The analysis portion of PARSHL returns a set of amplitudes , frequencies , and phases , for each frame index , with a ``triad'' ( ) for each track . From this analysis data the program has the option of generating a synthetic sound.
The synthesis is done one frame at a time. The frame at hop , specifies the synthesis buffer
The parameter interpolation across time used in PARSHL is the same as that used by McAulay and Quatieri [12]. Let ( ) and ( ) denote the sets of parameters at frames and for the th frequency track. They are taken to represent the state of the signal at time 0 (the left endpoint) of the frame.
The instantaneous amplitude is easily obtained by linear interpolation,
Frequency and phase values are tied together (frequency is the phase derivative), and they both control the instantaneous phase . Given that four variables are affecting the instantaneous phase: , and , we need at least three degrees of freedom for its control, while linear interpolation only gives one. Therefore, we need at least a cubic polynomial as interpolation function, of the form
(15) | |||
(16) |
Figure 7 shows the result of the analysis/synthesis process using phase information and applied to a piano tone.