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Compressive estimation of doubly selective channels in multicarrier systems: Leakage effects and sparsity-enhancing processing

Georg Tauboeck, Franz Hlawatsch, Daniel Eiwen, Holger Rauhut

arXiv:0903.2774v2cs.IT

TL;DR

The paper addresses compressed-sensing channel estimation for doubly selective multicarrier channels, where leakage weakens delay-Doppler sparsity and strong dispersion creates ISI/ICI. It proposes optimized sparsity-enhancing bases and an estimator for off-diagonal coefficients, with simulations showing considerable performance gains at moderate additional complexity.

  • Problem

    Delay-Doppler leakage limits the sparsity exploited by compressed sensing, while strongly dispersive channels require estimation of ISI/ICI off-diagonal coefficients.

  • Method

    The paper replaces the DFT with sparsity-enhancing basis expansions, optimizes the basis iteratively with or without channel statistics, and combines Fourier and prolate spheroidal sequences for ISI/ICI estimation.

  • Results

    Simulation results demonstrated considerable performance gains from sparsity-enhancing basis expansions and explicit ISI/ICI coefficient estimation.

  • Takeaways & Limitations

    The proposed processing improves compressive channel estimation while requiring moderate additional computational complexity, with bases precomputable before data transmission.

Abstract

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We consider the application of compressed sensing (CS) to the estimation of doubly selective channels within pulse-shaping multicarrier systems (which include OFDM systems as a special case). By exploiting sparsity in the delay-Doppler domain, CS-based channel estimation allows for an increase in spectral efficiency through a reduction of the number of pilot symbols. For combating leakage effects that limit the delay-Doppler sparsity, we propose a sparsity-enhancing basis expansion and a method for optimizing the basis with or without prior statistical information about the channel. We also present an alternative CS-based channel estimator for (potentially) strongly time-frequency dispersive channels, which is capable of estimating the "off-diagonal" channel coefficients characterizing intersymbol and intercarrier interference (ISI/ICI). For this estimator, we propose a basis construction combining Fourier (exponential) and prolate spheroidal sequences. Simulation results assess the performance gains achieved by the proposed sparsity-enhancing processing techniques and by explicit estimation of ISI/ICI channel coefficients.

I. INTRODUCTION

The paper applies compressed sensing to doubly selective channels in pulse-shaping multicarrier systems, exploiting delay-Doppler sparsity to reduce pilot requirements. It develops estimators for mildly and strongly dispersive channels, including leakage mitigation and ISI/ICI coefficient estimation.

  • I. INTRODUCTION: Compressed sensing estimates doubly selective multicarrier channels by exploiting their approximately sparse delay-Doppler representations.The framework includes OFDM as a special case and targets channels dominated by relatively few significant path clusters.
  • I. INTRODUCTION: The basic estimator targets diagonal channel coefficients and is intended for mildly dispersive channels where ISI and ICI are small.For pilot positions, interference is absorbed into an effective noise term before channel recovery.
  • I. INTRODUCTION: Leakage limits delay-Doppler sparsity, motivating a basis expansion that replaces the conventional DFT and an iterative basis-optimization procedure.The optimization can use prior statistical information about the channel or operate without it.
  • I. INTRODUCTION: An alternative estimator addresses potentially strongly time- and frequency-dispersive channels by estimating off-diagonal coefficients associated with ISI and ICI.These coefficients describe interference across symbols and subcarriers in the system channel.
  • I. INTRODUCTION: The study uses pulse-shaping multicarrier modulation, with CP-OFDM included as a special case.The system comprises modulation, a time-varying channel, filtering, demodulation, and equalization.

III. COMPRESSIVE CHANNEL ESTIMATION

The compressive estimation framework uses pilot observations and sparse recovery to reconstruct channel behavior from a reduced time-frequency sampling grid. It combines delay-Doppler support assumptions with established CS recovery methods and coherence-based pilot requirements.

  • III. COMPRESSIVE CHANNEL ESTIMATION: The basic estimator recovers diagonal channel coefficients, which are sufficient when the channel is mildly dispersive.Pilot observations use the relation r_l,k = H_l,k p_l,k plus effective noise and interference.
  • III. COMPRESSIVE CHANNEL ESTIMATION: Time-frequency subsampling reduces the estimation problem's dimensionality and can improve estimation performance.Pilots are selected from a subsampled grid containing the assumed effective delay-Doppler support.
  • III. COMPRESSIVE CHANNEL ESTIMATION: The receiver uses CS recovery to estimate the delay-Doppler spreading function instead of interpolating channel coefficients across the grid.The recovered spreading function is then transformed into estimates of all channel coefficients.
  • III. COMPRESSIVE CHANNEL ESTIMATION: CoSaMP offers a compromise among low complexity, practical performance, and provable performance guarantees for real-time channel estimation.The passage contrasts it with BP and OMP and reports a faster LSQR implementation with only slightly poorer performance than OMP.

