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One-Bit OFDM Receivers via Deep Learning

Eren Balevi, Jeffrey G. Andrews

arXiv:1811.00971v2cs.ITcs.LGeess.SPmath.ITstat.ML

TL;DR

One-bit quantization reduces receiver power but makes OFDM channel estimation and data detection difficult, in part because its high peak-to-average power ratio leads to severe inter-carrier interference. The paper develops a generative supervised channel estimator and a sequentially trained autoencoder; it reports better channel-estimation MSE than unquantized LS and lower BER than theoretical uncoded OFDM within specified SNR ranges.

  • Problem

    One-bit ADCs lower receiver power but make accurate OFDM channel estimation and data detection difficult, with high peak-to-average power ratio producing severe inter-carrier interference.

    Source: §I · p. 1

  • Method

    The paper uses a generative supervised DNN for pilot-based channel estimation and an autoencoder that jointly learns a precoder and decoder through two-step sequential training.

    Source: B. Contributions · p. 2

  • Results

    Up to 14 dB average SNR, one-bit channel estimation has lower MSE than unquantized LS; with G = 4, AE-OFDM beats theoretical uncoded OFDM BER in Rayleigh fading up to 10 dB.

    Source: B. Contributions · p. 2, B. Data Detection · p. 9

  • Takeaways & Limitations

    The authors conclude that reliable channel estimation and satisfactory detection are achievable despite one-bit quantization.

    Source: §V · p. 9

    For higher subcarrier counts, model parameters grow quadratically, making simple dimension scaling unreasonable; processing large blocks in subblocks is left to future work.

    Source: A. Channel Estimation · p. 9

Abstract

from arXiv · show

This paper develops novel deep learning-based architectures and design methodologies for an orthogonal frequency division multiplexing (OFDM) receiver under the constraint of one-bit complex quantization. Single bit quantization greatly reduces complexity and power consumption, but makes accurate channel estimation and data detection difficult. This is particularly true for multicarrier waveforms, which have high peak-to-average ratio in the time domain and fragile subcarrier orthogonality in the frequency domain. The severe distortion for one-bit quantization typically results in an error floor even at moderately low signal-to-noise-ratio (SNR) such as 5 dB. For channel estimation (using pilots), we design a novel generative supervised deep neural network (DNN) that can be trained with a reasonable number of pilots. After channel estimation, a neural network-based receiver -- specifically, an autoencoder -- jointly learns a precoder and decoder for data symbol detection. Since quantization prevents end-to-end training, we propose a two-step sequential training policy for this model. With synthetic data, our deep learning-based channel estimation can outperform least squares (LS) channel estimation for unquantized (full-resolution) OFDM at average SNRs up to 14 dB. For data detection, our proposed design achieves lower bit error rate (BER) in fading than unquantized OFDM at average SNRs up to 10 dB.

I. INTRODUCTION

One-bit quantization reduces receiver power but makes OFDM channel estimation and detection difficult, while prior low-resolution OFDM work has not resolved one-bit reception. The paper proposes task-specific learning methods for channel estimation and data detection, including a generative estimator grounded in analysis of one-bit observations.

I. INTRODUCTION

  • One-bit ADCs can reduce power consumption by more than two orders of magnitude in some cases, but OFDM’s high peak-to-average power ratio produces severe inter-carrier interference.Most prior work considered single-carrier communication, although OFDM is widely used in modern high-rate wireless systems.Source: §I · p. 1

A. Related Work

  • Prior low-resolution ADC studies cover frequency-selective channels and OFDM, but cited work reports 4–6-bit ADCs to approach unquantized OFDM performance and a 2–3-bit detector ineffective with one-bit ADCs.Related work also includes one-bit linear OFDM receivers for massive MIMO under an infinite-channel-taps condition and sparse-channel estimation for few-bit ADCs.Source: A. Related Work · p. 1

B. Contributions

  • Up to 14 dB average SNR, the proposed one-bit channel estimator achieves lower MSE than least squares (LS) channel estimation with unquantized OFDM samples.The estimator is a generative supervised DNN trained using a labeled dataset and a limited number of training symbols.Source: B. Contributions · p. 2
  • For detection, an autoencoder jointly learns a precoder and decoder using two-step sequential training, and can beat theoretical uncoded unquantized QPSK BER up to 10 dB with hidden-layer dimension increased fourfold.The non-differentiable quantization layer prevents end-to-end backpropagation; the decoder is learned offline, then the precoder online with the channel.Source: B. Contributions · p. 2

II. CHANNEL ESTIMATION WITH ONE-BIT ADCS

  • Reliable one-bit OFDM channel estimation is challenging because of increased inter-carrier interference; the proposed supervised model uses theoretical analysis to form a suitable labeled dataset.The paper treats channel estimation and data detection as separate receiver tasks and adapts different deep-learning models to each.Source: §II · p. 2, B. Contributions · p. 2

A. One-Bit OFDM Signal Analysis

The signal analysis models pilots and data over a block-fading channel and relates one-bit observations to frequency-domain channel taps. It uses quantization-distortion properties to motivate channel estimation by averaging pilot-weighted observations.

