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Neural Network-Based Delay-Doppler-Assisted Channel Estimation for OFDM
Mingcheng Nie, Hao Chang, Shuangyang Li, Haiyao Yu, Jiafu Hao, Yonghui Li
TL;DR
Conventional OFDM channel estimation becomes unreliable in high-mobility channels because Doppler-induced ICI invalidates its element-wise TF model. This paper derives an ICI-aware relation, recovers an equivalent DD channel with an NN under pilot-data mismatch, and reports lower NMSE and BER than conventional estimators.
Problem
Conventional single-tap estimation and TF interpolation become unreliable in high-mobility OFDM channels because Doppler-induced ICI invalidates the element-wise TF model.
Method
The proposed NN-based DD-assisted framework derives an ICI-aware TF relation, recovers an equivalent DD channel from contaminated pilot observations, and reconstructs the TF channel.
Results
The proposed method achieves lower NMSE and BER than conventional single-tap LS and LMMSE estimators and the LASSO-based DD-assisted estimator.
Takeaways & Limitations
DD-assisted recovery can improve OFDM channel estimation under high Doppler while accommodating fractional-delay and Doppler leakage without requiring strict DD sparsity.
Abstract
from arXiv · showhide
Conventional orthogonal frequency division multiplexing (OFDM) channel estimation relies on single-tap estimation and time-frequency (TF) interpolation, which becomes unreliable in high-mobility channels because Doppler-induced inter-carrier interference (ICI) invalidates the underlying element-wise TF model. This paper proposes a neural-network-based delay-Doppler (DD)-assisted channel estimation framework for OFDM over doubly selective channels. We first derive an ICI-aware TF domain input-output relation and formulate channel estimation as a DD recovery problem. Unlike conventional sparse recovery approaches, the proposed framework does not require the equivalent DD domain channel vector to be strictly sparse, thereby accommodating the leakage induced by fractional delay and Doppler shifts. Since the data symbols are unknown during channel estimation, the sensing matrix is constructed using only the known pilot symbols. As a result, data-induced interference is not explicitly modeled, leading to a structured mismatch in the pilot observations. To tackle this challenge, the adopted network iteratively exchanges observation- and channel-domain features through the sensing matrix to learn the mapping from these contaminated observations to the equivalent DD domain channel, which is subsequently used to reconstruct the TF-domain channel. Simulation results show that the proposed method achieves lower normalized mean-square error and bit-error rate than conventional OFDM estimators.
I. Introduction
High mobility disrupts the orthogonality that makes conventional OFDM channel estimation simple, while DD representations offer a way to capture rapidly varying channels. The paper therefore develops an NN-based DD-assisted estimator compatible with conventional OFDM pilots.
- Motivation: High mobility and high carrier frequencies undermine subcarrier orthogonality, causing conventional TF channel estimation to degrade.With limited pilot overhead, conventional methods can experience significant performance degradation in highly dynamic channels.
- Motivation: DD representations can describe rapidly time-varying channels in a more compact, quasi-static, and sparse form.This motivates using DD-domain structure for channel-estimation design.
- Motivation: Embedded DD pilots are not directly compatible with practical OFDM frames because they may spread across the TF domain and interfere with data symbols.The resulting design question is whether DD channel representation can be used with conventional TF-domain OFDM pilots.
- Contribution: The proposed NN-based DD-assisted estimator derives an ICI-aware OFDM relation and learns channel recovery under pilot-data interference and ICI.It estimates DD channel parameters and uses nonlinear mapping rather than relying only on conventional sparse-recovery procedures.
II. OFDM Transmissions over Doubly-Selective Channel
The paper models OFDM transmission through CP insertion, a time-varying multipath channel, matched filtering, and CP removal before transforming the received signal to the TF domain. The resulting TF channel has per-symbol diagonal terms and off-diagonal ICI terms.
- Transmission model: OFDM uses M subcarriers and N symbols under critical sampling, with TF symbols converted to time-domain signals by an IDFT.The subcarrier spacing and symbol duration satisfy Δf · T = 1.
- Transmission model: A cyclic prefix is appended to each OFDM symbol using a CP addition matrix before pulse shaping and continuous-time transmission.The CP addition matrix includes the last LCP columns of the M × M identity matrix.
- Doubly-selective channel: The time-varying channel is represented by paths with coefficients hp, delays τp, and Doppler shifts νp, whose normalized delay and Doppler indices need not be integers.Noninteger indices capture fractional delay and Doppler locations.
- Receiver model: Matched filtering produces a discrete-time received vector, after which stacking and CP removal yield a compact vector representation involving the effective channel matrix.The received signal also includes complex AWGN.
- TF-domain relation: The TF-domain channel matrix is block diagonal across OFDM symbols, while each block’s diagonal entries describe the symbol channel and off-diagonal entries account for ICI.Thus, Doppler-induced coupling appears within each TF-domain block rather than as an element-wise channel product.
