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Efficient DCT-Based Estimation and Compensation of Nonlinear Channels for OFDM Systems

Marc Martinez-Gost, Ana Pérez-Neira, Miguel Ángel Lagunas

arXiv:2608.25847v1eess.SP

TL;DR

Nonlinear OFDM channels require joint estimation of multipath, amplitude, and phase effects without excessive computational or training costs. The paper proposes an ML framework with a compact DCT representation and reports accurate compensation down to approximately 15 dB SNR for both nonlinearities and -10 dB for amplitude-only distortion.

  • Problem

    Existing nonlinear-channel estimation methods lack a unified, computationally efficient procedure for jointly handling multipath, amplitude, and phase distortions.

  • Method

    The framework uses maximum-likelihood estimation with a compact DCT-based parameterization of nonlinear amplitude and phase responses.

  • Results

    Accurate estimation is maintained to approximately 15 dB SNR with amplitude and phase nonlinearities, and to -10 dB SNR with amplitude distortion alone.

  • Takeaways & Limitations

    The estimator integrates with multiple nonlinear compensation strategies while supporting fast convergence, low complexity, and limited training overhead.

Abstract

from arXiv · show

This paper proposes a maximum-likelihood (ML) framework for estimating nonlinear frequency-selective channels in orthogonal frequency-division multiplexing (OFDM) communication systems. The nonlinear distortions are modeled using a Discrete Cosine Transform (DCT)-based representation, which results in a well-conditioned estimation problem with favorable convergence properties. The proposed method combines a compact parameterization with low computational complexity, enabling fast adaptation and efficient real-time implementation. Numerical results show that the proposed channel estimation can be used for multiple nonlinear compensation methods and achieve near-ideal BER performance with very limited training overhead. This performance is maintained down to approximately 15 dB SNR in the presence of both amplitude and phase nonlinearities, and down to -10 dB when only amplitude distortions are present.

I. INTRODUCTION

The paper addresses computationally demanding nonlinear channel estimation in OFDM, where PAPR-driven PA distortion complicates compensation and adaptive operation. It proposes an ML framework using a DCT-based nonlinear model for compact, fast, real-time-capable estimation.

  • Motivation: High PAPR pushes OFDM power amplifiers into nonlinear operation, creating a power-efficiency trade-off that motivates nonlinear compensation.Avoiding distortion requires substantial input back-off, reducing PA efficiency.
  • Motivation: Accurate nonlinear channel identification is increasingly difficult because wider bandwidths and stronger nonlinearities require higher sampling rates and more complex behavioral models.Adaptive DPD also introduces additional training overhead and transmitter-side power consumption.
  • Motivation: Existing DPoD methods depend on accurate nonlinear-channel estimation, while common approaches assume separately known or independently estimated multipath channels.Neural-network alternatives typically require large architectures and extensive datasets.
  • Motivation: Receiver-side estimation can capture aggregate nonlinear behavior across the transmission chain, rather than only PA characteristics.This is presented as attractive for uplink systems with computationally and energetically constrained user equipment.
  • Contribution: The proposed ML framework jointly models multipath, amplitude, and phase distortions using a compact DCT parameterization with favorable conditioning and fast convergence.The formulation avoids step-size tuning and second-order-statistics estimation, supporting low-complexity real-time adaptation.

B. Nonlinear Channel

The system model combines memoryless amplitude and phase nonlinearities with frequency-selective multipath propagation and noise. A DCT representation parameterizes the nonlinear responses while improving conditioning and simplifying adaptive estimation.

  • B. Nonlinear Channel: The channel is modeled as a block-fading FIR filter with i.i.d. complex Gaussian taps and L multipath components.The coefficients are normalized to provide unit channel power.
  • B. Nonlinear Channel: A memoryless power amplifier applies amplitude-to-amplitude and amplitude-to-phase distortions before frequency-selective channel propagation and additive Gaussian noise.The nonlinear transformation is applied element-wise to consecutive transmitted samples.
  • C. The DCT-based model for nonlinearities: The DCT models nonlinear functions with Q coefficients over a discrete index domain whose resolution is set by NDCT.A sufficiently large NDCT, such as 512, supports interpolation without additional DCT hyperparameters beyond Q.
  • C. The DCT-based model for nonlinearities: Mapping the normalized magnitude domain to the DCT grid enforces odd symmetry, guarantees ˆf(xn = 0) = 0, and yields a sparse odd-indexed parameterization.The symmetry reflects the constraint that no distortion occurs without an input signal.
  • C. The DCT-based model for nonlinearities: Orthogonal and bounded DCT kernels diagonalize the feature correlation structure, producing predictable convergence and minimum eigenvalue spread for adaptive estimation.The resulting framework can be integrated to jointly estimate linear and nonlinear channel distortions.

