’ Doppler spectrum and 3GPP power-delay profiles.
Motivation: With imperfect receiver channel knowledge, the shaping gains previously observed on AWGN and static channels vanish.The paper therefore searches for other performance improvements in realistic fading channels.
Pilotless transmission: The paper jointly optimizes transmitter and receiver components to eliminate orthogonal pilots using either superimposed pilots added to QAM or a learned constellation and bit labeling.The learned constellation is constrained to zero mean, so it avoids a DC offset and is not interpreted as containing superimposed pilots.
Evaluation: The study benchmarks neural approaches against LMMSE channel estimation and iterative estimation, demapping, and decoding receivers.The baselines use perfect tempo-spectral covariance knowledge for LMMSE estimation.
Neural receiver: A neural receiver jointly processing multiple subcarriers and OFDM symbols enables lower BER with sparse orthogonal pilot patterns, especially in high-mobility scenarios.This extends end-to-end learning benefits without necessarily changing standard QAM transmission.
Results: The two jointly learned approaches achieve BERs similar to the neural-receiver QAM baseline with orthogonal pilots across different speeds and delay spreads, while removing all orthogonal pilots yields 4%–8% throughput gains depending on scenario.All resource elements can carry modulated data rather than orthogonal reference signals.

II. CHANNEL MODEL AND RECEIVER BASELINES

The paper models OFDM over frequency- and time-selective fading channels and benchmarks neural approaches against non-iterative and iterative LMMSE-based receivers.

The system uses nS subcarriers across nT consecutive OFDM symbols, forming transmission frames after cyclic-prefix removal and discrete Fourier transformation.
The channel vector is modeled as complex Gaussian with covariance R describing temporal and spectral correlation.The correlation is assumed separable into frequency and time components.
Temporal correlation depends on receiver speed, carrier frequency, and OFDM-symbol duration, while frequency correlation depends on delay spread and the power-delay profile.
The non-iterative baseline performs LMMSE channel estimation from pilots followed by Gaussian soft-demapping.
The iterative baseline adds IEDD, repeatedly using decoder information to refine channel estimation and demapping.

B. LMMSE channel estimation and Gaussian demapping

The baseline receiver estimates the channel from pilot resource elements, then performs Gaussian soft-demapping and decoding; IEDD iteratively feeds decoder information back into estimation and demapping.

Pilot resource elements are selected from the received frame to form a vectorized observation for LMMSE channel estimation.
Imperfect channel knowledge produces residual interference combined with noise, represented as an effective disturbance for demapping.
The demapper computes bit LLRs under a Gaussian effective-noise assumption before deinterleaving and channel decoding.The text explicitly notes that the effective disturbance is not generally Gaussian.
IEDD iterates channel estimation, demapping, and decoding, using decoder-generated LLRs as prior information for subsequent processing.
The iterative formulation assumes independent transmitted symbols when constructing prior distributions, although channel coding can introduce redundancy.

III. END-TO-END LEARNING FOR OFDM

The paper extends end-to-end learning to OFDM fading channels by jointly optimizing transmitter and receiver parameters against an achievable-rate objective.

End-to-end training jointly optimizes transmitter and receiver parameters for a specified channel model.
Training minimizes total binary cross-entropy estimated through Monte Carlo sampling over transmitted bits and received channel outputs.
The achievable-rate objective separates an ideal bit-metric-decoding rate from rate loss caused by an imperfect receiver.
Minimizing the training loss maximizes the achievable rate, and the paper states that this rate is achievable under mismatched bit-metric decoding.
The learned transmitter includes constellation geometry and bit labeling, extending this approach from prior settings to OFDM channels.

A. Learning of geometric shaping and bit-labeling

The paper learns a centered constellation and neural receiver for OFDM, and separately learns superimposed-pilot allocations jointly with receiver parameters to improve resource use.

