A Deep Iterative Refinement Receiver for OTFS Symbol Detection in Doubly-Dispersive Channels

I. INTRODUCTION

• OTFS spreads symbols across time and frequency to exploit channel diversity, but delay and Doppler spreading make detection difficult and high Doppler degrades OFDM through intercarrier interference.The motivating settings include vehicle-to-vehicle networks, low Earth orbit satellites, and high-speed rail.Source: §I · p. 1
• MAP/ML search is exponential, MP can falter on cyclic graphs under fractional Doppler,, VB can underestimate variance, and LS/MMSE require costly inversions.Low-complexity alternatives to LS/MMSE aim for performance comparable to MMSE at lower computational cost –.Source: §I · p. 1
• Prior CNNs augment inputs with LS/MMSE/MP estimates – or learned channel embeddings,, while signal-only architectures omit explicit channel knowledge.Some approaches use depthwise separable convolutions, wavelet decomposition, or MMSE priors with an RCAN.Source: §I · p. 2
• Existing CNN receivers generally incorporate channel knowledge implicitly and estimate symbols in one pass, burdening a single inference with resolving doubly-dispersive interference.Prior estimate-based methods also depend on the quality of conventional initial estimates.Source: §I · p. 2

A. Contributions

• The proposed two-stage CNN detector combines channel-aware initialization with iterative refinement that updates a detection state using prior estimates and the received signal.The stages are designed to improve the initial estimate's robustness and progressively update detection under doubly-dispersive channels.Source: A. Contributions · p. 2
• The initializer uses a delay–Doppler alignment transform and channel-conditioned kernels to incorporate channel structure and generalize reliably to unseen channel realizations.It provides a robust initial symbol estimate for the refinement stage.Source: A. Contributions · p. 2
• Gated iterative updates use prior symbol probabilities, reconstructed signals, and received-signal feedback to suppress interference; experiments report lower BER than MMSE, MP, and single-shot CNN detectors.The authors also report improved initializer generalization to unseen channels, progressively reduced residual errors, a lower error floor, and inference-time comparisons under single-frame and batched settings.Source: A. Contributions · p. 2

B. Notation

• Bold lowercase letters denote vectors or probability tensors, bold uppercase letters denote matrices or higher-order tensors, and indexed non-bold variables denote individual elements.The notation also defines conjugate, expectation, modulo, real-part, imaginary-part, real and complex sets, circularly symmetric complex Gaussian distributions, and the Dirac delta function.Source: B. Notation · p. 2

II. SYSTEM MODEL

• The model considers a point-to-point single-input single-output OTFS system with rectangular transmit and receive pulses.It covers the transmit signal, doubly-dispersive channel, and received signal models.Source: §II · p. 2

A. OTFS Transmit Signal Model

• Each OTFS frame uses N time slots and M subcarriers, with slot duration T = TsM, subcarrier spacing ∆f = 1/T, frame duration Tf = TN, and bandwidth B = ∆fM.Ts is the sampling period.Source: A. OTFS Transmit Signal Model · p. 3
• The delay–Doppler grid has M × N locations, holding MN information symbols from an alphabet of size Q as XDD[ℓ, k].The indices run over ℓ = 0, . . . , M − 1 and k = 0, . . . , N − 1.Source: A. OTFS Transmit Signal Model · p. 3
• The transmitter maps delay–Doppler symbols to time–frequency samples using the inverse symplectic fast Fourier transform (ISFFT).The transformed samples are then passed through a Heisenberg transform parametrized by the transmit pulse gtx(t) to form the continuous-time signal.Source: A. OTFS Transmit Signal Model · p. 3

B. Doubly-Dispersive Channel

• The doubly-dispersive channel is represented in the delay–Doppler domain by P propagation paths, each with complex gain hi, delay τi, and Doppler shift νi.The model also defines normalized delay and Doppler parameters for the paths.Source: B. Doubly-Dispersive Channel · p. 3
• Each path has integer delay and Doppler indices ℓi and ki, with fractional Doppler component −0.5 < κi ≤ 0.5.Source: B. Doubly-Dispersive Channel · p. 3
• The received signal is defined using the delay–Doppler channel representation and includes additive noise w(t).Source: B. Doubly-Dispersive Channel · p. 3