III. NONLINEAR CHANNEL ESTIMATION

The estimation problem is formulated under the ML principle by modeling the full effective nonlinear channel in the time domain. Because the resulting objective is nonconvex, the paper uses block-coordinate descent for iterative estimation.

  • III. NONLINEAR CHANNEL ESTIMATION: Time-domain estimation treats the full OFDM symbol as a pilot observation for jointly estimating the channel and nonlinearities.This avoids relying on an analytical frequency-domain expression for magnitude-distortion effects and their induced ICI.
  • III. NONLINEAR CHANNEL ESTIMATION: Block-coordinate descent alternates optimization over channel taps, magnitude DCT coefficients, and phase DCT coefficients to compute a stationary point.Each parameter set is optimized while the others remain fixed.
  • III. NONLINEAR CHANNEL ESTIMATION: The ML estimator minimizes MSE between the observed received signal and a replica generated by the Hammerstein channel model.The replica incorporates the estimated channel and nonlinear response.
  • III. NONLINEAR CHANNEL ESTIMATION: The receiver represents magnitude and phase distortions with DCT coefficients when constructing the estimated received signal and instantaneous error.The phase contribution is applied through an element-wise complex exponential.
  • III. NONLINEAR CHANNEL ESTIMATION: Nonconvexity arises from multiplicative coupling between channel taps and magnitude coefficients and from phase parameters inside a complex exponential.These couplings prevent a closed-form global solution using standard convex optimization tools.

B. Frequency-selective Channel

For frequency-selective channel estimation, the method initializes nonlinearities simply and exploits DCT-induced isotropy to simplify MMSE and adaptive updates. This removes costly matrix operations and improves conditioning.

  • B. Frequency-selective Channel: The procedure initializes magnitude distortion as linear and phase distortion as zero, with the channel length assumed known.The paper reports this initialization as sufficient for reliable convergence across considered scenarios.
  • B. Frequency-selective Channel: The channel-tap subproblem is quadratic and reduces to standard MMSE estimation when nonlinear coefficients are fixed.The effective input to the linear estimator incorporates the estimated magnitude distortion.
  • B. Frequency-selective Channel: The DCT structure yields Ru = (||fAM||2/2)I, so the effective-input correlation depends only on magnitude-coefficient energy.This isotropic form replaces general correlation estimation with a structured expression.
  • B. Frequency-selective Channel: DCT-induced isotropy removes matrix inversion and reduces the MMSE solution to scaling the cross-correlation vector.The simplification lowers complexity and improves numerical stability without eigenvalue decomposition.
  • B. Frequency-selective Channel: The DCT feature covariance becomes diagonal with eigenvalue spread equal to one, providing uniform convergence across channel taps.An online normalized-LMS implementation can update the channel from current observations.

C. Nonlinear Magnitude Distortion

The magnitude and phase distortion estimators exploit DCT parameterizations to obtain linear or well-conditioned adaptive updates. Phase estimation additionally requires a unit-circle error formulation and sufficiently high SNR because phase extraction alters the noise structure.

  • Magnitude distortion: The magnitude-distortion problem is linear in fAM and reduces to MMSE estimation using the effective input signal vn.The effective input is defined as vn = CnDnĥ*; the resulting optimal filter coefficients use its autocorrelation and cross-correlation with the received signal.
  • Magnitude distortion: DCT orthogonality induces a diagonal covariance matrix, enabling stochastic-gradient magnitude estimation without explicit covariance estimation or matrix inversion.The corresponding LMS recursion preserves the adaptive channel-estimation benefits for nonlinear magnitude distortion.
  • Phase distortion: Phase parameters are linear after reformulating the estimation problem in the phase domain, avoiding the nonconvex dependence created by their appearance inside a complex exponential.The phase of the equalized observation supplies the ML estimate of the underlying signal phase when residual ISI is negligible and AWGN dominates.
  • Phase distortion: The phase error is defined on the unit circle because direct angular errors can become arbitrarily large near the ±π discontinuities.The DCT coefficients are optimized to reduce this phase error through a stochastic gradient procedure because the resulting problem remains nonconvex.
  • Phase distortion: The phase-domain update retains DCT conditioning advantages, including diagonal covariance structure and uniform convergence behavior.This benefit is conditional: phase extraction destroys additive Gaussian noise structure, so the reformulation requires approximately 10 dB SNR.

