PaperlayerView exact revision on arXiv

Source-linked AI summary

Exact Payload-Decoupling Conditions for Pilot-Only BEM Channel Estimation With Application to OTFS

Gianmarco Romano, Francesco A. N. Palmieri, Stefano Buzzi, Giovanni Di Gennaro, Amedeo Buonanno

arXiv:2609.00937v1cs.IT

TL;DR

Unknown payload symbols can contaminate conventional matched-pilot BEM channel estimates in high-mobility doubly dispersive links. The paper proves an exact zero pilot–data interference condition, gives a basis-independent pilot/guard/data placement rule, and applies it to OTFS, where the resulting estimator approaches perfect-CSI performance in the reported settings.

  • Problem

    Unknown payload symbols generally contaminate conventional matched-pilot channel estimates, limiting pilot-only BEM estimation in high-mobility doubly dispersive links.

  • Method

    The paper proves a necessary-and-sufficient ZPDI condition and derives a basis-independent disjoint-support rule for pilot, guard, and data placement.

  • Results

    Under ZPDI, matched-pilot LS coincides with ML for the reduced statistic; without ZPDI, deterministic channel-scaled payload bias persists at high SNR.

  • Takeaways & Limitations

    ZPDI enables a precomputed pilot-only projection, while the pilot Gram matrix separately governs coefficient identifiability, conditioning, and noise enhancement.

Abstract

from arXiv · show

In high-mobility doubly dispersive links, basis expansion models (BEMs) reduce channel dimensionality, yet unknown payload symbols generally contaminate conventional matched-pilot channel estimates. This paper establishes the exact conditions under which such estimates become payload-independent and derives a pilot, guard, and data-placement rule that guarantees these conditions. We prove a necessary-and-sufficient zero pilot--data interference (ZPDI) condition under which the matched-pilot least-squares (LS) solution coincides with the maximum-likelihood (ML) estimator for the reduced pilot statistic. When ZPDI holds, estimation requires a single precomputed projection. When it does not, the estimate contains a deterministic, channel-scaled payload bias that persists at high signal-to-noise ratio. A disjoint-support rule, independent of the selected basis, realizes ZPDI through pilot, guard, and data placement. We then specialize the framework to orthogonal time--frequency space (OTFS) and examine its structural and performance consequences. With the generalized complex-exponential BEM (GCE-BEM), the ZPDI estimator remains within about $2$~dB of the perfect channel state information benchmark in bit error rate at speeds up to 500~km/h.

I. INTRODUCTION

The paper addresses payload contamination in BEM-based channel estimation for doubly dispersive, high-mobility links and establishes exact conditions for eliminating it. It provides a basis-independent placement rule and applies the framework to OTFS.

  • High-mobility doubly dispersive channels create rapid variation, large Doppler spreads, and severe inter-carrier interference, challenging conventional OFDM.
  • The paper proves necessary-and-sufficient conditions for matched-pilot LS estimation to be payload-independent and equal to ML estimation for the reduced statistic.
  • The framework covers arbitrary BEM subspaces, unitary observation transforms, and cyclic-prefix, zero-padding, and chirp-periodic-prefix guard structures.
  • When ZPDI holds, the estimator uses precomputable matrices and has online complexity linear in frame length, while residual error separates into noise and BEM modeling error.
  • A disjoint-support rule places pilots, guards, and data so zero pilot–data interference holds independently of the selected BEM basis.
  • BEMs reduce channel estimation from NL apparent parameters to QL coefficients by representing the channel in a finite-dimensional basis, with modeling error as the trade-off.

III. PILOT-PROJECTED CHANNEL ESTIMATION AND ZERO PILOT–DATA INTERFERENCE

This section shows why the conventional matched-pilot projection is contaminated by unknown payload symbols and derives the exact condition that removes this interference. Under ZPDI, the resulting pilot-only estimator is a closed-form LS/ML solution for the reduced observation.

  • BEM channel estimation recovers coefficient vector γ from a reduced observation, but the receiver cannot reconstruct terms involving unknown payload symbols before detection.
  • The matched-pilot filter projects onto the known pilot subspace, yet its statistic retains a pilot–data interference term caused by cross-correlation between pilot and payload sensing matrices.
  • The interference lies in the same QL-dimensional recovery subspace as the useful pilot term, so linear filtering of the pilot statistic cannot separate it.
  • At high SNR, the PDI remains as a deterministic, channel-scaled payload bias; increasing pilot power attenuates but does not cancel it, and noise averaging leaves it unchanged.
  • ZPDI requires the pilot–payload sensing cross-correlation to vanish for every admissible payload, eliminating the PDI term for every channel realization.
  • With ZPDI, the pilot Gram matrix yields a closed-form matched-pilot LS/ML estimator that is independent of payload realization and assumed payload distribution.

