The shift-invariant discrete wavelet transform and application to speech waveform analysis.

The discrete wavelet transform may be used as a signal-processing tool for visualization and analysis of nonstationary, time-sampled waveforms. The highly desirable property of shift invariance can be obtained at the cost of a moderate increase in computational complexity, and accepting a least-squares inverse (pseudoinverse) in place of a true inverse. A new algorithm for the pseudoinverse of the shift-invariant transform that is easier to implement in array-oriented scripting languages than existing algorithms is presented together with self-contained proofs. Representing only one of the many and varied potential applications, a recorded speech waveform illustrates the benefits of shift invariance with pseudoinvertibility. Visualization shows the glottal modulation of vowel formants and frication noise, revealing secondary glottal pulses and other waveform irregularities. Additionally, performing sound waveform editing operations (i.e., cutting and pasting sections) on the shift-invariant wavelet representation automatically produces quiet, click-free section boundaries in the resulting sound. The capabilities of this wavelet-domain editing technique are demonstrated by changing the rate of a recorded spoken word. Individual pitch periods are repeated to obtain a half-speed result, and alternate individual pitch periods are removed to obtain a double-speed result. The original pitch and formant frequencies are preserved. In informal listening tests, the results are clear and understandable.