C. OTFS Received Signal Model

• The receiver obtains the time–frequency representation using the cross-ambiguity function between the matched filter and received signal.Source: C. OTFS Received Signal Model · p. 3
• The continuous time–frequency representation is sampled before further processing.Source: C. OTFS Received Signal Model · p. 3
• Applying the symplectic fast Fourier transform (SFFT) to the sampled time–frequency representation yields the delay–Doppler representation.Source: C. OTFS Received Signal Model · p. 3
• Under fractional Doppler shifts, each transmitted symbol spreads along the Doppler dimension, causing inter-Doppler interference (IDI).The spreading is approximated by retaining the N_i dominant leakage components on each side of the peak, where N_i << N [7].Source: C. OTFS Received Signal Model · p. 3

III. PROPOSED TWO-STAGE FRAMEWORK

• The IRN iteratively refines initial symbol estimates to progressively correct residual errors in the delay–Doppler domain.The framework places this refinement stage after channel-conditioned initialization.Source: §III · p. 4, Figure 1 · p. 4
• The two-stage receiver combines channel-conditioned initialization with iterative refinement and is reported to improve on conventional and single-shot CNN-based detectors across varied channel conditions.The initial estimates are refined by the IRN over T iterations.Source: §III · p. 4, Figure 1 · p. 4

A. Channel-Conditioned Initialization

• Align-Combine organizes and combines received signals using channel information and delay–Doppler shift relationships, giving the detection network channel-aware features.The initialization module feeds these features to an RDN and classifier head to produce initial symbol probabilities and a detection state.Source: A. Channel-Conditioned Initialization · p. 4
• An RDN extracts and refines channel-conditioned features into the initial detection state, and a classifier head maps that state to symbol probabilities.The initial estimates and state start the subsequent IRN; the RDN backbone is motivated by prior OTFS detection results on dense connectivity and feature reuse.Source: A. Channel-Conditioned Initialization · p. 4, Figure 1 · p. 4
• Align-Combine applies predetermined circular shifts for all supported integer delay–Doppler displacements, assuming known maximum delay and Doppler support.The shifts correspond to inverse shifts of the channel displacements in the OTFS input–output relationship.Source: A. Channel-Conditioned Initialization · p. 4
• The aligned copies are concatenated across feature channels, with real and imaginary components represented separately.The resulting stack organizes the received signal according to supported channel-tap shifts.Source: A. Channel-Conditioned Initialization · pp. 4–5
• Channel-conditioned convolution uses 1 × 1 kernels for integer Doppler shifts and 1 × (2Ni + 1) kernels for fractional shifts to capture Doppler leakage.Integer shifts do not spread energy across Doppler bins, whereas fractional shifts spread energy across neighboring bins.Source: A. Channel-Conditioned Initialization · p. 5

1) Integer Doppler Kernel Generation:

• The integer-Doppler input is a sparse estimated channel representation whose features concatenate the real and imaginary parts of each estimated tap.The feature vector is h_in ∈ R^2C.Source: 1) Integer Doppler Kernel Generation · p. 5
• Each integer delay–Doppler tap groups propagation paths whose integer delay and Doppler indices match that tap.P_c is the set of path indices i for which (ℓ_i, k_i) = (ℓ^(c), k^(c)).Source: 1) Integer Doppler Kernel Generation · p. 5
• A three-layer MLP processes the integer-Doppler channel representation to generate the corresponding 1 × 1 convolution kernel.The architecture is described in Table I.Source: 1) Integer Doppler Kernel Generation · p. 5

2) Fractional Doppler Kernel Generation:

