Synthesis algorithm

Figure 1 shows a block diagram for a packet-based formant synthesizer. Phases are generated by a phase accumulator (at the top) which repeats at the rate $\omega/2$ where $\omega$ is the desired fundamental frequency of the output. The table on the left hand side holds a packet, which is considered as a sampled period of a (real-valued) waveform with period $1$ . The phases of the partials of the stored waveform are all assumed to be aligned at $n=0$ :

$$p(t) = \sum_{k=0}^{N-1} {a_k}{{e^{2 \pi k t i }}}$$

where the coefficients $a_k$ are real-valued. The parameter $S$ is a shift factor; if $S=1$ each period of the phase generator swipes through the table twice, so that the wavetable is scanned as a phase increment corresponding to a fundamental frequency of $\omega$ . The notation $p(t)$ reflects our assumption that the wavetable (and others below) are sampled with sufficiently high resolution and/or interpolated so that their values may be regarded as depending on a continuous parameter $t$ .

Figure 1: GRANSAMP: Synthesis by windowed wave packets.

The right-hand side table lookup is a windowing function, necessary in case the value of $S$ is not a half-integer. The windowing function is assumed to be zero outside $[-0.5,0.5]$ . In these units, the Hann window for example is $w(t) = 0.5 + 0.5\cdot \cos(2\pi t)$ .

The Hann window provides perfect reconstruction of the waveform $p(t)$ if we make two overlapping copies of the diagram of Figure 1, one-half cycle out of phase, provided further that the parameter $S$ is an integer so that the two copies make an in-phase cross-fade of the waveform $p(n)$ . This can be done as diagrammed in Figure 2.

Figure 2: Combining two out-of-phase copies of ``GRANSAMP" from Figure 1.