E. Joint Estimation Algorithm

The joint estimator addresses the separate-estimation assumption through alternating optimization of channel, magnitude-distortion, and phase-distortion parameters. It applies normalized stochastic-gradient updates with one shared step-size parameter across the update blocks.

  • Joint estimation: The joint framework is introduced because separate derivations assume that the other parameter set is known.Alternating optimization removes this limitation by estimating the coupled parameter sets together.
  • Joint estimation: Algorithm 1 alternates updates of the multipath channel coefficients, magnitude DCT coefficients, and phase DCT coefficients over all Nsamples OFDM samples.The receiver uses the training sequence {xn} and corresponding received samples {yn}.
  • Joint estimation: A single step-size parameter α serves all three updates because the gradients are properly normalized.The receiver architecture feeds the estimated channel and nonlinear parameters into the equalizer.
  • Joint estimation: Parameter normalization improves numerical stability and convergence by normalizing the bounded amplitude-modulation coefficients after each update.The normalization follows from the assumed non-decreasing amplitude-distortion function and its boundary behavior.

F. Computational Complexity

The algorithm has low per-sample complexity because its three adaptation blocks use DCT-based operations and LMS-style updates. A compact DCT representation, such as Q = 6 coefficients, further reduces computation and avoids matrix inversions.

  • Per-block complexity: The linear channel adaptation stage requires O(Q^2 + L) operations per sample.This combines matrix-vector and diagonal-matrix products, an L-length error inner product, and length-L update operations.
  • Per-block complexity: The fAM adaptation step requires O(QL) operations per sample.Its dominant cost comes from two vector products; error computation and the LMS update each scale as O(Q).
  • Per-block complexity: The fPM adaptation stage requires O(L + Q) operations per sample.Equalization contributes O(L), while phase-error computation and the LMS update contribute O(Q).
  • Overall efficiency: A compact DCT model, for example Q = 6, reduces computational cost by capturing nonlinear responses with few coefficients.The DCT formulation also supports a simple LMS implementation that avoids matrix inversions.

IV. NONLINEAR COMPENSATION TECHNIQUES

The section reviews transmitter- and receiver-side nonlinear compensation for OFDM and introduces a DCT-based, self-supervised predistortion framework. It estimates the forward distortion, derives its inverse in a controlled manner, and supports low-complexity adaptation.

  • ML decoding is infeasible for OFDM because exhaustive symbol search grows exponentially with the number of subcarriers.
  • Predistortion: Predistortion independently modifies OFDM magnitude and phase so the predistorter–nonlinearity cascade approximates an identity mapping.
  • Predistortion: Phase predistortion directly cancels AM–PM distortion, whereas magnitude predistortion estimates the inverse AM–AM characteristic through nonlinear regression.
  • Predistortion: The proposed inverse-learning framework processes a reference signal through the estimated distortion and a parametric inverse, then compares the output with the input.
  • Predistortion: The DCT inverse model preserves decorrelated coefficients, bounded gradients, and stable convergence, while a more complex inverse may require more coefficients Q.
  • Predistortion: Direct receiver-side inverse estimation violates the ML structure and can degrade high-SNR performance by transforming Gaussian noise into signal-dependent non-Gaussian noise.
  • Predistortion: Receiver-side adaptation can reuse existing digital processing resources, avoiding dedicated transmitter feedback hardware, analog-to-digital conversion, and synchronization.

B. Postdistortion

Postdistortion corrects nonlinear distortion at the receiver, where compensation methods use channel and distortion information to refine symbol estimates. The reviewed approaches include iterative decoding, neural networks, clipping-based methods, and compressive sensing.

