DAFX: Digital Audio Effects, Second Edition

11.2 Warping

11.2.1 Time Warping

Suppose that we want to change the shape of a periodic waveform s(t) by moving the amplitude values attained by the signal to other time instants. One can achieve this by plotting the signal on an elastic sheet and by stretching and/or compressing the sheet at different points along its horizontal direction. The waveshape appears as if the original time axis had been deformed. Instants of time that were equidistant now have a different time distribution. This deformation of the time axis called time warping is characterized by a warping map θ(t), mapping points of the original t-axis onto points of the transformed axis. An example of time warping a sinewave is shown in Figure 11.1. The figure is obtained by plotting the original signal along the ordinates and transforming time instants into new time instants via the warping map, obtaining the signal plotted along the abscissa axis. Notice that to one point in the original signal there can correspond more points in the warped signal. These points are obtained by joining time instants of the original signal to points on the warping characteristic θ(t) using horizontal lines. The corresponding warped time instants are the value(s) of the abscissa corresponding to these intersection point(s). The time-warped signal is obtained by plotting the corresponding amplitude values at the new time instants along the abscissa. In this example the signal sin(θ(t)) may be interpreted as a phase modulation of the ...