A. One-Bit OFDM Signal Analysis

  • Pilots estimate the channel at the beginning of each coherence interval, before data transmission; the channel changes independently among blocks.The pilots are IDFT-modulated and transmitted through the dispersive channel with a cyclic prefix.Source: A. One-Bit OFDM Signal Analysis · p. 2
  • OFDM time-domain samples are approximated as Gaussian, allowing Bussgang’s theorem to express nonlinear quantization distortion using the original signal; the matrix A makes received pilots and distortion uncorrelated.The analysis also states that quantization distortion and pilots are uncorrelated.Source: A. One-Bit OFDM Signal Analysis · pp. 2–3
  • With many pilots and known instantaneous channel, pilot, and noise powers, averaging pilot-weighted one-bit observations after the normalized DFT yields Λ up to a scalar; its diagonal entries are frequency-domain channel taps.The passage states that estimating Λ is equivalent to estimating the channel.Source: A. One-Bit OFDM Signal Analysis · pp. 2–3

B. Supervised Learning Model

The supervised channel estimator uses a DNN to learn from pilot-derived labeled data, aiming to reduce the pilots needed for channel estimation. After training, it generates samples from the learned output distribution and averages them to estimate the channel.

B. Supervised Learning Model

  • The DNN is trained with labeled data produced from pilot symbols, their one-bit quantized observations, and the DFT matrix.The model is motivated by the possibility of perfect estimation with many pilots, while pilots should be minimized and instantaneous channel power is not practical to know.Source: B. Supervised Learning Model · p. 3
  • The architecture maps pilot inputs to outputs z_p using two hidden layers, a compromise between approximation capacity and computational complexity.The input and output layers have dimension 2N, hidden layers use ReLU, and the output layer uses a linear activation for regression.Source: B. Supervised Learning Model · p. 3, Figure 2 · p. 3
  • Training minimizes mean squared error between network outputs and labeled data, approximating the distribution of the labeled outputs.After training, random inputs generate output samples whose average estimates the channel.Source: B. Supervised Learning Model · p. 3
  • Generated inputs and samples consume no channel bandwidth, but the model must be retrained with N_t pilots whenever the channel changes.The number M of generated samples is limited by processing complexity rather than bandwidth.Source: B. Supervised Learning Model · p. 4

III. DATA DETECTION WITH ONE-BIT ADCS

One-bit quantization disrupts OFDM subcarrier orthogonality, making detection difficult even with perfect channel estimation and equalization. The paper addresses this by jointly learning a precoder and decoder with an autoencoder and a sequential training policy.

III. DATA DETECTION WITH ONE-BIT ADCS

  • Even with perfect channel estimation and equalization, one-bit quantization disrupts subcarrier orthogonality and causes severe intercarrier interference that hinders reliable QPSK detection.The illustrated case uses QPSK over a 10-tap frequency-selective channel at 20 dB SNR.Source: §III · p. 4, Figure 3 · p. 4
  • The proposed autoencoder jointly learns an OFDM precoder and decoder to improve symbol detection with one-bit ADCs.The quantized samples enter a non-differentiable layer, so the model uses a two-step sequential training policy.Source: §III · p. 4

A. Autoencoder Based OFDM

AE-OFDM adapts an autoencoder to one-bit OFDM by treating the precoder, modulation, channel, noise, and quantizer as an encoder and post-quantization processing as a decoder. It learns these blocks to detect transmitted symbols without explicit receiver-side DFT or equalization.

