PaperlayerView exact revision on arXiv

Source-linked AI summary

Adaptive DOA Estimation With KL-Based Kernel Bandwidth Adaptation Under Impulsive Noise

Mojtaba Hajiabadi, Saeid Pakravan, Ghosheh Abed Hodtani, Ming Zeng

arXiv:2610.14623v1eess.SP

TL;DR

Impulsive noise can make DOA estimators relying on second-order statistics susceptible to outliers, while fixed kernel bandwidths impose a robustness tradeoff. This paper combines kernel-weighted spatial-filter adaptation with online KL-based bandwidth adjustment. Its simulations report improved accuracy in challenging conditions, with the strongest gains at low-to-moderate SNRs.

Abstract

from arXiv · show

Direction-of-arrival (DOA) estimation can degrade markedly in impulsive noise, where methods relying on Gaussian or second-order statistical assumptions are particularly susceptible to large-amplitude outliers. This paper develops an adaptive DOA estimator that employs Gaussian kernel weighting to attenuate the influence of large prediction errors while adjusting the kernel bandwidth online according to a Kullback--Leibler (KL) divergence criterion. A sample-based approximation of the KL criterion yields a recursive bandwidth update that adapts the degree of outlier suppression to the observed error statistics rather than prescribing a fixed bandwidth a priori. The spatial-filter coefficients and kernel bandwidth are updated recursively, and the DOAs are recovered from the resulting spatial spectrum. Mean and mean-square analyses are developed under standard adaptive-filtering approximations to characterize the coefficient convergence and steady-state behavior. Simulation results under Gaussian-mixture impulsive noise demonstrate stable coefficient adaptation and improved DOA estimation accuracy over representative competing methods in several challenging operating regimes, with the most pronounced gains occurring at low-to-moderate signal-to-noise ratios. The results also indicate reduced sensitivity to the initial kernel bandwidth through online bandwidth adaptation.

I. INTRODUCTION

Second-order methods can be sensitive to impulsive disturbances, and fixed kernel bandwidths impose a robustness–adaptation tradeoff. The paper combines reference-sensor spatial-filter adaptation with online KL-based bandwidth adjustment; its array model considers multiple narrowband far-field sources received by a sensor array.

I. INTRODUCTION

  • Prior work spans robust statistics, learning, and adaptive filtering, with performance depending on the criterion, estimation architecture, and tuning parameters –.The introduction describes variational Bayesian, fractional lower-order, nonlinear, characteristic-function, learning-based, and adaptive approaches.Source: §I · p. 1
  • Fixed kernel bandwidths trade stronger attenuation of large residuals against preserving informative samples, convergence, and steady-state accuracy.The tradeoff may be inadequate when error statistics vary during adaptation.Source: §I · p. 1
  • The estimator combines reference-sensor spatial-filter adaptation with KL-based bandwidth updates driven by recent prediction errors, avoiding a bandwidth prescribed a priori.DOAs are recovered from the spatial spectrum formed using the adapted filter.Source: §I · p. 2
  • The analysis develops mean and mean-square characterizations, while numerical experiments assess DOA accuracy, adaptation behavior, and parameter sensitivity under Gaussian-mixture impulsive noise.The analyses use standard adaptive-filtering approximations.Source: §I · p. 2

A. Array Signal Model

  • The signal model considers Q narrowband far-field sources from distinct directions received by a uniform linear array of M sensors.The observations include source signals and additive sensor noise.Source: A. Array Signal Model · p. 2

B. Adaptive DOA Formulation

The DOA formulation uses one sensor as a reference and the remaining sensors as an auxiliary subarray for adaptive prediction. After filter convergence, source directions are selected from peaks in the resulting spatial spectrum.

B. Adaptive DOA Formulation

  • The sensor indexed m = 0 is the reference, and the other M − 1 sensors form the auxiliary subarray used by the complex spatial filter.The filter predicts the reference-sensor observation from the auxiliary-subarray observations.Source: B. Adaptive DOA Formulation · p. 2
  • The reference-sensor response is unity, so directions satisfying the adapted prediction relation produce small spatial-spectrum values.The prediction compares the reference response with that predicted from the auxiliary subarray.Source: B. Adaptive DOA Formulation · p. 3
  • After convergence, the DOAs are estimated from dominant local maxima; when Q is known, the Q largest distinct peaks are selected.The search is over the prescribed angular region.Source: B. Adaptive DOA Formulation · p. 3
  • Robust adaptation of the spatial-filter coefficients is the remaining challenge under impulsive noise, motivating a kernel-weighted recursion and online bandwidth adjustment.Source: B. Adaptive DOA Formulation · p. 3

III. PROPOSED KL-ADAPTIVE DOA ESTIMATOR

The proposed estimator uses Gaussian kernel weighting to limit large prediction errors' influence on spatial-filter adaptation. It adapts the kernel bandwidth online because the bandwidth controls the robustness–adaptation tradeoff.

