An efficient algorithm for the calculation of a constant <i>Q</i> transform

TLDR

The method calculates kernels that are applied to each DFT to produce a constant‑Q transform. The algorithm efficiently converts a DFT into a constant‑Q transform using few multiples, adding minimal overhead while fully exploiting FFT speed, as demonstrated on clarinet and violin signals.

Abstract

An efficient method of transforming a discrete Fourier transform (DFT) into a constant Q transform, where Q is the ratio of center frequency to bandwidth, has been devised. This method involves the calculation of kernels that are then applied to each subsequent DFT. Only a few multiples are involved in the calculation of each component of the constant Q transform, so this transformation adds a small amount to the computation. In effect, this method makes it possible to take full advantage of the computational efficiency of the fast Fourier transform (FFT). Graphical examples of the application of this calculation to musical signals are given for sounds produced by a clarinet and a violin.