A. Zero Pilot–Data Interference Condition

The paper characterizes exactly when matched-pilot estimation is independent of unknown payload symbols and provides a practical pilot, guard, and data-placement rule that guarantees this property.

  • Exact condition: The matched-pilot estimator is payload-independent for every admissible payload and channel coefficient vector if and only if ZPDI holds.Under ZPDI, the estimator coincides with the ML estimator for the reduced pilot-projected observation under Gaussian noise.
  • Failure mode: If ZPDI fails, the matched-pilot estimate contains a deterministic, channel-scaled payload bias that persists at high SNR.The necessity proof follows because exact recovery for every payload and coefficient vector requires the payload cross-term to vanish identically.
  • Exact condition: ZPDI requires the pilot and payload column spaces to be orthogonal after every admissible delay shift.Equivalently, all L^2 pilot–data cross-correlation blocks must vanish, not merely one delay-pair block.
  • Practical consequences: The ZPDI estimator needs no payload decisions or channel priors and reduces online processing to one precomputed matrix–vector product.Its remaining error is attributed to thermal noise, pilot Gram conditioning, finite BEM modeling error, and support or synchronization mismatch.
  • Practical design rule: A basis-independent sufficient rule requires every independently delay-shifted pilot and data vector to have disjoint support.The proof uses D^H D = I and the elementwise product of shifted pilot and data vectors to force every cross-correlation block to zero.
  • Practical design rule: The guard must span the full delay support on either side: a narrower guard permits shifted pilot–data overlap, while a wider guard only consumes resources.This rule is stronger than merely separating pilot and data positions in the transmitted block.

IV. IDENTIFIABILITY, CONDITIONING, AND COMPLEXITY OF THE ZPDI ESTIMATOR

After ZPDI removes payload interference, the pilot Gram matrix determines whether the BEM coefficients are identifiable and how the estimator behaves numerically.

  • Role of the Gram matrix: Once ZPDI holds, Rpp governs coefficient recovery and sensitivity because it describes the inner geometry of the channel-shifted pilot waveforms.The guard annihilates the pilot–data cross-term, whereas Rpp determines identifiability and inversion sensitivity.

A. Identifiability

ZPDI removes payload contamination but does not by itself ensure unique recovery of the QL BEM coefficients; full column rank of the pilot matrix remains necessary.

  • Full-rank requirement: The ZPDI estimator is well defined only when Ψp has full column rank, equivalently when the pilot Gram matrix is invertible.Thus payload decoupling and coefficient identifiability are distinct requirements.
  • BEM-order limit: The BEM order is limited because pilots provide only Nobs independent observations while the model contains QL unknown coefficients.The basis cannot create observations that the pilot excitation does not supply.
  • BEM-order limit: Choosing Q above the direct counting bound makes Ψp rank deficient, but satisfying the count is necessary rather than sufficient.The retained basis functions may still be linearly dependent under the pilot operator, so the operative condition remains full column rank.

B. Conditioning and Noise Enhancement

With ZPDI established, conditioning controls noise sensitivity, while additional pilot-structure conditions can reduce offline inversion cost without changing the online complexity order.

  • Conditioning and noise: Under ZPDI, estimation accuracy depends on how Rpp^-1 shapes matched noise; near-singularity causes large errors despite unique recovery.The condition number κ(Rpp) measures this sensitivity, while a scaled-identity Gram matrix treats coefficient directions comparably.
  • Pilot structure: ZPPL nulls cross-Gram blocks between distinct delay taps, decoupling their coefficient groups once ZPDI is established.When each diagonal Q × Q block is nonsingular, the estimator separates into L per-tap estimators.
  • Pilot structure: Rpp is diagonal only when ZPPL also combines with pilot-weighted orthogonality among BEM functions within each delay tap.Even then, unequal diagonal entries mean diagonality alone does not guarantee ideal conditioning.
  • Complexity: A dense Rpp requires O(Q^3L^3) inversion, whereas ZPPL reduces the cost to O(LQ^3) and enables parallel block inversions.If the additional within-tap orthogonality condition holds, inversion reduces to QL scalar reciprocals.
  • Complexity: The projection can be precomputed once, leaving each frame with O(NQL) online complexity.ZPPL makes the offline inversion cheaper and parallelizable but does not change the online order unless further pilot sparsity is exploited.