III. DD-Assisted Channel Estimation
The proposed framework formulates OFDM channel estimation as recovery of an equivalent delay-Doppler channel, then reconstructs the time-frequency channel through a neural estimator. It avoids strict sparsity and iteratively exchanges observation and channel features through the sensing matrix.
- Framework formulation: The framework uses an ICI-aware TF input-output relation and exploits DD parameters beyond diagonal pilot-position coefficients estimated by conventional STE.This extends channel estimation from element-wise pilot estimates to a DD-assisted reconstruction framework.
- DD representation: Fractional delay and Doppler are discretized on a finite DD grid, whose grid-point components form a dictionary for the TF-domain channel.The grid contains Q points, each associated with delay and Doppler values.
- DD representation: Unlike sparse recovery methods, the framework does not impose sparsity on the equivalent DD vector because off-grid components leak energy across neighboring grid points.This leakage makes the grid-defined equivalent channel generally non-sparse even when the physical DD channel is sparse.
- Neural estimator: The neural network takes real-valued sensing and observation representations as input, estimates the DD channel vector, and reconstructs the TF channel as ˆg = ¯Dˆh.The network uses feature representation, processing, and exchange modules.
- Neural estimator: Feature exchange repeatedly projects observation features into the estimation domain and estimation features back through the sensing matrix, with residual updates across iterations.Shared MLPs process the observation and channel-estimation features, while the final channel feature produces the DD estimate.
- Training: The network is trained by minimizing mean-square error between the true and reconstructed TF-domain channel vectors.The feature-processor parameters are optimized through backpropagation.
A. System Settings
The simulations use an 8×8 OFDM system with three paths, fractional delay and Doppler indices, all-one lattice pilots, and QPSK data. Conventional STE baselines estimate pilot CSI and interpolate across time and frequency.
- Simulation configuration: The system uses M = N = 8, P = 3 channel paths, and delay and Doppler indices sampled from [0, 3] and [−2, 2].The indices are not restricted to integer values.
- Simulation configuration: Lattice pilots occupy both time and frequency dimensions, are set to all ones, and are evaluated with independently generated unit-power QPSK data.This pilot structure is intended for doubly selective channel estimation.
- Evaluation metrics: The received signal power is defined from the noiseless TF-domain signal y0 = HTFxTF, and normalized MSE is used for evaluation.The SNR is defined with respect to received signal power.
- Baselines: Conventional baselines apply STE at pilot locations and recover nonpilot CSI through nearest-pilot linear interpolation along time and frequency.The LS and LMMSE variants are denoted ST-LS and ST-LMMSE.
B. Training Settings
The feature processors are shared MLPs that update observation and channel-estimation features over multiple iterations, with a scalar output at the final channel-estimation stage.
- Network training settings: Each feature processor uses an input layer, a hidden layer of dimension d_h, and an output layer, with d = 12, d_h = 64, L = 10, and α = 0.5.The final Ψ^(T) processor outputs a scalar coefficient rather than a d-dimensional feature.
C. Simulation results
Conventional single-tap estimators degrade severely in high-mobility channels because Doppler-induced ICI violates their element-wise TF model. DD-assisted estimation improves NMSE and BER, with the proposed NN method performing best among practical schemes.
- Conventional OFDM estimators suffer severe NMSE degradation in high-mobility environments because Doppler shifts introduce ICI that STE neglects.STE ignores off-diagonal interference components in the TF channel matrix.
- DD-assisted schemes achieve better NMSE by estimating DD channel parameters and reconstructing the corresponding TF channel.LASSO partially accounts for ICI but remains limited by pilot-data interference and sparse-representation mismatch.
- The proposed NN-based DD-assisted estimator achieves the most robust NMSE performance by exploiting structured DD information while mitigating ICI and pilot-data interference.
- The proposed NN-based DD-assisted scheme achieves the best BER among practical estimators, although a gap remains relative to perfect CSI.The remaining gap indicates that channel-estimation accuracy still limits detection performance.
V. Conclusion
The paper proposes an NN-based DD-assisted OFDM channel-estimation framework that reconstructs the TF channel from an equivalent DD representation under high mobility. Simulations show lower NMSE and BER than conventional and LASSO-based estimators.
- The framework derives an ICI-aware relation, formulates estimation as equivalent DD-channel recovery, and reconstructs the TF channel.
- Unlike conventional sparse recovery, the framework does not require strict DD sparsity and accommodates leakage from fractional delay and Doppler shifts.
- The network addresses pilot-only model mismatch by exchanging observation- and channel-domain features through the sensing matrix.The sensing matrix uses known pilots, while data-induced interference contaminates the observations.
- Simulation results show lower NMSE and BER than conventional single-tap LS and LMMSE estimators and the LASSO-based DD-assisted estimator.