Multi-Channel PQMF Filter Banks

• Multi-channel PQMFs are cosine-modulated filter banks that achieve near-perfect signal reconstruction through critical sampling, optimal frequency subband decomposition, and minimal aliasing.
• They employ matrix factorization and cosine modulation techniques to design flexible, multidimensional filters applicable to neural vocoding, audio compression, and source separation.
• Their construction supports both FIR and IIR implementations, enabling artifact-free processing in advanced signal analysis tasks such as wavelet transforms and anisotropic data decomposition.

A multi-channel pseudo-quadrature mirror filter (PQMF) is a critically sampled, cosine-modulated filter bank framework engineered for near-perfect (or perfect) signal reconstruction, efficient frequency subband decomposition, and minimized aliasing. Extending fundamental QMF and PQMF architectures, the multi-channel PQMF supports an arbitrary number of channels (M > 2), high overlap factors, and flexible adaptation to a wide array of signal processing and neural audio modeling tasks. With mathematical underpinnings in matrix factorization and spectral prototype design, multi-channel PQMFs form a core component in applications ranging from source separation and wavelet transforms to artifact-free neural vocoding and high-fidelity neural audio compression.

1. Mathematical Structure and Synthesis

Multi-channel PQMFs employ a linear-phase prototype filter $h[n]$, whose spectral properties determine the anti-aliasing and reconstruction performance of the entire system. For $M$ frequency subbands, the set of subband filters $\{h_k[n]\}_{k=0}^{M-1}$ is generated via cosine modulation:

$$h_k[n] = 2\, h[n] \cos\left( \frac{\pi}{M}(k+0.5)n + \varphi_k \right)$$

where $\varphi_k$ is typically selected as $(-1)^k \pi/4$, providing near-orthogonality and enforcing minimal overlap among subbands (Zhang et al., 21 Sep 2025, Bak et al., 2022). The prototype $h[n]$ itself is obtained as a spectral factor of a full-band filter $F(z)$, such that:

$F(z) = h(z)\, h(z^{-1})$

The analysis and synthesis polyphase matrices, $P^a(z)$ and $P^s(z)$, are derived from $h[n]$ and its modulated variants, with the perfect or near-perfect reconstruction condition ideally enforced by:

$P^s(z) = [P^a(z)]^{-1} z^{-d}$

where the integer $d$ delays ensure causality in implementation (Mimilakis et al., 2017).

For abstract signal processing tasks involving multivariate or multidimensional signals, PQMF matrices can be structured via tensor products of univariate QMF filters and unimodular coordinate transformations, as elucidated in Smith factorization techniques (Cotronei et al., 2018). This allows PQMF systems to be adapted for anisotropic dilations, generalized shearlets, and higher-dimensional analysis.

2. Factorization and Multiresolution Implementation

The PQMF can be elegantly represented and factorized using matrix algebra, treating the combined operation of analysis filtering, downsampling, modulation, and synthesis as a global $M \times M$ matrix $A(z)$ with entries in a suitable function ring (Jorgensen et al., 2014). The factorization into lower and upper triangular (lifting) matrices:

$$A(z) = L_p(z) U_p(z) \cdots L_1(z) U_1(z) A^{(1)}(z)$$

decomposes signal transformation into a sequence of elementary lifting steps. Each step isolates and processes specific frequency band pairs, supporting the modular design of the filter bank. The connection to Cuntz algebra isometries,

$(S_i f)(z) = z^{i} f(z^M)$

frames downsampling and upsampling as operators acting naturally within this algebraic structure, ensuring that sampling operations commute with the factorization process.

The flexibility of this matrix-based construction is crucial, as it supports:

• Both FIR and IIR filter designs,
• A variety of non-polynomial filterbank entries for enhanced frequency tiling,
• Explicit critical sampling, as well as non-separable multidimensional extensions.

3. Aliasing and Spectral Leakage Control

PQMFs are designed with windowing and spectral optimization strategies that suppress alias components between adjacent subbands. Suppression is achieved by selecting prototype $h[n]$ to satisfy frequency-domain constraints such as:

$$|H(e^{j\omega})|^2 + |H(e^{j(\pi/M - \omega)})|^2 = 2, \qquad 0 < |\omega| < \pi/(2M)$$

$|H(e^{j\omega})|^2 = 0, \qquad \omega > \pi/M$

These design conditions minimize leakage between frequency bins, supporting high disjointness and sparsity in time-frequency (T-F) representations (Mimilakis et al., 2017). Filter banks with high stopband attenuation (exceeding 100 dB) provide effective anti-aliasing performance, which is critical when employed in neural vocoder subsystems or source separation pipelines (Bak et al., 2022).

