NSAF-NKP-II: Type-II NKP-Based Adaptive Filtering

• The paper introduces NSAF-NKP-II, an adaptive filtering algorithm that uses nearest Kronecker product decomposition to reduce computational overhead and enhance performance.
• NSAF-NKP-II employs a critically sampled subband model, enabling robust handling of correlated inputs, impulsive noise, and nonlinear system responses.
• It offers significant improvements in convergence, stability, and efficiency compared to conventional NSAF and earlier NKP-based variants.

The type-II nearest Kronecker product-based normalized subband adaptive filter (NSAF-NKP-II) algorithm is an advanced adaptive filtering scheme that achieves rapid convergence and substantial computational efficiency by integrating the principles of nearest Kronecker product (NKP) decomposition into the normalized subband adaptive filter (NSAF) framework. NSAF-NKP-II facilitates robust identification and signal processing in scenarios with highly correlated inputs, impulsive noise, and nonlinear system responses, including sparse system identification, echo cancellation, and active noise control. Its design offers a marked reduction in computational overhead compared to both conventional NSAF and the earlier NSAF-NKP-I variant, while providing explicit stability and steady-state performance guarantees (Ye et al., 15 Jan 2026).

1. Subband Signal and Filter Model

The NSAF-NKP-II algorithm operates by transforming the adaptive filtering task into critically sampled subband domains. At time $r$, the system models the relationship between the input vector $x_r \in ℝ^D$, the unknown system $m₀ \in ℝ^D$, and additive noise $v_r$ via:

$d_r = x_r^T m₀ + v_r$

where $x_r = [x_r, x_{r-1}, ..., x_{r-D+1}]^T$. The analysis filter bank decomposes the input into $N$ subbands using filter matrix $F = [f_1, ..., f_N] \in ℝ^{L \times N}$, yielding subband inputs and desired signals at decimated time $r$:

$$x_{r,j} = f_j^T [x_r, ..., x_{r-L+1}]^T \ d_{r,j} = f_j^T [d_r, ..., d_{r-L+1}]^T$$

Aggregated subband signals are given by $\bar X_r = X_r F$ and $d_r^{(s)} = \hat d_r F$.

2. Nearest Kronecker Product Decomposition Principle

NKP decomposition approximates the unknown filter $m₀$ as a sum of Kronecker products when $D = D_1 D_2$:

$$m₀ \approx \sum_{p=1}^P m_{2,p}^o \otimes m_{1,p}^o$$

with $m_{1,p}^o \in ℝ^{D_1}$ and $m_{2,p}^o \in ℝ^{D_2}$. Equivalently, $m₀ \approx \mathrm{vec}(M₁ M₂^T)$ for suitable $M₁ \in ℝ^{D_1 \times P}$ and $M₂ \in ℝ^{D_2 \times P}$. The best rank-$P$ approximation uses the leading $P$ singular vector pairs from the matricization $M₀ \in ℝ^{D_1 \times D_2}$ of $m₀$. This structure forms the mathematical foundation for efficient adaptive filtering.

3. NSAF-NKP-II Algorithmic Derivation

The NSAF-NKP-II algorithm parameterizes the adaptive filter at iteration $r$ as:

$$\hat m_r = \sum_{p=1}^P \hat m_{r,2,p} \otimes \hat m_{r,1,p}$$

This representation enables efficient arithmetic based on Kronecker identities, facilitating flexible filter updates. Subband error signals are computed as:

$e_{r,j} = d_{r,j} - \hat m_r^T x_{r,j}$

which can be reformulated using projections into NKP parameter subspaces:

$$e_{r,j} = d_{r,j} - \hat m_{r,1}^T x_{r,j,2} = d_{r,j} - \hat m_{r,2}^T x_{r,j,1}$$

The normalized subband cost functions are:

$$J_1(\hat m_{r,1}) = \frac{1}{2} \sum_{j=1}^N \frac{e_{r,j}^2}{\|x_{r,j,2}\|^2 + \delta}, \quad J_2(\hat m_{r,2}) = \frac{1}{2} \sum_{j=1}^N \frac{e_{r,j}^2}{\|x_{r,j,1}\|^2 + \delta}$$

