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Bias-Compensated Complex-Valued Bilinear Filtering
Bernhard Plaimer, Andreas Meingassner-Lang, Yannis Kowalewski, Matthias Wagner, Oliver Lang, Mario Huemer
TL;DR
Noisy inputs introduce bias into complex-valued bilinear filters, a system class not directly covered by earlier bias-compensation methods. This paper derives bias-compensated C-BWF and C-BLMS filters that account for input-noise statistics; simulations report substantial improvements over standard and state-of-the-art methods.
Problem
Input noise induces bias in complex-valued bilinear filters, while earlier bias-compensated methods do not directly cover systems bilinear with respect to their coefficients.
Source: §1 · p. 3Method
The paper derives bias-compensated C-BWF and C-BLMS filters using an adapted cost function that accounts for input-noise statistics.
Source: §1 · p. 3, §6 · p. 17Results
The proposed filters substantially outperform their standard counterparts and state-of-the-art bias-compensated algorithms across a wide range of input-noise levels in simulations on a complex-valued MISO system.
Source: §6 · p. 18Takeaways & Limitations
The filters address input-noise-induced bias in complex-valued bilinear system identification without increasing the number of coefficients.
Source: §1 · p. 3A wrongly estimated input-noise variance can cause the derived filter to diverge.
Source: §4 · p. 12
Abstract
from arXiv · showhide
Complex-valued optimal and adaptive filters are widely used to identify unknown systems in a broad range of practical applications. Recently, a number of complex-valued bilinear filters, such as the complex-valued bilinear Wiener filter (C-BWF), and the complex-valued bilinear least mean squares (C-BLMS) filter, have been proposed to model and identify complex-valued systems, which are bilinear with respect to their coefficients. However, if the input signals are contaminated with noise, the performance of these complex-valued bilinear filters degrades significantly due to an additional noise-induced bias. To overcome this issue, we propose novel bias-compensated complex-valued bilinear filters that estimate and account for input-noise statistics. Specifically, a bias-compensated C-BWF and a bias-compensated C-BLMS filter are derived in this work. We further include a convergence analysis for the latter. Since practical applications require knowledge of the input-noise variance, we briefly present an adapted method from literature to estimate this quantity. Finally, simulation results demonstrate the superior performance of the proposed filters compared to their standard (non-bias-compensated) counterparts and several state-of-the-art methods.
1 INTRODUCTION
Noisy inputs bias complex-valued bilinear filter coefficients, while existing bias-compensation methods do not directly cover this system class. The paper derives bias-compensated bilinear filters and reports superior simulation performance.
1 INTRODUCTION
- System identification is used in applications including wireless channel estimation, acoustic echo-path identification, control-plant modeling, and radar channel modeling.Source: §1 · p. 1
- Existing complex-valued bias-compensation methods remain restricted to linear or widely linear systems, leaving complex-valued systems bilinear in their coefficients uncovered.Source: §1 · p. 2
- Noisy-input complex-valued bilinear identification arises in IQ-imbalance compensation and automotive-radar channel calibration.Source: §1 · p. 2
- Input noise adds bias to complex-valued bilinear filters, whose earlier proposed forms assumed noise-free input data.Source: §1 · p. 3
- The work derives bias-compensated C-BWF and C-BLMS filters, analyzes C-BLMS convergence, and presents an adapted method to estimate input-noise variance.Source: §1 · p. 3
- The proposed bias-compensated filters substantially outperform standard counterparts and several state-of-the-art methods in noisy-input complex-valued MISO system identification.Source: §1 · p. 3
2 Complex-Valued Bilinear Model
The model identifies an unknown complex-valued bilinear system using coefficient vectors h and g. With noisy inputs, estimates based on the observed input can become biased; a linear reformulation is possible but uses more coefficients.
2 Complex-Valued Bilinear Model
- The complex-valued bilinear model uses coefficient vectors h and g to identify an unknown system from its input signal matrix and output.For the illustrated MISO setting, h is a complex-valued FIR channel and g contains per-channel scaling factors.Source: §2 · pp. 3–4
System identification with noisy input data
- Bilinear identification has a scaling ambiguity: h and g are identifiable only up to a complex-valued scalar factor.A linear reformulation estimates f and removes this ambiguity, but requires LM coefficients instead of L + M.Source: System identification with noisy input… · pp. 4–5
- In practice, the observed input may be X̄_k = X_k + V_k, and minimizing a cost based on the resulting error can bias the coefficient estimates.The proposed filters address this input-noise-induced bias.Source: System identification with noisy input… · p. 5
3 Bias-Compensated Complex-Valued Bilinear Filters
The paper derives bias-compensated C-BWF and C-BLMS filters to address the bias caused by noisy inputs. The C-BWF corrects its cost, while the C-BLMS includes adaptive updates and a convergence analysis with step-size constraints.