• For fractional Doppler, the method first constructs a channel gain-weighted per-tap fractional Doppler spread.Source: 2) Fractional Doppler Kernel Generation · p. 5
• The fractional-Doppler representation concatenates real and imaginary components along the spatial dimension, then uses a two-layer MLP and a small CNN to learn spread features and capture interactions across feature channels.The MLP mixes real and imaginary components independently across taps; its output is reshaped to R^(1 × (2N_i+1) × 2C) before the 1 × 1 convolutions.Source: 2) Fractional Doppler Kernel Generation · p. 5
• The generated integer- and fractional-Doppler kernels are convolved with the extended input Y_ext, and the resulting channel-conditioned features are processed by the RDN to produce the initial detection state.The classifier head maps that state to the initial symbol estimates.Source: 2) Fractional Doppler Kernel Generation · p. 5
• The overall module comprises Align-Combine alignment, channel-conditioned kernel generation, convolution parameterized by W, an RDN, and a classifier head.These operators are denoted T(·), f_θ(·), F_W(·), G_ϕ(·), and C_ψ^(0)(·), respectively.Source: 2) Fractional Doppler Kernel Generation · p. 6

B. Iterative Refinement Network

• The IRN addresses residual errors in the single-shot initializer by carrying a detection state across iterations, applying learned residual corrections, and passing refined symbol probabilities forward.Refinement parameters are not shared across iterations, allowing each step to learn a distinct correction.Source: B. Iterative Refinement Network · p. 6
• At each step, previous symbol probabilities yield soft symbol estimates that reconstruct the received signal and per-tap features indexed by fixed delay–Doppler shifts.The per-tap channels associate feature channels with predefined physical shifts, structuring representation of each channel tap’s contribution.Source: B. Iterative Refinement Network · p. 6
• The refinement input retains the original received signal and explicitly supplies the observation–reconstruction residual and its magnitude to expose mismatch and error severity.Retaining the original observation guards against information loss in the reconstructed signal.Source: B. Iterative Refinement Network · p. 7
• Each refinement layer uses three residual dense blocks, concatenates and fuses their outputs with a 1 × 1 convolution, and combines the update with the prior state using a learnable scalar.Dense and residual connections support feature reuse across heterogeneous inputs.Source: B. Iterative Refinement Network · p. 7
• A classifier head converts each updated detection state into symbol probabilities; its architecture and parameter count match those of the initial-estimation heads.Classifier heads are provided within the iterative refinement network.Source: B. Iterative Refinement Network · p. 7

C. Training

• Each epoch uses 1,800 channel realizations with 65 OTFS frames each, uniformly sampled symbols and SNRs from 5–25 dB, and Adam with a cosine learning-rate schedule.Frames are generated under the doubly-dispersive channel model described in Section II.Source: C. Training · p. 7
• Cross-entropy loss supervises symbol-probability estimates at initialization and every refinement stage, with uniform weighting that does not prioritize any stage.The objective uses the predicted symbol distribution at iteration t, the ground-truth labels, and the total number of refinement iterations.Source: C. Training · p. 7
• Figure 4 compares initialization-stage cross-entropy loss with and without the Align–Combine module for 16-QAM.Source: Figure 4 · p. 7

IV. SIMULATION RESULTS

• Simulations use a 4 GHz carrier, 15 kHz subcarrier spacing, and 32 delay and Doppler bins; the default channel has five multipath components and maximum Doppler index 2.Delays are uniformly drawn from [0, 3], Doppler shifts follow Jakes’ model, and channel gains are CN(0, 1/P).Source: §IV · p. 7
• The simulation section assesses the proposed receiver’s performance in an OTFS system under doubly-dispersive channel conditions.Unless otherwise specified, the setup uses the channel parameters described above.Source: §IV · p. 7

A. Architectural Evaluation

• Removing Align-Combine prevents the RDN initializer from generalizing to unseen channels, showing the importance of channel-aware feature construction.Training and validation use disjoint channel realizations, and the comparison evaluates the initialization stage on raw received signals.Source: A. Architectural Evaluation · p. 7
• The proposed initializer gives the refinement network a more reliable starting state and lower early-iteration BER than the IRN-only variant at 15 dB and 20 dB.Both variants approach similar error floors at higher iteration counts, but the proposed initialization requires fewer iterations to reach a given BER.Source: A. Architectural Evaluation · pp. 7–8, Figure 5 · p. 8
• The decomposition evaluation compares full reconstruction, per-tap reconstruction, and per-symbol reconstruction, with the latter inspired by MP detection.Full reconstruction separates real and imaginary components; per-tap separates delay–Doppler tap contributions; per-symbol assigns each symbol’s contribution its own feature-map channel.Source: A. Architectural Evaluation · p. 8
• Per-tap reconstruction converges faster and reaches a lower BER floor than full or per-symbol reconstruction at both SNR levels.Its feature channels preserve a fixed mapping to delay–Doppler taps, supporting effective iterative refinement.Source: A. Architectural Evaluation · p. 8

