In this talk we review the concepts associated to frequency warping and show applications in several directions, ranging from adapted signal representations to physical models of stiff systems and from digital audio effects to expression tools.
Frequency warping is a signal transformation obtained by remapping the frequency axis according to a prescribed law. While by far the most common application of the transformation is in filter design (Constantinides transformation), its use in audio signal processing and sound synthesis has been rediscovered. In recent applications the technique has been used in order to design perceptual filter banks or to dynamically change the frequency damping characteristics (loop filter) in the physical model of guitar strings. However, the most interesting implications of frequency warping rest on warping signals rather than filters.
In the mid 60's P. Broome introduced the Laguerre sequences. The set gives rise to orthogonal signal expansions that can be computed in terms of a chain of all-pass filters. The expansion coefficients can be interpreted as a frequency warped version of the original signal, with the reconstruction formula equivalent to unwarping. The concept was subsequently employed by Oppenheim, Johnson and Braccini in order to perform non-uniform bandwidth spectral analysis based on a frequency warped DFT. In recent work we employed frequency warping and iterated frequency warping in the construction of arbitrary scale factor or perceptually based wavelets. Moreover, an extension of Laguerre sequences leads to reversible time-varying frequency warping algorithms interesting in audio effects.
The computation of the Laguerre transform is intensive [O(N^2)] and inherently non-causal. Furthermore, the warping characteristics are constrained to belong to a one-parameter family of curves. However, a new approximate algorithm is proposed that makes accurate real-time computation of frequency warping possible even with arbitrary warping maps.