Online SPM in FxLMS Testbed

• The paper demonstrates online secondary path modeling integrated with FxLMS to adaptively estimate S(z) without noise injection, ensuring stability amidst rapid environmental changes.
• It employs concurrent NLMS adaptation, mode-switching, and meta-learning strategies to reduce recovery time and improve estimation accuracy.
• Performance evaluations in real-time testbeds highlight significant gains in noise reduction, convergence speed, and computational efficiency.

Online secondary path modeling in the FxLMS testbed targets the concurrent identification of the secondary acoustic path $S(z)$ within adaptive feedforward active noise control (ANC) systems utilizing the filtered-x least mean square (FxLMS) algorithm. Accurate, real-time modeling of the secondary path is critical, as $S(z)$ is necessary to generate the filtered reference for the control filter adaptation. Modern methodologies enable this identification without destabilizing auxiliary noise injection, support time-varying scenarios, and are evaluated on rigorous testbeds under rapid environmental changes (Hu et al., 2018, Ji et al., 2023, Yang et al., 20 Jan 2026).

1. Problem Definition and Challenges

The key challenge in online secondary path modeling is to estimate the impulse response vector $s = [s(0), ..., s(N_s-1)]^\top$ of $S(z)$ adaptively and concurrently with controller operation, using only the usual reference $r(n)$ and error $e(n)$ signals, and—ideally—without explicit test noise injection. The secondary path estimate $\hat{s}(n)$ must adapt alongside the control filter $w(n)$ while maintaining stability and high noise reduction. Environmental changes (e.g., movement of error microphone or loudspeaker, structural shifts) can cause $S(z)$ to change abruptly, requiring rapid re-identification. Traditional identification approaches degrade ANC performance by requiring additive test noise; modern solutions circumvent this while preserving or enhancing stability and computational efficiency (Hu et al., 2018, Ji et al., 2023, Yang et al., 20 Jan 2026).

2. Mathematical Formalism

Let $w(n)$ be the control filter of length $N_c$ and $\hat{s}(n)$ the secondary path estimate of length $N_s$. The controller output is $y(n) = w(n)^\top r(n)$, with $r(n) = [r(n), r(n-1), ..., r(n-N_c+1)]^\top$. The true filtered reference is $x(n) = S(z) r(n)$. The adaptation dynamics center on the joint autocorrelation matrix

$$R = E\{u(n)u(n)^\top\},\quad u(n) = \begin{bmatrix} r(n) \ x(n) \end{bmatrix} \in \mathbb{R}^{N_c + N_s}$$

with $R$ in block form:

$$R = \begin{pmatrix} R_{rr} & R_{rx} \ R_{xr} & R_{xx} \end{pmatrix}$$

where typical definitions hold.

The identifiability of $s$ depends on the rank of $R$. For static $w(n)$, $R$ is full rank if and only if $N_c > N_p$ (primary path length). When $N_c \leq N_p$, $x(n)$ is a linear combination of past $r(n)$, causing rank deficiency, hence non-unique solutions. However, when $w(n)$ is time-varying, as in standard adaptive ANC, the continual update of $w(n)$ decorrelates $r(n)$ and $x(n)$ over time, ensuring that $R$ is full rank on average and $s$ remains identifiable even if $N_c \leq N_p$ (Hu et al., 2018).

In the modified FxLMS paradigm, key update equations are:

• Controller update: $w(n+1) = w(n) + \mu_w e(n) x_f(n)$, with $x_f(n) = [x * \hat{s}](n)$
• Secondary path model: $$\hat{s}(n+1) = \hat{s}(n) + \mu_s e(n) x(n) / (\|x(n)\|^2 + \delta)$$ or, in computation-efficient LMS updating: $$\hat{s}(n+1) = \hat{s}(n) - \mu_s y_{\vec{n}} e_s(n)$$, where $e_s(n)$ is the inner error quantifying SPM mismatch (Ji et al., 2023).

3. Algorithmic Strategies and Testbed Realizations

Recent developments converge on algorithmic architectures that avoid explicit additive noise for identification, and/or implement mode-switching strategies to rapidly adapt after path changes. Three prominent methodologies are deployed in testbeds:

Method 1: Concurrent Adaptation Without Noise Injection

• ANC and secondary path estimates are jointly adapted using normalized LMS (NLMS) or recursive least squares (RLS), taking advantage of the time-variation in $w(n)$ to ensure identifiability (Hu et al., 2018).
• Initialization: $w(0)$ is a short impulse, $\hat{s}(0) = 0$.
• Step sizes and filter lengths are empirically chosen (e.g., $N_s = 48$, $N_c \approx 64$, $\mu_w = 0.01$, $\mu_s = 0.002$).
• Monitoring: Real-time error power $E[e^2(n)]$, tracking of $\|\hat{s}(n) - \hat{s}(n-1)\|$.