• With more than four iterations, the proposed detector beats all baselines across integer-Doppler SNR; at four iterations, it beats MP at low-to-moderate SNR but MP is slightly stronger at high SNR.Source: Figure 7 · p. 8, B. Detection Performance · pp. 8–9
• Under fractional Doppler, four iterations outperform RDN, RCAN, MP, and MMSE, while additional iterations progressively lower BER across the SNR range.RDN beats MMSE and MP at low SNR but degrades at mid-to-high SNR; RCAN improves upon its MMSE prior across SNR.Source: Figure 8 · p. 9, B. Detection Performance · p. 9
• The proposed detector maintains nearly identical performance under integer and fractional Doppler, while MP degrades under fractional Doppler and MMSE improves slightly.RCAN also improves under fractional Doppler because its augmented input includes the received signal and initial MMSE estimate.Source: B. Detection Performance · p. 9

C. Robustness to Multipath Variations

• As paths increase from P = 5 to 9, the proposed detector gains more than MMSE, MP, and RCAN, especially at high SNR where unresolved multipath interference dominates.MMSE shows only marginal improvement, while MP and RCAN show moderate gains; the proposed detector’s larger gains are most pronounced at high SNR.Source: C. Robustness to Multipath Variations · p. 9
• With more propagation paths, iterative refinement yields substantially larger gains than RCAN’s single-pass estimate, indicating greater benefit from exploiting additional multipath diversity.RCAN concatenates an MMSE symbol estimate with the raw received signal as auxiliary information; the proposed detector refines estimates over multiple stages.Source: C. Robustness to Multipath Variations · p. 9
• Under mixed channel-path-count training, the proposed neural detector achieves BER comparable to fixed P = 5 training and consistently outperforms considered baselines across the reported range of P.This demonstrates suitability for practical scenarios where the number of multipath components is unknown a priori and may vary over time.Source: C. Robustness to Multipath Variations · pp. 9–10

D. Robustness to Power Delay Profile Variations

• Across constant and exponential PDPs with α ∈ {0.2, 0.5, 1}, the detector reliably detects transmitted bits, with gradual degradation as α increases and effective multipath diversity falls.Larger decay factors concentrate more channel energy in early delay taps and diminish weaker multipath components.Source: Figure 10 · p. 10, D. Robustness to Power Delay Profile… · p. 10
• For PDP robustness, training uses Rayleigh-fading synthetic channels with constant and normalized exponential profiles, sampling α uniformly from [0.1, 1] for each realization.Source: D. Robustness to Power Delay Profile… · p. 10

E. Generalization to Standardized Channel Models

• On unseen 3GPP EVA and ETU channels, the synthetic-trained detector achieves BER comparable to oracle models across the evaluated SNR range and slightly outperforms them at high SNRs.The oracle models are trained exclusively on their respective EVA or ETU channels.Source: Figure 11 · p. 10, E. Generalization to Standardized… · p. 10

F. Robustness to Doppler Variations

• Across the considered velocity range, the proposed detector maintains nearly constant BER at Eb/N0 = 15 dB and 20 dB, while MMSE is also relatively stable and MP improves as Doppler spread increases.Each Proposed curve corresponds to a detector instance trained for the corresponding maximum Doppler shift.Source: F. Robustness to Doppler Variations · pp. 10–11

G. Complexity

• At 5 iterations, the proposed method reaches BER 1.2 × 10−4, while MP falls from 2.2 × 10−1 at 5 iterations to 2.3 × 10−3 at 15.The proposed method is close to its BER saturation level at 5 iterations and improves only marginally at higher iteration counts; MP requires more iterations to approach convergence.Source: Table IV · p. 11, G. Complexity · p. 11
• MP runtime grows sub-linearly with iterations and is higher than the proposed method at 5 iterations, but the proposed method is slower beyond 10 iterations.The proposed method's runtime increases approximately linearly, whereas MP's runtime is affected by fixed overheads and early stopping.Source: Table IV · p. 11, G. Complexity · p. 11