3.1 Bias-compensated C-BWF
- In the scalar real-valued case, increasing input-noise variance shifts the standard C-BWF minimum toward the axes, biasing its estimates.The illustration contrasts the total-cost minima with the noise-free term's minimum curve, on which the true h and g lie.Source: §3.1 · p. 6, Figure 3 · p. 7
- The proposed C-BWF subtracts the additional input-noise term from its cost function to compensate for bias.The derivation uses Wirtinger calculus and an alternating algorithm to address dependency between the coefficient estimates.Source: §3 · p. 5, §3.1 · pp. 6–7
3.2 Bias-compensated C-BLMS filter
- The bias-compensated C-BLMS uses steepest-descent update equations for the coefficient estimates.The paper analyzes convergence under assumed input- and output-noise covariance forms, deriving step-size bounds and an additional constraint on their combination.Source: §3.2 · pp. 8–9
- The convergence analysis imposes an additional restriction on the step-size boundaries through the condition Δ > 0.When input-noise variance approaches zero, the proposed filter's step-size boundaries match those of standard C-BLMS.Source: §3.2 · p. 11
4 Estimation of Input Noise Variance
The variance-estimation method adapts a strategy from prior work to estimate unknown input-noise variance, assuming the output-noise variance is known. It smooths noisy instantaneous estimates, while an incorrectly estimated input-noise variance may cause divergent adaptation.
4 Estimation of Input Noise Variance
- The input-noise variance estimator assumes known output-noise variance, which the derivation does not estimate.After filter convergence, the cost function equals the output-noise variance; the method adapts a strategy from to estimate input-noise variance.Source: §4 · pp. 11–12
- An exponential moving average with smoothing factor β ∈ (0, 1) replaces noisy instantaneous numerator and denominator estimates.The estimator initializes Φ_e²,0 = σ_n² and Φ_f²,0 ≥ 0.Source: §4 · p. 12
- A wrongly estimated input-noise variance may cause divergent adaptation, so the method derives an upper bound for the estimate.The bound is used as an upper limit for the estimated input-noise variance.Source: §4 · pp. 12–13
5 Simulations
Simulations show that the proposed bias-compensated filters estimate input-noise variance reliably and maintain lower normalized misalignment across tested noise levels. The C-BWF converges several orders faster than the C-BLMS filters, with an extra settling period when it uses an estimated variance.
5 Simulations
- The simulations identify a complex-valued bilinear MISO system with L=30 and M=20, and compare the proposed filters with four state-of-the-art algorithms on an equivalent linear model.The four reference algorithms require known input- and output-noise variances; the proposed filters are also evaluated with estimated input-noise variance.Source: §5 · p. 13
5.1 Estimated Noise Variance
- For σ²_v=0.5, estimated input-noise variance converges close to the true value for both bias-compensated filters at σ²_n=0.2 and σ²_n=1.C-BLMS estimates are shown over sample index k and C-BWF estimates over iteration n.Source: §5.1 · p. 14, Figure 4 · p. 14, Figure 6 · p. 15
- After convergence, both filters’ estimated input-noise variance closely tracks the true value across the tested σ²_v range for σ²_n=0.2 and σ²_n=1.The estimate-versus-truth results average 100 Monte Carlo runs.Source: §5.1 · pp. 14–15, Figure 5 · p. 14
5.2 Normalized Misalignment
- The C-BWF filters converge several orders faster than the C-BLMS filters, except that estimated-variance C-BWF requires a short additional adaptation period.That extra period lasts until the input-noise variance estimate settles.Source: §5.2 · pp. 15–16
- At σ²_n=0.2, bias-compensated C-BLMS achieves significantly lower steady-state NM than the standard filter and outperforms the state-of-the-art algorithms.At σ²_n=1 and σ²_v=0.5, ACGDTLS has slightly lower steady-state misalignment, but that advantage does not persist at larger input-noise variances.Source: §5.2 · p. 15, Figure 7 · p. 16, Figure 9 · p. 17
5.3 Steady-state performance
- Across σ²_v=0–3, both proposed filters maintain substantially lower NM than uncompensated filters and state-of-the-art references, using true or estimated input-noise variance.The sweep averages 100 Monte Carlo runs; the plotted comparison at σ²_n=0.2 is shown after convergence.Source: Figure 8 · p. 16, §5.3 · p. 17
6 Conclusion
The work derives bias-compensated complex-valued bilinear filters to address input-noise bias, and reports that they outperform standard and state-of-the-art alternatives across a wide range of input-noise levels.
6 Conclusion
- The work derives bias-compensated C-BWF and C-BLMS filters using an adapted cost function that accounts for input-noise statistics.Source: §6 · p. 17
- Across a wide range of input-noise levels, the proposed bias-compensated filters substantially outperform standard counterparts and state-of-the-art bias-compensated algorithms.Fig. 10 plots normalized misalignment after convergence over the true input-noise variance, with output-noise variance set to 1.Source: Figure 10 · p. 18, §6 · p. 18
- The C-BLMS analysis yields step-size boundaries, and the work investigates estimating input-noise variance from observed data.Source: §6 · p. 18