Transforms model a signal as a collection of waveforms of a particular form: sinusoids for the Fourier transform, mother wavelets for the wavelet transforms, periodic basis functions for the periodicity transforms. All of these methods are united in their use of inner products as a basic measure of the similarity and dissimilarity between signals, and all may be applied (with suitable care) to problems of rhythmic identification. A slightly early version of this chapter is also available electronically.

5.1 Inner Product: The Angle Between Two Signals5.3.1 Frequency via the DFT/FFT5.2 Correlation and Autocorrelation 5.3 The Fourier Transform 5.3.2 Three Mistakes 5.3.3 Short-time Fourier Transform 5.3.4 The Phase Vocoder 5.4 Wavelet Transforms5.5.1 Periodic Subspaces5.5 Periodicity Transforms 5.5.2 Projection onto Periodic Subspaces 5.5.3 Algorithms for Periodic Decomposition 5.5.4 Signal Separation 5.5.5 Choice of Sampling Rate 5.5.6 Discussion of PT 5.6 Summary |