Method 2: Computation-Efficient Mode-Switching SPM

• System alternates between adaptive ANC (Mode 1) and SPM (Mode 2).
• Divergence detector triggers SPM mode when reference-to-error power ratio drops below threshold; SPM mode is exited when model error slope stagnates (Ji et al., 2023).
• The SPM update cost per sample is $L_s$ multiplies (50% reduction compared to classical dual-filter approaches).
• Efficacy: After a secondary path change, converges in $1.8\pm0.2$ s with steady-state ERLE $41.8\pm0.5$ dB; zero ANC gap on switching. See Table 1.

Method	Steady-State ERLE (dB)	SP Remodel Time (s)	ANC Gap (s)
Proposed	41.8 ± 0.5	1.8 ± 0.2	0
5-stage (Pradhan)	41.3 ± 0.7	2.0 ± 0.3	0.5 ± 0.1
Akhtar VSS-LMS	35.2 ± 1.1	1.2 ± 0.1	0

Method 3: Meta-Learned Co-Initialization

• Control filter and secondary path model are co-initialized via model-agnostic meta-learning (MAML) trained over a small set of measured paths (Yang et al., 20 Jan 2026).
• On detection of abrupt path/environment changes, parameters are reset to their learned initial values for rapid recovery.
• This yields lower early-stage MSE (by 5 dB), 30% faster time-to-target, ~20% less auxiliary-noise energy, and $\sim2$ s recovery time after path switch versus $\sim5$ s baseline.

4. Practical Implementation in ANC Testbeds

A typical ANC–FxLMS–OSPM testbed comprises:

• Microphone(s) (error/reference), control loudspeaker, ADCs/DACs, real-time DSP or low-latency real-time PC/FPGA.
• Sampling rates range from 1–48 kHz, with 13–16 kHz common for testbeds (Hu et al., 2018, Ji et al., 2023, Yang et al., 20 Jan 2026).
• Filter lengths: $N_c, N_s$ often 48–512 taps, selected based on measured path duration/spectral content.
• Software: MATLAB simulation, dSPACE/Simulink, or custom embedded code.
• Monitoring: Error power, impulse response convergence, real-time visualization of secondary path estimate.

Meta-learning-based initialization employs RWTH Aachen PANDAR database (for in-ear headphones) to meta-train initializations over diverse secondary path scenarios, optimizing for rapid adaptation (Yang et al., 20 Jan 2026).

5. Performance Evaluation and Representative Results

Testbed results consistently demonstrate that online secondary path modeling without noise injection outperforms additive-noise baselines. For instance (Hu et al., 2018):

• No-noise NLMS reaches $0.06$ RMSE on $\hat{s}$ in $\sim3$ s with $17$ dB ANC attenuation, outperforming additive-noise NLMS (RMSE $0.15$, $16$ dB attenuation).
• Adaptive methods with time-varying $w(n)$ can eliminate the traditional filter length restriction.

Mode-switching and meta-learned initializations reduce recovery time and error power:

• Mode-switching SPM achieves zero ANC downtime during secondary path reidentification (Ji et al., 2023).
• MAML-initialized FxLMS recovers target error rates $30\%$ faster using less auxiliary identification energy (Yang et al., 20 Jan 2026).

Figures in (Hu et al., 2018) and (Ji et al., 2023) further illustrate convergence curves, learning curves under path switching, and the evolution of error power.

6. Current Trends and Research Directions

Recent focus centers on improving early-stage adaptation and recovery speed after abrupt environmental changes, eliminating additive noise without degrading accuracy, and minimizing computational resources. Meta-learning approaches suggest that transfer of initialization knowledge across environments meaningfully reduces startup and adaptation times. Mode-switching control with computationally efficient SPM is now favored for practical, low-latency ANC devices (Ji et al., 2023, Yang et al., 20 Jan 2026). Investigations into the impact of secondary-path dispersion for meta-learned initializations indicate that task diversity in training promotes robust generalization in unseen acoustic scenarios (Yang et al., 20 Jan 2026).

A plausible implication is that future systems will increasingly integrate meta-learned priors and principled divergence detection, dynamically balancing ANC performance and online system identification in highly nonstationary environments.

7. Summary Table: Comparison of Approaches

Approach	Noise Injection	ID Trigger/Mode Switch	Recovery/Adaptation
Classical Additive-Noise	Yes	N/A	Slower, steady plateau
Simultaneous NLMS (No Noise)	No	N/A	Rapid, optimal RMSE
Mode-Switching SPM	No	Reference-error ratio, slope	1.8 s, zero ANC gap
MAML Co-Initialization	Optional	Error-jump detector	~2 s, low startup MSE

All methods above have been implemented in real-time testbeds and validated under both synthetic and measured environmental changes, confirming their applicability to embedded and consumer ANC deployments (Hu et al., 2018, Ji et al., 2023, Yang et al., 20 Jan 2026).