A. Learning of geometric shaping and bit-labeling: The transmitter learns 2^m complex constellation points, then centers and normalizes them to enforce zero mean and unit average power.
A. Learning of geometric shaping and bit-labeling: Centering prevents embedded superimposed pilots, so the receiver must exploit constellation geometry without orthogonal pilots.
A. Learning of geometric shaping and bit-labeling: The learned constellation is trained offline for a range of SNRs, Doppler spreads, and delay spreads, then replaces conventional geometries such as QAM.
A. Learning of geometric shaping and bit-labeling: The neural receiver operates over multiple subcarriers and OFDM symbols, with hardware acceleration identified as a way to reduce detection cost.
B. Learning of superimposed pilots: Superimposed pilots split each resource element's energy between a reference signal and data, allowing every resource element to carry modulated data.
B. Learning of superimposed pilots: The pilot allocation matrix and neural receiver are jointly optimized for achievable rate rather than channel-estimation MSE.

C. Receiver architecture

The receiver is a convolutional residual neural network that processes complete OFDM frames and directly outputs bit-level log-likelihood ratios, replacing conventional channel estimation and demapping. It is evaluated against standard receivers across pilot patterns, mobility, and delay-spread conditions.

Receiver design: The convolutional residual NN takes received baseband samples across subcarriers and OFDM symbols and outputs LLRs for channel decoding.The network substitutes the estimator and demapper, with real and imaginary components stacked as an additional input dimension.
Architecture details: Dilated and separable convolutional layers expand the receptive field and reduce the number of weights without significant performance loss.Zero-padding preserves input and output dimensions throughout the convolutional layers.
Evaluation setup: The evaluation compares the NN receiver with non-iterative and iterative baselines using conventional QAM and 5G NR pilot patterns across speed ranges.Training uses TDL-B and TDL-C profiles, while TDL-A is reserved for evaluation to avoid overfitting to a particular profile.
Evaluation setup: The study uses one- and two-symbol 5G NR pilot patterns, with the second pattern allocating additional pilots for high-mobility scenarios.The data-element ratios are 162/168 for 1P and 156/168 for 2P.
Receiver design: The receiver operates on the entire OFDM frame, matching the frame-wide processing used for baseline channel estimation.The baseline computations involve 1008 × 1008 matrices, whereas the NN avoids iterative processing.

B. Evaluation of the NN-based receiver

The NN-based receiver is most advantageous with sparse pilots and high mobility, where it remains close to the perfect-channel-knowledge bound while conventional receivers degrade. Its frame-wide processing enables substantial pilot reduction without a significant BER increase.

BER performance: Only the NN-based receiver achieves BERs within 2 dB of the perfect-channel-knowledge bound across all speed ranges with the 1P pattern.At low speed, its BER is less than 0.5 dB from that bound.
BER performance: Conventional baselines are less robust as speed increases, and the iterative scheme’s low-speed gains vanish at high speeds.With the 2P pattern, all schemes achieve BERs close to the perfect-channel-knowledge bound.
Channel aging: With 1P pilots, channel aging creates increasing mismatch between estimated and true channel coefficients for resource elements farther from the pilots.The mismatch persists even at infinite SNR and is amplified by channel noise.
Channel aging: A second pilot at the end of the slot reduces channel-aging effects and leads to much lower error rates with the 2P pattern.This explains why the 2P results approach the perfect-channel-knowledge reference more closely.
Pilot reduction: The NN receiver remains effective with very sparse pilots, in some cases reducing the pattern to a single subcarrier carrying two pilots without significant BER increase.Using more pilots can lower BER for conventional baselines, but reduces throughput.

C. Evaluation of end-to-end learning

Jointly learning the transmitter and receiver enables pilot-free operation through either superimposed pilots or a learned constellation geometry. These schemes retain BER comparable to QAM with orthogonal pilots while improving goodput by avoiding pilot resource elements.

Pilotless designs: The optimized QAM-SIP allocation assigns less than 10% of each resource element’s energy to superimposed pilots.Edge resource elements receive a somewhat higher SIP energy ratio, but variation across the frame is small.
BER performance: The GS and QAM-SIP schemes achieve BERs similar to QAM with orthogonal pilots and the NN receiver across both pilot-pattern baselines.This similarity holds across all three considered speed ranges.
Goodput: The GS and QAM-SIP schemes improve goodput by avoiding orthogonal reference signals while maintaining comparable BER.Goodput counts successfully received bits per frame and accounts for the fraction of data-carrying resource elements.
Goodput: 4 % to 8 % additional gains are obtained from end-to-end learning, depending on the pilot pattern used by the baselines.The schemes approach perfect-channel-knowledge goodput, especially above Es/σ2 = 15 dB.
Pilotless designs: The learned constellation geometry is zero mean, so it contains no orthogonal or superimposed reference signal.Suppressing all orthogonal pilots requires joint optimization of the transmitter and receiver.

