Exact Factorisation and Fast Computation of Invertible Constant-Q Transforms

2026-09-24

AI Summary

This study addresses the computational challenges of the Constant-Q Transform (CQT), where unequal frequency band lengths hinder GPU parallelization and constrain memory efficiency. To overcome these limitations, this work proposes a fixed-mapping factorization method that optimizes CQT computation by integrating non-stationary Gabor frames, spectral selection, and conjugate window functions. The resulting framework supports both streaming processing and backpropagation while significantly reducing arithmetic depth. This approach enables efficient and invertible audio spectral analysis. Experimental results demonstrate that the proposed method accelerates round-trip computation by 2–8×, reduces peak memory consumption by over 30%, and achieves a signal-to-noise ratio of 130 dB for single-precision signal reconstruction, establishing it as a highly effective solution for differentiable audio processing.

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.

Problem

• Constant-Q Transform
• GPU computation
• invertibility
• audio machine learning
• spectral analysis

Innovation

• Constant-Q Transform
• Exact Factorisation
• GPU Acceleration
• Nonstationary Gabor
• Differentiable Audio