C. Basic Compressive Channel Estimator

The basic estimator converts pilot-based channel observations into a sparse reconstruction problem in a 2-D DFT basis. Recovery of the sparse coefficient vector yields the delay-Doppler representation and, consequently, the channel coefficients.

  • C. Basic Compressive Channel Estimator: The estimator assumes the delay-Doppler spreading function is approximately S-sparse, making the channel representation suitable for compressed sensing.The same approximate sparsity is inherited by the frequency-domain representation F[m, i].
  • C. Basic Compressive Channel Estimator: A 2-D DFT expansion represents subsampled time-frequency channel coefficients using a unitary basis matrix U.The coefficient vector is related to the delay-Doppler representation and is sampled at pilot positions.
  • C. Basic Compressive Channel Estimator: Restricting the expansion to pilot positions produces a CS measurement equation with M = JD variables and Q = |P| measurements.The measurement matrix is formed from pilot-selected rows of U and normalized columns.
  • C. Basic Compressive Channel Estimator: Sparse recovery estimates the coefficient vector x, from which F[m, i] and then all channel coefficients H_l,k are reconstructed.This replaces conventional interpolation of pilot-based channel estimates.
  • C. Basic Compressive Channel Estimator: The required pilot count is governed by sparsity and the dimensions of the delay-Doppler grid under the CS recovery guarantee.The sufficient bound uses the DFT basis coherence μ_U = 1 and randomly selected pilot positions.

IV. DELAY-DOPPLER SPARSITY AND LEAKAGE EFFECT

Finite bandwidth and blocklength spread specular channel paths across the delay-Doppler plane, weakening the sparsity assumed by the basic estimator. The paper therefore introduces generalized basis expansions to improve sparsity.

  • IV. DELAY-DOPPLER SPARSITY AND LEAKAGE EFFECT: Even under specular scattering, the discrete delay-Doppler spreading function is not composed of Dirac-like points at the scatterers' locations.Leakage is characterized by Λ^(ν)(x, y) = φ^(ν)(x)ψ(y).
  • IV. DELAY-DOPPLER SPARSITY AND LEAKAGE EFFECT: Leakage results from finite transmit bandwidth and finite blocklength and produces poorer delay-Doppler sparsity.A larger blocklength reduces leakage but can make the constant-parameter specular model less accurate.
  • IV. DELAY-DOPPLER SPARSITY AND LEAKAGE EFFECT: The Doppler leakage factor ψ decays linearly, or polynomially of order 1, with distance from the Doppler location modulo N_r.This slow decay prevents choosing the effective sparsity parameter extremely small.
  • IV. DELAY-DOPPLER SPARSITY AND LEAKAGE EFFECT: The delay-Doppler spreading function remains approximately sparse because the leakage function is compressible, although each path occupies multiple coefficients.For P specular paths, the model gives an effective sparsity proportional to P N_Λ.
  • IV. DELAY-DOPPLER SPARSITY AND LEAKAGE EFFECT: Replacing the DFT with a generalized orthonormal basis is proposed to enhance sparsity while retaining the DFT as a special case.The generalized expansion can be inserted into the same CS estimation framework.

A. 1-D and 2-D Basis Expansions

The paper replaces the conventional 2-D DFT representation with a generalized orthonormal basis whose 1-D components can better concentrate delay-Doppler coefficients. This basis preserves Doppler structure while potentially improving sparsity in the other index.

  • Generalized basis construction: The generalized basis replaces the 1-D DFT component with orthonormal functions b_m,i[λ] selected to produce sparse coefficients across Doppler shifts.The bases do not depend on the specific scatterer parameters, including P, η_p, τ_p, and ν_p.
  • Basis structure: The generalized 2-D basis retains the DFT dependence on κ but replaces its λ dependence with the optimized functions b_m,i[λ].This changes the transform only in the index where the new basis is intended to improve concentration.
  • Sparsity enhancement: For a single scatterer, the new coefficients β_m,i can be potentially sparser in the i direction than coefficients obtained with the 1-D DFT basis.The improvement targets the poor coefficient decay caused by leakage in the DFT expansion.
  • Multiple scatterers: For P scatterers, if each single-scatterer coefficient vector is S-sparse, the combined coefficient vector is PS-sparse.The basis construction itself remains independent of the number and parameters of the scatterers.
  • Computational trade-off: The generalized basis increases computational complexity because FFT acceleration applies only with respect to κ, although the added cost is small when J is not too large.Optimal designs for the 1-D bases are developed later in the paper.

