Additive Synthesis from Sinusoidal Peak Tracks

In the analysis phase, sinusoidal peaks are measured over time in a sequence of FFTs, and these peaks are grouped into tracks across time. If the time advance from one FFT to the next is fixed (5ms is a typical choice for speech analysis), then we obtain uniformly sampled amplitude and frequency trajectories as the result of the analysis. The sampling rate of these amplitude and frequency envelopes is equal to the frame rate of the analysis. (If the time advance between FFTs is $\Delta t=5$ms, then the frame rate is defined as $1/\Delta t = 200$ Hz.) For resynthesis using inverse FFTs, these data may be used unmodified. For resynthesis using a bank of sinusoidal oscillators, on the other hand, we must somehow interpolate the envelopes to create envelopes at the signal sampling rate (typically $44$ kHz or higher).

It is typical in computer music to linearly interpolate the amplitude and frequency trajectories from one frame to the next [248]. Higher order interpolations of so-called envelope break-points were also developed at CCRMA in the late 1970s (e.g., using cubic splines), but for tonal sounds, linearly interpolation is usually sufficient, and the higher-order envelopes did not see much use, presumably due to the greater complexity of dealing with them coupled with the lack of significant benefit. Let's call the piecewise linear upsampled envelopes ${\hat A}_k(n)$ and $\hat{F}_k(n)$, defined now for all $n$ at the normal signal sampling rate. For steady-state tonal sounds, the phase may be discarded at this stage and redefined as the integral of the instantaneous frequency when needed:

$$\hat{\Theta }_k(n) \isdef \hat{\Theta }_k(n-1) + 2\pi T \hat{F}_k(n).$$

When phase must be matched in a given frame, such as when it is known to contain a transient event, the frequency can instead move quadratically across the frame to provide cubic polynomial phase interpolation [158], or a second linear breakpoint can be introduced somewhere in the frame for the frequency trajectory (in which case the area under the triangle formed by the second breakpoint equals the added phase at the end of the segment).