V. OTFS CHANNEL ESTIMATION AND GCE-BEM ANALYSIS

The paper specializes its pilot-only BEM estimation framework to OTFS in the delay–Doppler domain, using a single pilot with a full guard. This pattern satisfies ZPDI when the guard placement also prevents cyclic wrap-around interference.

  • OTFS model: OTFS represents symbols on a K × M delay–Doppler grid spanning N = KM time samples.The reduced-CP variant prepends one cyclic prefix to the complete frame before transmission.
  • OTFS model: The received OTFS model has the same block-linear structure as the general formulation after identifying its transformation and channel operators.The time-domain doubly dispersive channel is represented by H, with s obtained by vectorizing the delay–Doppler grid and w denoting additive noise.
  • Pilot and guard placement: The analyzed pattern embeds one pilot in the delay–Doppler grid and surrounds it with null guards spanning all Doppler bins and the channel’s required delay extent.The guard contains (2L −1)M −1 null symbols, while payload occupies positions outside the guard region.
  • Pilot and guard placement: The pattern reserves a fraction (2L −1)/K of grid cells for channel estimation, so overhead increases with the channel delay-spread parameter L.The overhead remains moderate for moderate delay spreads but becomes more costly as L increases.
  • ZPDI condition: The full-guard pattern satisfies the zero pilot–data interference condition and therefore admits an exact, interference-free ZPDI estimator.This conclusion follows from the disjoint-support verification for pilot and shifted payload vectors.
  • Pilot and guard placement: The pilot placement must satisfy K > 2L −2 and keep the delay guard inside the grid to prevent cyclic wrap-around interference.The condition is equivalent to requiring the upper and lower guard boundaries not to overlap after modulo-K wrapping.

D. Pilot–Pilot Leakage

The OTFS full-guard pattern also prevents pilot–pilot leakage across delay taps. Consequently, its pilot Gram matrix becomes block diagonal and channel estimation separates into independent per-tap solves, subject to BEM identifiability.

  • ZPPL condition: The single-pilot full-guard pattern satisfies ZPPL because shifted pilot vectors have disjoint supports across different delay taps.The result holds for any BEM basis and is stronger than ordinary vector orthogonality.
  • Pilot energy: Each shifted pilot vector has energy Ep, with Ep = |sp|2, so the pilot excitation is preserved across delay taps.Every shifted vector contains M nonzero entries whose combined energy equals the pilot energy.
  • Gram structure: Disjoint pilot supports make the pilot Gram matrix Rpp = Ψp^HΨp block diagonal.Its blocks are the per-delay-tap Gram matrices of the BEM basis sampled at the corresponding pilot positions.
  • Per-tap estimation: The estimator therefore separates into L independent per-tap solves, each governed by a Q × Q basis Gram block Tℓ.For each tap, the block scales as Rℓℓ = (Ep/M)Tℓ.
  • Identifiability: Each per-tap block has rank at most M, so invertibility requires the BEM order Q not to exceed the number of Doppler bins.The single-pilot full-guard pattern supplies Nobs = M independent observations per delay tap.

E. GCE-BEM Analysis

The GCE-BEM converts the per-tap pilot Gram blocks into Hermitian Toeplitz matrices governed by a Dirichlet kernel. Oversampling improves fractional-Doppler resolution but worsens conditioning, whereas R = 1 yields ideal conditioning.

  • GCE-BEM construction: The GCE-BEM uses complex exponentials with frequencies ωq = 2π/(NR), where R ≥1 controls Doppler-grid spacing.R = 1 gives the conventional CE-BEM grid, while larger R refines the grid for fractional Doppler components.
  • Per-tap Gram blocks: For the single-pilot pattern, each per-tap Gram block Tℓ is obtained by summing basis products over the geometric pilot positions.The resulting entries admit a closed form involving the Dirichlet kernel DM(x) = sin(Mx)/(M sin x).
  • Per-tap Gram blocks: Each Tℓ is Hermitian Toeplitz with diagonal entries 1/K, while oversampling populates off-diagonal terms by reducing basis orthogonality.The rank bound rank(Tℓ) ≤ M gives the GCE-BEM identifiability ceiling Q ≤ M.
  • Resolution–conditioning trade-off: For R > 1, larger oversampling reduces fractional-Doppler modeling error but raises κ(Rpp) and amplifies high-SNR noise.Thus Doppler resolution and numerical conditioning trade off through the same loss of basis orthogonality.
  • Conventional CE-BEM case: When R = 1 and Q ≤ M, Rpp = (Ep/N)IQL, its condition number is κ(Rpp) = 1, and no matrix inversion is required.In this case, the per-tap estimator reduces to a scaled matched filter.

