The PhISEM approach uses the results of ``off-line'' particle interaction simulations to derive statistical collision distributions.
From a numerical simulation of a maraca, statistics were collected with respect to frequency, waiting time, and sound intensity of bean collisions with the outer gourd wall.
In general, a short-term energy input produced an exponential decay in intensity of collisions.
The likelihood of a significant sound-producing collision was found to be roughly constant, until all beans were nearly at rest.
A process with constant event probability at any given time has an exponential distribution in time (i.e., the time duration between events decays away exponentially over time).
The collected statistics could then be used to control a model in which events trigger distinct overlapping, decaying signals.
However, a more efficient approach is found by noting the noise-like nature of the resulting sound from a maraca. This suggests the use of decaying noise to represent collision events and a single second-order digital filter to model the gourd resonance.
Because the sum of exponentially decaying random noise is equal to a single noise source multiplied by a decaying value, only a single exponential decay and a single noise source are required to compute the total sound.
The maraca synthesis algorithm requires only two random-number calculations (for collision-event likelihood and noise sound source), two exponential decay multiplications (for net system energy and sound level), and one biquad filter, as shown in Fig. 9.