  • Postdistortion corrects nonlinear distortion after it has affected the transmitted signal, using receiver-side processing.
  • The received frequency-domain signal retains the linear channel as a multiplicative per-subcarrier factor H_k, while nonlinear coupling appears as inter-carrier interference.
  • One-tap equalization removes the linear multipath effect, but nonlinear distortion remains embedded in the ICI terms.
  • Many compensation methods assume an accurate frequency-selective channel estimate; iterative decoding additionally assumes perfect knowledge of the nonlinear response.
  • Neural-network postdistortion learns the inverse nonlinear channel and uses the frequency-selective channel response as an input.
  • Decision-aided reconstruction, clipping-noise cancellation, and compressive sensing respectively recover clipped peaks, estimate and subtract clipping distortion, or exploit time-domain sparsity.
  • Iterative decoding hard-decodes subcarriers, estimates additive distortion in the time domain, and repeatedly uses it to refine symbol decisions.

V. SIMULATION RESULTS

The simulations evaluate nonlinear channel estimation and its use in transmitter- and receiver-side compensation for a satellite communication scenario. They use OFDM with two HPA models and assume estimation and communication occur within one coherence block.

  • The study targets satellite communications, where near-saturation HPA operation and OFDM’s high PAPR create amplitude and phase distortion while input back-off reduces power efficiency.
  • The resulting channel estimates are evaluated first for nonlinear channel estimation and then within two compensation schemes at the transmitter and receiver.
  • Simulation setup: The OFDM setup uses 1024 subcarriers, 16-QAM, a cyclic prefix of 16, channel length L = 3, and an assumed receiver length L̂ = 6.
  • Simulation setup: The configuration transmits 25,000 samples, uses five iterations, and represents amplitude and phase distortions with Q = 6 and Q = 12 DCT coefficients, respectively.
  • Simulation setup: The experiments use TWTA and SSPA HPA models; the SSPA case includes amplitude distortion only by setting f_PM(x) = 0.
  • Prior comparisons reported substantially lower MSE and faster convergence for the DCT model than polynomial and neural-network representations across tested configurations.
  • Scope assumption: Estimation and compensation are assumed to occur within the same coherence block so that the nonlinear channel response remains unchanged.

B. Nonlinear Channel Estimation

The proposed estimator jointly recovers nonlinear frequency-selective channel components and supports compensation across nonlinear configurations. It remains accurate at practical SNR levels while enabling low-overhead implementation.

  • Nonlinear channel estimation: At approximately 15 dB SNR, the TWTA estimator still accurately recovers the nonlinear channel components, although phase nonlinearity has the largest estimation error.At 30 dB, all components are recovered with negligible error.
  • Nonlinear channel estimation: At SNR = −10 dB, SSPA estimation errors remain small enough to ensure excellent nonlinear compensation performance.Reliable estimation is achieved down to 0 dB, and only the nonlinear response needs illustration because the multipath response is perfectly estimated across the considered SNR regimes.
  • Nonlinear channel estimation: 2 OFDM symbols, or 2048 time samples, accurately capture the SSPA nonlinear frequency-selective channel, compared with 24 symbols in the general setup.
  • Robustness and scope: Underestimating the number of multipath components prevents accurate recovery, whereas overestimating the channel length is safer because redundant components can be suppressed.The estimator drives coefficients of additional components to zero without introducing artificial paths.
  • Nonlinear channel compensation: The framework supports predistortion and iterative decoding, with predistortion using an inverse nonlinear response and iterative decoding feeding estimated phase distortion back to the transmitter.The iterative procedure runs for Nstep = 5 iterations.
  • Nonlinear channel compensation: Predistortion achieves BER performance nearly indistinguishable from an ideal linear OFDM system for TWTA compensation, whereas iterative decoding has a small degradation.The degradation is attributed to the quasi-ML procedure being inherently suboptimal.

APPENDIX A

The appendix derives autocorrelation expressions and the optimal solution for a quadratic estimation problem. The derivations use independence, phase-distribution, and DCT-orthogonality assumptions.

  • Autocorrelation derivation: The autocorrelation derivation uses independence between phase and amplitude processes in the OFDM signal.
  • Autocorrelation derivation: Modeling phase as an i.i.d. variable uniformly distributed on [0, 2π) yields zero off-diagonal autocorrelation entries.
  • Autocorrelation derivation: The diagonal autocorrelation terms follow from the orthogonality of the DCT basis and independent realizations across time.
  • Optimal solution: The optimal solution of (P3) is obtained by expanding the quadratic cost function and setting its gradient to zero.The effective-input autocorrelation matrix and cross-correlation vector define the quantities used in the derivation.
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