D. PAPR study

The proposed pilotless schemes preserve conventional-QAM PAPR while extending reliable detection and throughput gains across realistic channel conditions. Their main scope boundary is that future MIMO settings may still require orthogonal pilots for multiuser channel estimation.

PAPR results: QAM-SIP and GS produce nearly the same PAPR distributions as conventional QAM.The evaluation generates 7 × 10^6 OFDM symbols per scheme and computes PAPR from their time-domain signals.
PAPR results: The proposed schemes therefore are not expected to increase transmitted-signal distortion relative to conventional QAM.The paper identifies this as encouraging for implementation and use.
Conclusion: With a few orthogonal pilots, the NN receiver achieves lower BER, especially in high-mobility scenarios, while joint optimization enables reliable detection without orthogonal pilots.Removing pilot resource elements permits throughput gains.
Conclusion: Pilot suppression can use SIPs linearly combined with QAM data or a learned zero-mean constellation.Both approaches avoid orthogonal and superimposed reference signals in the learned-geometry case.
Scope and implications: The learned constellations and SIP patterns work across SNR, Doppler, and delay-spread ranges, supporting removal of DMRS-related control signaling overhead.The paper positions these schemes as candidates for beyond-5G systems.
Limitations and future work: Future MIMO extensions may still require orthogonal pilots to estimate channels for different users and compute equalization and precoding matrices.The paper also identifies channels with subcarrier interference or no cyclic prefix as future extensions.

APPENDIX

The appendix derives symbol priors from bit-level LLR information, using softmax to recover the symbol distribution. The derivation is exact under independent bit levels but becomes approximate when decoder-derived priors induce dependence.

The prior information LLRP(k, i) is associated with the ith transmitted bit on resource element k.
Equation (26) is exact when bits mapped to the same resource element are independent.
Decoder-derived prior information is conditioned on the channel output, so bit levels are generally dependent.
The exact symbol prior cannot generally be computed from marginal bit distributions alone.
The resulting expression is only an approximation when the independence condition is not satisfied.
The vector hPm represents unscaled log-probabilities from which PXk is recovered using softmax.

1 LLRP(k, i), · · · ,

This appendix derives LMMSE channel estimation and bit-level probability expressions for OFDM symbols with prior or extrinsic information. It models channel-estimation error and maps bit subsets to constellation-symbol distributions.

The LMMSE derivation rewrites the OFDM channel transfer function in vector form and derives the channel estimate and estimation-error covariance.
The channel model assumes transmitted symbols follow the prior distribution PXk, with their means computed under that distribution.
The channel response is decomposed into an LMMSE estimate and estimation error, with Gaussian noise wk and error variance specified.
Extrinsic bit LLRs LLRE are used to compute the a priori distribution of transmitted symbols.
Constellation subsets Ci,0 and Ci,1 contain points whose ith labeling bit equals 0 and 1, respectively.

B. Proof of Proposition 1

The proof uses random coding and a bitwise decoding metric over received OFDM frames. Under the stated stationarity and ergodicity assumptions, the resulting rate is achievable.

B. Proof of Proposition 1: The proof constructs a random binary codebook, maps messages to codewords, and indexes bits by frame, resource element, and bit position.
B. Proof of Proposition 1: Stationary and ergodic channel, noise, and codeword processes permit use of the mean ergodic theorem as the number of frames grows.
B. Proof of Proposition 1: The receiver decodes each codeword using non-negative bitwise metrics operating jointly on data-carrying resource elements.
B. Proof of Proposition 1: The codeword may span multiple frames for the proof, although practical systems usually contain it within one frame.
B. Proof of Proposition 1: The proof bounds decoding error using the union bound, Markov inequality, random code construction, and independent uniform codeword bits.
B. Proof of Proposition 1: Qk,i(b|yl) normalizes the bitwise metric into a probability mass function over the two possible bit values.
B. Proof of Proposition 1: The derived rate is achievable: for any Rc < R, the probability of error can be made arbitrarily low.
B. Proof of Proposition 1: The rate expression can be rewritten using the divergence between the true posterior PBk,i and the decoding distribution Qk,i.