B. Generalized Compressive Channel Estimator

The generalized compressive estimator maps pilot observations into a normalized sparse-recovery problem using the new basis. Its sparsity benefit must be balanced against increased basis coherence, while Doppler spacing determines whether conventional DFT leakage occurs.

  • Estimator formulation: The generalized expansion produces the measurement model h(p) = Φx, allowing recovery of the length-JD vector x from pilot observations.The matrix Φ is formed from the basis restricted to pilot positions and column-normalized by D.
  • Pilot requirements: The required number of pilots scales at most linearly with delay-Doppler sparsity S and poly-logarithmically with J and D.Pilot positions are selected uniformly at random within the subsampled time-frequency grid and fixed during data transmission.
  • Recovery trade-off: Better sparsity can be offset by larger coherence μ_V, because the generalized basis has μ_V ≥ 1 whereas the DFT basis has μ_U = 1.Thus, basis design affects both the sparsity level and the recovery condition.
  • Canonical spacing: At canonical Doppler spacing ν_Δ = 1/(T_sN_r), the 1-D DFT coefficients are 1-sparse and exhibit no leakage effect.The corresponding sequence simplifies to a periodic unit sample, concentrating each coefficient at one index.

B. Statistical Basis Optimization

The statistical basis-optimization framework chooses orthonormal basis functions to minimize average coefficient ℓ1-norm under channel statistics. It can incorporate prior distributions of delay, Doppler, and path-gain magnitude, but mismatched statistics produce nonoptimal bases.

  • Statistical objective: With random delay, Doppler, and path gain, the basis is optimized to make the expansion coefficients maximally sparse on average.The average sparsity objective is represented using the expected ℓ1-norm of the coefficient vector.
  • Optimization formulation: The statistical optimization reduces to minimizing an expected coefficient objective over orthonormal 1-D bases or, equivalently, unitary basis matrices.Known channel statistics determine the functions used in the optimization.
  • Numerical evaluation: The statistical framework samples the Doppler range through a Riemannian sum, with Monte Carlo evaluation offered as an alternative.Monte Carlo computation is especially advantageous when the maximum Doppler frequency is unknown.
  • Statistical mismatch: Mismatched channel statistics make the resulting basis matrices differ from the truly optimal ones.The paper analyzes the resulting change in average sparsity under an incorrect statistical model.

C. Basis Optimization Algorithm

Because unitary-basis optimization is nonconvex, the paper uses iterative small updates represented by Hermitian matrices and matrix exponentials. Each accepted update preserves unitarity and guarantees a monotonically decreasing cost sequence.

  • Optimization challenge: The basis optimization problems are nonconvex because the feasible set of unitary matrices is not convex.Consequently, standard convex optimization cannot be applied directly to the original problems.
  • Iterative update: The algorithm starts from the DFT basis and updates each unitary basis through a matrix exponential of a small Hermitian perturbation.The first-order approximation B ≈ I_J + jA is used to construct a convex subproblem, while actual updates remain unitary.
  • Convex subproblem: At each iteration, the algorithm solves a convex constrained problem for a small Hermitian update matrix.The constraint ∥A∥_∞ ≤ ρ_r controls approximation accuracy and keeps the new basis close to the current one.
  • Update acceptance: The algorithm accepts an update only when it lowers the cost, guaranteeing a monotonically decreasing cost sequence.Rejected updates leave the current basis unchanged while the iteration continues under the prescribed constraint level.
  • Termination and use: The iteration terminates when the constraint level falls below a threshold or when a maximum iteration count is reached.The optimized bases are computed once before channel estimation because they do not depend on the received signal.
  • Observed effect: For one channel realization in the CP-OFDM scenario, deterministic basis optimization yields a significant enhancement of sparsity over the DFT basis.The comparison uses α_m,i for the DFT basis and β_m,i for the optimized basis.

VII. CHANNEL ESTIMATION FOR STRONGLY DISPERSIVE CHANNELS

For strongly dispersive multicarrier channels, the estimator expands the discrete-time channel into orthonormal temporal basis functions to recover all system channel coefficients, including ISI/ICI terms.

  • The estimator targets all channel coefficients, including off-diagonal ISI/ICI coefficients that are non-negligible for strongly dispersive channels.The off-diagonal coefficients characterize intersymbol and intercarrier interference.
  • The discrete-time channel impulse response is expanded over orthonormal basis functions indexed by temporal basis index i.The expansion uses m-dependent coefficients T_h[m, i].
  • The generalized spreading function T_h[m, i] extends the discrete delay-Doppler representation, which is recovered with complex exponential basis functions.
  • The model assumes causal channels with maximum delay D−1 and restricts the generalized spreading-function support to delay indices 0 through D−1.The limiting cases D = K and J = N_r are allowed.

B. Compressive Channel Estimator

The compressive estimator iteratively combines known pilots with reliable decision-directed virtual pilots, then uses sparse recovery and ISI/ICI equalization to refine channel estimates.

