Exact Factorisation and Fast Computation of Invertible Constant-Q Transforms
Abstract
The constant-Q transform (CQT) represents audio on a logarithmic frequency axis. Its nonstationary Gabor formulation is exactly invertible, but the unequal numbers of time coefficients in its bands complicate GPU computation. An exact factorisation combines spectral selection, conjugation, windowing, and reordering into a fixed map between one packed Fourier transform and the shorter band inverse transforms. The factors give waveform reconstruction, real adjoints for backpropagation, and bounds on arithmetic depth and block width; overlapping slices permit streaming with bounded memory. Tests on two GPU models show that Flash-CQT reduces analysis-synthesis round-trip time by factors of two to eight relative to a baseline computing the same CQT. The proposed implementation also uses over 30% less peak temporary workspace and reaches a negligible reconstruction error, with a signal-to-noise ratio of about 130 dB, in single-precision floating-point arithmetic. These advances make Flash-CQT a practical, computationally efficient front end for spectral analysis and modern audio machine-learning systems.
Related
- Prime factorisation of stable-matching instances: uniqueness, simultaneous products, and an exact census
- A stochastic Carleson embedding theorem with constant e and the vector of Riesz transforms
- Dimension-free weak (1,1) estimates for Riesz transforms with constant drift
- Exact value functions and sensitivities for energy storage by fast Legendre transforms
- Exact Covariance Structure and Efficient Computation for L-Comoments
BibTeX
@misc{franchino2026exact,
title = {{Exact Factorisation and Fast Computation of Invertible Constant-Q Transforms}},
author = {Facundo Franchino and Eloi Moliner and Vesa Välimäki},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.29119},
eprint = {2609.29119},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.29119},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Exact Factorisation and Fast Computation of Invertible Constant-Q Transforms
AU - Facundo Franchino
AU - Eloi Moliner
AU - Vesa Välimäki
PY - 2026
DA - 2026/09/24
PB - arXiv
DO - 10.48550/arxiv.2609.29119
UR - https://arxiv.org/abs/2609.29119
AB - The constant-Q transform (CQT) represents audio on a logarithmic frequency axis. Its nonstationary Gabor formulation is exactly invertible, but the unequal numbers of time coefficients in its bands complicate GPU computation. An exact factorisation combines spectral selection, conjugation, windowing, and reordering into a fixed map between one packed Fourier transform and the shorter band inverse transforms. The factors give waveform reconstruction, real adjoints for backpropagation, and bounds on arithmetic depth and block width; overlapping slices permit streaming with bounded memory. Tests on two GPU models show that Flash-CQT reduces analysis-synthesis round-trip time by factors of two to eight relative to a baseline computing the same CQT. The proposed implementation also uses over 30% less peak temporary workspace and reaches a negligible reconstruction error, with a signal-to-noise ratio of about 130 dB, in single-precision floating-point arithmetic. These advances make Flash-CQT a practical, computationally efficient front end for spectral analysis and modern audio machine-learning systems.
ER -