A. Kernel-Weighted Spatial-Filter Adaptation

  • Gaussian kernel weighting makes the coefficient correction approach zero for large prediction errors, limiting their influence on spatial-filter adaptation.The weight approaches unity for small errors and decreases rapidly as error magnitude increases.Source: A. Kernel-Weighted Spatial-Filter… · p. 3
  • The coefficient update scales the instantaneous correction by exp[−|e(n)|^2/(2h^2)], with μw as the coefficient step size.Source: A. Kernel-Weighted Spatial-Filter… · p. 3
  • A smaller bandwidth attenuates large residuals more strongly, whereas a larger bandwidth weakens attenuation and approaches a conventional error-driven update.Because error statistics may evolve during adaptation, a fixed bandwidth may be inadequate.Source: A. Kernel-Weighted Spatial-Filter… · p. 3

B. KL-Based Kernel-Bandwidth Adaptation

The bandwidth is adapted online rather than fixed in advance, using recent prediction errors to approximate a KL-based criterion. The resulting bandwidth controls the Gaussian weighting used in the spatial-filter update.

B. KL-Based Kernel-Bandwidth Adaptation

  • The method adapts bandwidth from recent prediction errors using a sample-based approximation of a KL-divergence criterion.The prediction-error density is unknown, so the criterion’s expectation is approximated from a recent window of errors.Source: B. KL-Based Kernel-Bandwidth Adaptation · p. 3
  • A leave-one-out kernel density estimate excludes each error sample from its own density estimate.The errors are treated as samples of a complex prediction-error random variable, and the density estimate uses an isotropic Gaussian kernel on the complex plane.Source: B. KL-Based Kernel-Bandwidth Adaptation · p. 3
  • Gradient ascent updates the bandwidth, which then sets the Gaussian weighting and lets residual attenuation evolve during adaptation.A positive minimum bandwidth prevents the value from approaching zero.Source: B. KL-Based Kernel-Bandwidth Adaptation · p. 4

C. Overall Adaptive DOA Procedure

The procedure updates the spatial-filter coefficients and bandwidth using prediction errors, then uses the adapted coefficients to form a spatial spectrum. DOA estimates are selected from its dominant local maxima.

C. Overall Adaptive DOA Procedure

  • The online procedure computes prediction errors, updates bandwidth and kernel weights, and recursively updates the spatial-filter coefficients.During initial acquisition, before L errors are available, the bandwidth remains at its prescribed initial value.Source: C. Overall Adaptive DOA Procedure · p. 4
  • The current prediction error determines both the bandwidth adjustment and the subsequent kernel-weighted coefficient correction.The bandwidth is updated once a window of L errors is available; before then, the initial bandwidth is retained.Source: C. Overall Adaptive DOA Procedure · p. 4
  • After the final snapshot, the adapted coefficients form the spatial spectrum, and the Q largest distinct local maxima provide the DOA estimates when Q is known.The procedure evaluates the bandwidth update over an L-sample error window, while the coefficient update uses the current array snapshot.Source: C. Overall Adaptive DOA Procedure · p. 4

IV. PERFORMANCE ANALYSIS

The analysis characterizes coefficient behavior under adaptive-filtering approximations because exact analysis of the joint coefficient–bandwidth recursion is generally intractable.

IV. PERFORMANCE ANALYSIS

  • Exact stochastic analysis of the joint coefficient–bandwidth recursion is generally intractable because the kernel weight depends on prediction error and the bandwidth is adaptive.The analysis therefore treats bandwidth as locally quasi-stationary over the coefficient-analysis time scale.Source: §IV · p. 4
  • The coefficient analysis assumes bandwidth is locally quasi-stationary over its time scale.This is the approximation used to characterize coefficient dynamics.Source: §IV · p. 4
  • Under these approximations, the analysis provides conditions for mean stability and an approximate characterization of mean-square behavior.Source: §IV · p. 4

A. Mean Behavior and Stability

The mean-behavior analysis adopts local adaptive-filtering approximations to obtain a tractable coefficient model. Its stability characterization is local to fixed or slowly varying bandwidth, not a global guarantee for the joint recursion.

A. Mean Behavior and Stability

  • The mean model uses local independence and a reference vector chosen to satisfy a weighted-residual orthogonality condition.It neglects, to first order near steady state, dependence between coefficient error and weighted instantaneous quantities.Source: A. Mean Behavior and Stability · p. 5
  • For a fixed or locally quasi-stationary bandwidth, the coefficient recursion is mean stable under the stated condition.The condition is presented for the approximate frozen-bandwidth mean model.Source: A. Mean Behavior and Stability · p. 5
  • The stability condition describes local coefficient dynamics, not global stability of the joint coefficient–bandwidth recursion.For time-varying bandwidth, a stronger uniform contraction condition on the mean transition matrices is required.Source: A. Mean Behavior and Stability · p. 5

B. Mean-Square Behavior

The section characterizes coefficient mean-square behavior under stated adaptive-filtering approximations, including a sufficient stability condition for time-varying bandwidth and an approximate steady-state MSD.