  • At the first iteration, the estimator uses randomly positioned known pilots; later iterations augment them with virtual pilots from equalized and quantized symbol decisions.The extended pilot set includes neighboring symbols to capture dominant interference.
  • For each iteration, the received observations are expressed as a noisy basis-expansion system and transformed into a compressed-sensing recovery problem.A CS recovery method estimates the expansion coefficients and hence the generalized channel representation.
  • The recovered generalized spreading function yields estimates of all channel coefficients, after which an ISI/ICI equalizer produces updated symbol decisions.
  • The algorithm adds detected symbols only when a reliability criterion indicates that their equalized estimates are sufficiently closer to the selected constellation point than alternatives.For QPSK, reliability is evaluated using a positive threshold ε.
  • The extended pilot sets generally grow across iterations, while termination occurs after channel estimates stabilize or after a fixed iteration limit.No convergence proof is provided, although convergence was observed for reasonably chosen parameters.
  • For weakly dispersive channels, setting V = {0} and using a one-tap equalizer reduces the method to an iterative decision-directed extension of the basic estimator.The extension can improve accuracy and permit smaller pilot sets, at the cost of additional complexity.

C. Sparsity-Inducing Basis Functions

The paper constructs basis functions that retain delay-Doppler sparsity while limiting leakage, combining DFT functions with DPSSs and orthonormalizing the result.

  • The basis functions are designed so the generalized spreading-function factor ϑ(ν)[i] is sparse across the relevant Doppler-frequency range.The DFT basis alone suffers leakage, while the proposed construction targets sparsity for all ν in the specified interval.
  • An alternative basis-optimization formulation minimizes the average ℓ1 norm of the expansion coefficients over a unitary matrix, but its cost is high for large N_r ≈ N_L.
  • A practical construction combines DPSSs, which concentrate energy in a finite interval, with DFT basis functions matched to the allowable Doppler range.The resulting functions have effective support within a small index interval for all relevant ν.
  • The combined set is Gram-Schmidt orthonormalized because some truncated DPSSs are not orthonormal to the DFT functions.
  • The construction preserves DFT-based sparsity for most indices, while only the small number J1−1 of remaining in-support indices need not be sparse.Because J1 is small, the overall sparsity is not significantly deteriorated.
  • For the reported example, the combined DFT-DPSS basis is sparsest: pure DPSS has no sparsity within the support interval, whereas pure DFT suffers strong leakage.The comparison uses νTs = 0.115/2048 and maximum Doppler frequency equal to 20% of the subcarrier spacing.

VIII. SIMULATION RESULTS

The simulations evaluate gains from sparsity-enhancing basis expansions and explicit ISI/ICI estimation against the basic compressive estimator using three sparse-recovery algorithms.

  • The experiments assess performance gains from sparsity-enhancing basis expansions and from estimating ISI/ICI channel coefficients explicitly.
  • The evaluated recovery algorithms are Lasso, OMP, and CoSaMP.Lasso is described as equivalent to basis-pursuit denoising.

A. Simulation Setup

Simulations evaluate compressive channel estimators using MSE and BER across SNR, pilot count, basis designs, and Doppler dispersion. The results show gains from sparsity-enhancing bases and explicit ISI/ICI estimation, while additional basis complexity remains moderate.

  • Simulation setup: Simulations use CP-OFDM systems with K ∈ {512, 1024, 2048}, a CP length ratio of 1/4, 4-QAM, rate-1/2 convolutional coding, and interleaving.Root-raised-cosine transmit and receive filters use roll-off factor ρ = 1/4.
  • Simulation setup: The simulated channel combines sparse specular scattering with a diffuse component having 20 dB less total power.The sparse part contains strong, medium, and weak scatterers, while additive white Gaussian noise sets the prescribed SNR.
  • Simulation setup: All estimators use a subsampled time-frequency grid with ΔK = 4 and ΔL = 1, with pilots selected uniformly at random.Performance is measured by normalized MSE and BER.
  • Basis-expansion performance: The DFT-DPSS basis achieves performance similar to the optimized basis and clearly outperforms the pure DFT basis, especially at high SNR.The gain is attributed to improved sparsity despite higher coherence than the DFT basis.
  • Basis-expansion performance: As pilot count increases, all estimators improve, while optimized and combined DFT-DPSS bases remain superior to the DFT basis.The tested pilot counts range from 512 to 8192, corresponding to 1.5625%–25% of all symbols.
  • ISI/ICI coefficient estimation: Increasing decision-directed iterations improves performance by estimating off-diagonal ISI/ICI coefficients and using virtual pilots, including for strongly dispersive channels.The method is also useful for mildly dispersive channels because it requires fewer pilots, improving spectral efficiency.
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