F. Robustness to Doppler Variations

• A model trained at kmax = 5 performs comparably to velocity-specific models in neighboring Doppler regimes, but degrades more noticeably at kmax = 2 and kmax = 1.The passage identifies generalization to nearby Doppler regimes as useful when maximum Doppler shift is unknown or varies over time.Source: F. Robustness to Doppler Variations · p. 11

G. Complexity

• Runtime tests used batch size one across 104 OTFS test frames on an NVIDIA A100, with vectorized MP updates and PyTorch tensor operations for MMSE.The implementations were designed to support a consistent GPU-based runtime comparison.Source: G. Complexity · p. 11
• At 10 iterations, proposed runtime is 106.6 ms per batch for B = 32 versus 108.8 ms for B = 1; MP rises from 111.5 ms to 418.4 ms.The proposed method maintains nearly identical runtime across these batch sizes, indicating efficient use of GPU parallelism.Source: G. Complexity · p. 11

V. CONCLUSION

• The Align-Combine module uses the known OTFS delay-Doppler relationship and channel-conditioned convolutional kernels, while the IRN iteratively refines symbol estimates to suppress multipath interference.The conclusion reports improved BER particularly in the mid-to-high SNR regime, with effectiveness across varying velocities, multipath environments, and unseen channel distributions.Source: §V · p. 11
• Future work could jointly refine channel and symbol estimates by updating channel state from pilots and evolving symbol estimates, rather than treating the channel estimate as fixed.The proposed joint refinement would feed the refined channel estimate into subsequent symbol detection.Source: §V · p. 11

APPENDIX

• Each FLOP counts one real multiplication or addition, with complex operations decomposed into real arithmetic for the complexity analysis.A complex multiplication counts as 4 real multiplications and 2 real additions; a complex addition counts as 2 real additions.Source: APPENDIX · p. 12
• Align-Combine alignment costs 2MNC FLOPs, while combining includes fractional-Doppler representation construction, kernel generation, and convolution over the delay-Doppler grid.The representation construction costs 10P(2Ni + 1) FLOPs; the passage also specifies separate MLP and CNN kernel-generation costs.Source: APPENDIX · p. 12
• A single RDB uses densely connected convolutional layers, local feature fusion, and a residual connection to aggregate its input and output over an M × N feature map.The passage denotes its total operation count as FLOPsRDB but does not include the full expression in the supplied text.Source: APPENDIX · p. 12

+ 2CRDB

• The RDN stacks NRDB RDB blocks followed by a feature-fusion convolution, with the fusion cost depending on the detection-state channel dimension Cstate.The supplied expression for the RDN cost is incomplete.Source: + 2CRDB · p. 12
• The classifier head costs 128MN(Cstate + Q) FLOPs.Source: + 2CRDB · p. 12
• Each IRN iteration comprises soft symbol estimation, signal reconstruction, three-RDB feature refinement, gated state update, and a classifier head.Their respective costs are 4MNQ, 10MN(2Ni + 1)P, FLOPsRDB1 + FLOPsRDB2 + FLOPsRDB3 + 2MN(3CRDB out)Cstate, 3MNCstate, and 128MN(Cstate + Q) FLOPs.Source: + 2CRDB · p. 12
• The optimized MP comparison counts only per-iteration message passing, excluding precomputed quantities and one-time initialization overhead without changing the algorithm or its asymptotic complexity.A single iteration sums mean and variance updates costing 2MNṔ(5 + 2Q) and MNṔ(2Q + 8) FLOPs, plus probability updates costing 15MNṔQ FLOPs; Ṕ ≤ P(2Ni + 1).Source: + 2CRDB · p. 12
• MMSE equalization is approximated at 32(MN)^3 + 16(MN)^2 + 2MN FLOPs, and the appendix says numerical counts are summarized in Table V.Counts use M = N = 32, Q = 16 (16-QAM), Ni = 10, and P = 5, along with the network hyperparameters specified in Section III.Source: + 2CRDB · p. 12