Gradient updates for step sizes $\mu_1$, $\mu_2$ follow:

$$\begin{align*} \hat m_{r+K,1} &= \hat m_{r,1} + \mu_1 \sum_{j=1}^N \frac{x_{r,j,2} e_{r,j}}{\|x_{r,j,2}\|^2 + \delta} \ \hat m_{r+K,2} &= \hat m_{r,2} + \mu_2 \sum_{j=1}^N \frac{x_{r,j,1} e_{r,j}}{\|x_{r,j,1}\|^2 + \delta} \end{align*}$$

After every $K$ subband updates, the fullband estimate is synthesized:

$$\hat m_{r+K} = \sum_{p=1}^P \hat m_{r+K,2,p} \otimes \hat m_{r+K,1,p}$$

4. Computational Complexity and Comparisons

The NSAF-NKP-II algorithm is engineered for computational efficiency. For $D = D_1 D_2$, decimation interval $k$, filter length $L$, number of subbands $N$, and Kronecker rank $P$:

Algorithm	Multiplications per $k$ samples	Big-O Complexity
NSAF	$k[3ND+2N] + LK(D+1)$	$O(kND + NDL)$
NSAF-NKP-I	$k[PD + 4DPL + ...]$ (see full formula)	$O(kPD(L+N) + kND(P(D_1+D_2)))$
NSAF-NKP-II	$PD + 4PND + 3PND_2 + 3PND_1 + 4N + (D+1)LN$	$O(PDN + NDL + PND)$

A plausible implication is that NSAF-NKP-II, for $P \ll D$, $N \ll D$, realizes a 50–90% reduction in multiplications relative to NSAF-NKP-I, primarily by eliminating $LPN$ terms.

5. Stability and Steady-State Analysis

Under independence and small-step assumptions, stability is dictated by:

$0 < \mu_1 + \mu_2 < 2$

For the symmetric choice $\mu_1 = \mu_2 = \mu$, this reduces to $0 < \mu < 1$, a stricter bound than the typical $0 < \mu < 2$ for NSAF, though practical choices $(\mu=0.1-0.5)$ remain feasible.

Theoretical steady-state excess mean-square error (EMSE) is given, with subband excess error $\zeta_{r,j} = x_{r,j}^T(m_0 - \hat m_r)$ and input orthogonality/noise variance $\sigma_v^2$:

$$\mathrm{EMSE} = \frac{(\mu_1 + \mu_2)\,\sigma_v^2}{2 - \mu_1 - \mu_2}$$

Special case $\mu_1 = \mu_2 = \mu$ yields:

$\mathrm{EMSE} = \frac{\mu\,\sigma_v^2}{1-\mu}$

6. Robust and Nonlinear NSAF-NKP-II Extensions

To address impulsive noise environments, robust variants are constructed:

• RNSAF-NKP-MCC: based on the maximum correntropy criterion, achieving stable convergence under $\alpha$-stable noise.
• RNSAF-NKP-LC: leveraging a logarithmic criterion for similar robustness.

Nonlinear extensions further generalize NSAF-NKP-II:

• TFLN-NKP-NSAF: incorporates trigonometric functional link networks for handling asymmetric nonlinear distortions.
• Volterra-NKP-NSAF: utilizes Volterra series expansion for higher-order nonlinear modeling.

In ANC scenarios, the filtered-x NSAF-NKP-II (NKP-FxNSAF) algorithm extends the framework for real-time adaptive noise cancellation.

7. Empirical Performance and Application Domains

Simulation experiments demonstrate NSAF-NKP-II's efficacy:

• Sparse system identification: Matches NSAF-NKP-I convergence (30% computational cost), surpassing NLMS, NSAF, NLMS-NKP, and RLS-NKP under highly correlated inputs.
• Acoustic echo cancellation: Delivers faster ERLE rise than alternatives, with comparable performance to NKP-APA but 2–3× fewer multiplications.
• Impulsive noise robustness: Standard NSAF-NKP-II diverges for heavy impulsive noise, but RNSAF-NKP-MCC and RNSAF-NKP-LC combat this effectively.
• Nonlinear filtering: TFLN-NKP-NSAF and Volterra-NKP-NSAF outperform fullband and other subband methods in convergence rate and MSE.
• Active noise control: NKP-FxNSAF achieves faster ANR growth than competing ANC algorithms with half the complexity of comparable methods.

In summary, type-II NSAF-NKP algorithms offer optimized convergence-speed and computational efficiency, consolidating adaptive Kronecker product decomposition with subband normalization for robust, high-performance adaptive signal processing (Ye et al., 15 Jan 2026).