A. Autoencoder Based OFDM

  • AE-OFDM treats the learned precoder, OFDM modulator, channel, noise, and quantizer collectively as the encoder, with post-quantization receiver processing as the decoder.The precoder and decoder are implemented as neural networks.Source: A. Autoencoder Based OFDM · p. 4, Figure 4 · p. 5
  • Pilot symbols are not precoded during channel estimation because the precoder is designed after channel estimation.The learned frequency-domain precoder is applied to data symbols before the IDFT and transmission over the dispersive channel.Source: A. Autoencoder Based OFDM · pp. 4–5
  • The decoder is a multilayer neural network that reconstructs the transmitted symbols from the one-bit quantized observations.Its layer activations and parameter-matrix dimensions are specified by the network design.Source: A. Autoencoder Based OFDM · p. 5
  • AE-OFDM learns the DFT and equalization implicitly, eliminating the need for explicit receiver-side DFT and equalization.The modulated symbols are the input layer and detected symbols are the output layer.Source: A. Autoencoder Based OFDM · p. 5

B. Training

One-bit quantization prevents end-to-end backpropagation through the AE-OFDM precoder, so the paper trains the decoder and precoder sequentially. The decoder is trained offline, while the precoder is learned with a supervised model using stored layer data.

B. Training

  • Because the quantizer’s derivative is zero almost everywhere, backpropagation cannot train neural layers before it, including the precoder.This motivates a training policy that avoids end-to-end optimization.Source: B. Training · p. 5
  • The two-step policy trains the decoder without the channel and OFDM modulator, then learns a compatible precoder with both included.The decoder can be trained offline; the precoder is learned online as the channel changes.Source: B. Training · p. 5
  • The end-to-end path precodes and normalizes symbols, applies the IDFT and channel, adds noise, quantizes, then decodes to constellation points.The layered architecture represents in-phase and quadrature components in separate halves of each layer vector.Source: B. Training · p. 6
  • The decoder shares parameters between real and imaginary signal parts, while supervised learning within the autoencoder trains the encoder under one-bit quantization.The encoder model uses the decoder’s l2 layer as labeled data for input l1.Source: B. Training · p. 6
  • The precoder’s supervised training uses l1 as input and l2 as labeled data, with the channel already known after estimation.The layer values are stored during decoder training to create the training dataset.Source: B. Training · p. 6

C. Practical Challenges

Under block fading, precoder parameters must be retrained whenever the channel changes, creating potential training-symbol overhead. The section describes receiver- and transmitter-side training alternatives, each with different practical costs.

C. Practical Challenges

  • Precoder retraining at every channel change can create excessive training-symbol overhead; the worst-case assumption requires at least 2N pilots, more than 1019 for N = 64.Although the DNN is trained with 5000 symbols, this is not a count of pilots sent over the channel.Source: C. Practical Challenges · p. 7
  • Receiver-side precoder training after channel estimation allows as many training samples as needed without adding pilot overhead, but adds receiver processing.The passage notes stochastic-computing hardware as a possible way to handle this complexity.Source: C. Practical Challenges · p. 7
  • Alternatively, the receiver can send channel information to the transmitter for precoder training, which is especially useful in downlink communication.This approach avoids receiver-side training; decoder parameters are trained offline.Source: C. Practical Challenges · p. 7

D. Implementation

AE-OFDM supports several practical implementations for increasing input dimension; this section focuses on time-domain oversampling. The oversampling factor expands the channel representation while the precoder remains N × N.

D. Implementation

  • AE-OFDM can increase input dimension through redundant subcarriers, multiple antennas, or time- or frequency-domain oversampling, with this section focusing on oversampling.The choice depends on how the input dimension is increased when G > 1.Source: D. Implementation · p. 7

1) Time Domain Oversampling:

  • Time-domain oversampling samples the received signal at G offsets per symbol-rate sample, with g ranging from 0 to G − 1.The sampling interval is Ts = T/N, and offsets are spaced by Ts/G.Source: 1) Time Domain Oversampling · p. 7
  • With time-domain oversampling, the channel matrix becomes GN × N while the precoder remains a complex N × N matrix.The oversampled channel matrix is expressed in terms of the symbol-rate sampled channel matrix.Source: 1) Time Domain Oversampling · p. 8
  • The decoder forms a real vector from the concatenated real and imaginary parts of s2 to detect the transmitted symbols.The received signal model includes time-domain channel taps and complex Gaussian noise.Source: 1) Time Domain Oversampling · pp. 7–8

2) Frequency Domain Oversampling:

Frequency-domain oversampling is implemented by zero-padding before the IDFT, increasing its block size by the oversampling factor. The section also describes combined time- and frequency-domain oversampling and tradeoffs between the two approaches.