VI. NUMERICAL ILLUSTRATIONS AND END-TO-END RESULTS

The numerical studies test ZPDI consequences, Gram-matrix structure, and end-to-end OTFS estimation under exact-BEM and practical model-mismatch settings. Full guarding eliminates pilot–data leakage and its high-SNR estimation floor, while identifiability and conditioning depend on BEM order and basis choice.

  • Simulation setup: A single pilot with a full guard uses 112 pilot-plus-guard cells and 1936 data cells on a 2048-cell OTFS grid.The resulting pilot overhead is approximately 5.5%.
  • Experimental scope: The simulations evaluate ZPDI decoupling, error decomposition, Gram-matrix properties, and end-to-end BER and MSE consequences for OTFS.The exact-BEM setup isolates payload-dependent bias and thermal noise before practical comparisons.
  • Payload decoupling: Only the full symmetric delay guard d = L −1 reduces worst-case pilot–data leakage to machine precision; deficient guards leave nonzero leakage.For deficient guards d ∈{0, 1, 2}, the reported leakage values are {8.2, 6.7, 4.7} × 10−2.
  • Error decomposition and interference floor: The ZPDI full-guard estimator follows the noise-only NMSE curve without a floor, whereas non-ZPDI bias creates a persistent high-SNR floor.Increasing pilot power attenuates the non-ZPDI floor as 1/Ep but does not eliminate it at any finite ratio.
  • Identifiability, conditioning, and structure: For CE-BEM, Rpp remains full rank through Q ≤M; for Q > M, the rank deficit is L(Q −M) and structural singularity begins at Q = M + 1.At Q = 9 and R = 2, the off-tap Gram energy is zero to machine precision, while κ(Rpp) ≈7.0 × 105; the CE-BEM limit has κ(Rpp) = 1.

B. ZPDI Estimator

The ZPDI receiver uses a pilot-only channel estimate and evaluates GCE-BEM choices across mobility conditions. At 500 km/h, the ZPDI GCE-BEM result remains close to perfect CSI, while basis mismatch produces floors for competing CE-BEM-based methods.

  • Receiver comparison: The standalone ZPDI receiver uses a TD-LMMSE equalizer built from the estimated channel, alongside comparisons with perfect CSI and prior estimators.The pilot-only initialization is also used in an iterative BEM receiver comparison.
  • BEM selection: The simulations consider 125 and 500 km/h, selecting GCE-BEM order within a lower modeling-coverage bound and the identifiability limit.Increasing Q reduces modeling error but enlarges the coefficient vector and Gram matrix, while the noise term grows linearly with Q.
  • BER results: At 500 km/h, the ZPDI GCE-BEM estimator with R = 2 and Q = 9 is within about 2 dB of perfect CSI at high SNR.At 125 km/h, GCE-BEM with Q = 3 stays close to perfect CSI.
  • BER and MSE results: CE-BEM and Liu-initialization methods exhibit high-SNR BER floors, consistent with residual basis mismatch, while the estimator of also floors at 500 km/h.The corresponding MSE curves account for these BER floors, whereas ZPDI GCE-BEM avoids a visible interference-induced floor.

C. ZPDI Initialization of a Decision-Directed Receiver

The pilot-only ZPDI estimate provides a deterministic initialization for decision-directed refinement at 500 km/h, using a precomputed projection before payload decisions are available. Two data-aided refinements substantially improve BER and MSE relative to standalone initialization and the Liu receiver.

  • Initialization and refinement: The pilot-only ZPDI estimate initializes iterative data-aided LS refinement before payload detection.Initialization requires only the precomputed projection Wpr; later refinements use detected payload symbols.
  • Initialization and refinement: At 500 km/h, the ZPDI-initialized receiver applies two data-aided LS refinements with the same basis.The ZPDI branch uses (R, Q) = (2, 9) in both stages, while the Liu receiver uses separate initialization and refinement settings.
  • Performance: Two refinements after ZPDI initialization bring BER close to perfect CSI and below the standalone pilot-only ZPDI curve.The corresponding MSE also improves relative to standalone ZPDI initialization.
  • Performance: The Liu receiver retains high-SNR BER floors after one and two iterations, while up to 20 iterations reduce but do not eliminate the floor.At high SNR, two ZPDI-based refinements match the 20-iteration Liu result in MSE.
  • Performance: At 125 and 500 km/h, GCE-BEM ZPDI estimation approaches perfect-CSI BER, whereas CE-BEM and reference estimators exhibit high-SNR floors.At 500 km/h, the ZPDI estimator has no visible floor over the simulated SNR range.
Loading 2609.00937v1…