4. Applications in Signal Processing and Machine Learning

Multi-channel PQMFs serve pivotal roles in several domains:

• Time-Frequency Transform for Source Separation: Enhanced T-F representations (e.g., $N=1024$ channels, $L=8$ overlap) achieve higher W-disjoint orthogonality and Gini-measured sparsity than STFT, outperforming alternatives like MDCT for quasi-harmonic sources (Mimilakis et al., 2017).
• Artifact-Free Neural Vocoding: PQMF filter banks in GAN-based vocoders (e.g., Avocodo) realize artifact-suppressed multi-band and subband discrimination, mitigating both aliasing and upsampling artifacts, verified in both subjective MOS and objective metrics (F0-RMSE, LSD-HF, PESQ) (Bak et al., 2022).
• Neural Audio Compression: In MBCodec, multi-channel PQMFs guide residual vector quantization (RVQ) codebooks to specialize in per-band acoustic features, enabling thorough disentanglement from semantic content and realizing compression ratios as high as $170\times$ (down to 2.2 kbps), with ablation studies confirming strong improvements in PESQ and Mel-spectrogram distance upon PQMF inclusion (Zhang et al., 21 Sep 2025).
• Wavelet and Anisotropic Analysis: PQMF frameworks embedded in tensor-product or Smith-decomposed QMF structures support orthogonal wavelet bases and generalized shearlets for directional, multidimensional data (Cotronei et al., 2018).

5. Perfect Reconstruction and Polyphase Optimization

Perfect or near-perfect reconstruction is central to PQMF utility. In multi-channel settings, polyphase matrix engineering is essential. The design approach mandates:

• Linearly constrained window optimization,
• Polyphase decomposition (with, for instance, $P^a_{n',k}(m)$ terms involving prototype modulation over frame, overlap, and subband indices),
• Integer delay alignment to ensure phase linearity and minimal distortion.

In FIR settings, as discussed in (Martin-Rodriguez et al., 2022), designing QMF (and by extension PQMF) banks with odd coefficients ensures linear phase and pure integer delay, facilitating lossless reconstruction in wavelet and multiband analysis applications.

6. Extensions: Anisotropy, Non-Polynomiality, and Multidimensionality

Advanced PQMF implementations generalize to cases with:

• Anisotropic scaling and shearing (for directional feature extraction), via Smith factorization and unimodular transformations (Cotronei et al., 2018).
• Non-polynomial filter banks (e.g., meromorphic entries in $A(z)$) that go beyond FIR constraints, offering improved frequency response flexibility and supporting perfect reconstruction over broader algebraic classes (Jorgensen et al., 2014).
• Multichannel, multidimensional settings via tensor products and block structures, delivering critically sampled, directionally sensitive decompositions suitable for image, video, and scientific data (Cotronei et al., 2018).

7. Empirical Evidence and Performance Metrics

Empirical studies demonstrate the effectiveness of multi-channel PQMFs:

Application area	PQMF Impact	Metrics/Results
Neural vocoding (Avocodo)	Suppresses aliasing/imaging artifacts; clean subband splits	MOS, F0-RMSE, LSD-HF, PESQ improved over baselines
Neural audio compression (MBCodec)	Enables disentangled, subband-specialized codebooks	PESQ: 3.83 (PQMF) vs. 2.34 (no PQMF); 2.2 kbps achieved
Source separation	Increases disjointness, sparsity vs. MDCT, STFT	Higher WDO, Gini index (esp. for harmonic sources)

Objective improvements are consistently correlated with PQMF-based guidance, as shown by ablation analyses in neural audio coding and T-F transform tasks (Zhang et al., 21 Sep 2025, Mimilakis et al., 2017, Bak et al., 2022).

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Multi-channel PQMFs represent a matured, mathematically grounded approach for frequency band decomposition, critical sampling, and artifact management in both traditional and neural audio processing. Their flexibility in configuration, robustness in aliasing suppression, and compatibility with advanced machine learning systems underscore their continued relevance in audio, image, and multidimensional signal representation contexts.