2) Frequency Domain Oversampling:

  • Zero-padding before the IDFT increases its block size by G, and the precoder is derived from the frequency-oversampled channel.The resulting Hfos and Ffos matrices are each GN × GN.Source: 2) Frequency Domain Oversampling · p. 8
  • Combined oversampling uses G = GtGf, where Gt and Gf are the time- and frequency-domain oversampling factors.The precoder can be determined accordingly for the combined implementation.Source: 2) Frequency Domain Oversampling · p. 8
  • Time-domain oversampling may suit sub-6-GHz systems with moderate bandwidth, whereas frequency-domain oversampling may suit mmWave systems with large bandwidth.The former can avoid much extra sampling power; the latter trades higher sampling-rate power cost for longer IDFT and DFT operations.Source: 2) Frequency Domain Oversampling · p. 8

IV. SIMULATIONS

The simulations evaluate channel estimation with synthetic QPSK OFDM data and compare the proposed DNN with LS and maximum-likelihood baselines. With 20 pilots, the DNN can outperform unquantized LS in the three-tap setting up to 12 dB, while higher subcarrier counts raise a scaling challenge.

IV. SIMULATIONS

  • The simulations use synthetic QPSK symbols, complex Gaussian channel taps with uniform power delay profile, and additive white Gaussian noise; performance is measured by MSE.Channel estimation is evaluated over 100 channel realizations, with 10,000 randomly generated inputs per estimate.Source: §IV · p. 8, A. Channel Estimation · p. 8

A. Channel Estimation

  • Up to 12 dB, the DNN beats unquantized LS for three complex Gaussian channel taps, and it outperforms one-bit LS across all SNRs.The comparison uses MSE; the DNN has seen only one-bit quantized samples.Source: A. Channel Estimation · p. 8, Figure 8 · p. 9
  • Twenty pilots are reasonable for 64 subcarriers: increasing pilots from 10 to 20 significantly improves performance, while a further increase has little impact.The study compared DNNs trained with 10, 20, or 25 pilots.Source: A. Channel Estimation · p. 8
  • Maximum-likelihood channel estimation has exponential complexity O(SN), whereas the proposed model has complexity O(W 2), with W = 32N 2.Source: A. Channel Estimation · p. 9
  • Increasing the channel taps to 10 slightly improves DNN performance, extending its advantage over unquantized LS to 14 dB.Source: A. Channel Estimation · p. 9
  • For many subcarriers, fully connected model size grows quadratically; processing large OFDM blocks in smaller kernel-based subblocks is left to future work.Source: A. Channel Estimation · p. 9

B. Data Detection

For one-bit OFDM data detection, the paper evaluates an autoencoder receiver against conventional OFDM in fading. With G = 4, it beats the theoretical uncoded OFDM error rate up to 10 dB; the paper also identifies scaling and acquisition as open challenges.

B. Data Detection

  • The evaluation compares conventional uncoded OFDM with unquantized and one-bit samples, using subcarrier basis detection and a theoretical unquantized Rayleigh-fading benchmark.The setup has 48 data, 4 pilot, and 12 guard subcarriers, with a cyclic prefix length of 16.Source: B. Data Detection · p. 9
  • With G = 4, AE-OFDM beats the theoretical uncoded OFDM error rate in Rayleigh fading up to 10 dB.Source: B. Data Detection · p. 9, Figure 9 · p. 10
  • Doubling G makes AE-OFDM competitive with unquantized OFDM up to 6 dB, while G = 1 slightly underperforms one-bit OFDM detection.Source: B. Data Detection · p. 9

V. CONCLUSIONS AND FUTURE WORK

  • The conclusion reports that one-bit ADCs can reduce receiver cost and power but cause performance loss with conventional methods; deep learning methods can beat unquantized OFDM performance.Source: §V · pp. 9–10
  • The study leaves larger-subcarrier architectures, multi-antenna channel estimation and combining, and initial time/frequency acquisition for future work.Fully connected layers are described as probably unreasonable for N ≫ 64.Source: §V · p. 10

APPENDIX PROOF OF THEOREM 1

The proof invokes Bussgang’s theorem for a zero-mean Gaussian input to a memoryless system, then uses Gaussian-distribution and matrix substitutions to complete the argument.

APPENDIX PROOF OF THEOREM 1

  • For zero-mean Gaussian input to the memoryless system Q(·), Bussgang’s theorem makes the input-output cross-correlation matrix proportional to the input auto-correlation matrix.Source: APPENDIX PROOF OF THEOREM 1 · p. 10
  • The appendix identifies A as a diagonal matrix and describes diag(Cypyp) as the diagonal matrix formed from Cypyp’s diagonal terms.Source: APPENDIX PROOF OF THEOREM 1 · p. 10
  • The proof applies the Gaussian probability distribution function, substitutes the resulting expression into earlier equations, and concludes with the final substitution.Source: APPENDIX PROOF OF THEOREM 1 · p. 10
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