A. Mean-Square Behavior

  • Under local-independence and quasi-stationarity approximations, the two cross terms in (32) are neglected in expectation.This yields the mean-square recursion used in the analysis.Source: B. Mean-Square Behavior · p. 5
  • For time-varying bandwidth, a sufficient uniform stability condition is that an induced matrix norm satisfies sup_n ∥F_n∥ < 1.This condition is stated separately from the frozen-bandwidth approximation.Source: B. Mean-Square Behavior · p. 5
  • With frozen bandwidth, convergence of F_n and Q_n and ρ(F∞) < 1 allow the steady-state relation to yield an approximate MSD for Σ = I_M−1.The analysis characterizes local second-order behavior rather than an exact closed-form solution of joint coefficient–bandwidth dynamics.Source: B. Mean-Square Behavior · p. 5

C. Computational Complexity

The section describes the estimator’s computational costs and introduces numerical evaluation under impulsive-noise conditions using the default simulation settings.

C. Computational Complexity

  • Coefficient prediction and update require O(M) operations per snapshot, while direct bandwidth-gradient evaluation over an L-error window requires O(L^2) operations per update.The stated overall adaptive-processing expression is truncated in the passage.Source: C. Computational Complexity · p. 5
  • Evaluating the spatial spectrum over P candidate angles after adaptation requires O(PM) operations.Source: C. Computational Complexity · p. 5
  • The extra cost relative to fixed-bandwidth coefficient adaptation comes primarily from pairwise bandwidth operations over a finite error window, and is controlled by L.Source: C. Computational Complexity · p. 5

V. NUMERICAL RESULTS

  • The numerical-results section evaluates the estimator under impulsive-noise conditions, using Table II settings unless stated otherwise.Source: §V · p. 5

A. Simulation Setup

The simulations use a Gaussian-mixture noise model and assess DOA accuracy alongside adaptation behavior and spatial spectra. Adaptive bandwidth departs from its initial value and settles, while spectra retain peaks aligned with the true source directions at two representative SNR levels.

A. Simulation Setup

  • The setup uses a half-wavelength-spaced ULA and mutually independent, unit-variance, zero-mean circular complex Gaussian source sequences, with Gaussian-mixture sensor noise.The mixture parameters include impulse probability ε and an impulsive-component control κ > 1.Source: A. Simulation Setup · pp. 5–6
  • DOA accuracy is measured by RMSE across independent trials, matching estimated DOAs to true DOAs before evaluating error.Source: A. Simulation Setup · p. 6

B. Adaptive Bandwidth and Spatial-Spectrum Behavior

  • The adaptive bandwidth departs from its initial value during the transient and settles within a stable range, with improved steady-state coefficient accuracy over fixed bandwidth.The comparison uses coefficient MSD and bandwidth evolution.Source: B. Adaptive Bandwidth and… · p. 6
  • At two representative SNR levels, dominant spectral peaks remain aligned with true source directions; higher SNR produces sharper, more clearly separated peaks.The lower-SNR condition is described as more challenging because of impulsive disturbance.Source: B. Adaptive Bandwidth and… · p. 6

C. DOA Estimation Performance

The estimator's strongest RMSE gains occur at low-to-moderate SNR, with robust performance also reported across array sizes, snapshot counts, and source separations. Parameter sweeps and coefficient-MSD trajectories show adaptation tradeoffs, while the analysis remains local to coefficient dynamics.

C. DOA Estimation Performance

  • The proposed estimator's largest RMSE improvement occurs at low-to-moderate SNR; it remains competitive across array sizes and snapshots and is more robust at close source spacing.As SNR rises, the performance gap narrows; differences among methods also diminish as angular separation increases.Source: C. DOA Estimation Performance · pp. 6–7, Figure 4 · p. 7

D. Parameter Sensitivity

  • Steady-state MSD varies relatively little with initial bandwidth, while window length and bandwidth step size govern bandwidth responsiveness; coefficient step size trades adaptation speed against steady-state error and stability.Excessively large coefficient step sizes can degrade stability, while satisfactory performance is maintained over a practical parameter range.Source: D. Parameter Sensitivity · p. 7, Figure 5 · p. 8

E. Mean-Square Convergence Behavior

  • Smaller coefficient step sizes converge more slowly but yield lower steady-state MSD, whereas larger steps accelerate transients at the cost of higher steady-state error; tested recursions remain empirically stable.These observations are consistent with approximate mean and mean-square conditions, not a global stability guarantee for the joint coefficient–bandwidth recursion.Source: E. Mean-Square Convergence Behavior · p. 7, Figure 6 · p. 8

VI. CONCLUSION

  • The analysis characterizes local coefficient dynamics rather than providing a global stability guarantee for the joint coefficient–bandwidth recursion; fuller joint stochastic analysis remains future work.The conclusion also reports improved DOA accuracy in challenging conditions, especially at low-to-moderate SNR, and reduced sensitivity to initial bandwidth.Source: §VI · p. 8
Loading 2610.14623v1…