Spectral warping revisited

Spectral warping is a time domain to time domain transformation on a signal that effectively warps the frequency content of the original signal. Here we present a matrix formulation of the spectral warping transformation. The transform matrix is decomposed into three steps. The first is a DFT to convert the time signal into the frequency domain. Step two is an interpolation matrix to calculate the signal content at the desired new frequency samples. This effectively provides the frequency warping. The final step is an inverse DFT to transform the signal back into the time domain. A direct consequence of this matrix representation is a direct FIR implementation of spectral warping, rather than the more commonly used IIR technique. We demonstrate that spectral warping is a generalisation of linear filtering, and show how the conventional all-pass spectral warping transformation can be generalised by using either arbitrary frequency mapping functions or different interpolation schemes. Finally, the conditions for the invertibility of the spectral warping transformation are derived.