Techniques for estimating parameters using function approximation have been discussed in previous papers, [4] [5] [1]. We will summarize this work here.
The basic problem is to find an inverse mapping of a sound-generating system. A physical model can be thought of as performing a mapping from parameters to a sound outcome; we wish to map from a sound intention to the physical parameters. These parameters comprise a description of the sound. By selecting the correct model and estimating the parameters we can extract physically meaningful information from the sounding world.
There are two basic architectures to consider for parameter estimation; dynamic and static parametric modeling. Dynamic modeling, [5], needs an explicit time stamp in order to give instantaneous parameter estimates as a function of time. In dynamic modeling it is assumed that the parameters vary significantly as a function of time, thus for each sound a set of time-varying parameters are extracted.\
One of the major drawbacks of dynamic parameter estimation is that it explicitly encodes for each moment of a sounds evolution in terms of articulator activity. Sounds evolve in time even if the parameters are static, such is the case in physical modeling where a set of stationary parameters give rise to a complex evolving signal.\
Figure 3: Distal Learning Architecture. A forward model of the physical environment
is used to constrain the learning of an inverse model. The parameters that the
inverse model estimates are then passed through to the physical environment and the
environment performance error is used to fine-tune the inverse estimator.
We chose to implement a static parameter estimator whose input is a dynamic sound representation and whose output is a set of stationary parameters; see Figure 3. The justification for this approach is that there are far fewer parameters that vary than those that are static. Much of the variation in musical instrument articulators is in expressive control; such as in the use of vibrato. These dynamic parameters often operate at very low frequency and can be described as a simple function of time. Thus, even dynamic parameters can be reduced to a static description if an appropriate function generator is selected.\
Static parameter estimators are more robust, simpler to implement and require less training data than dynamic estimators. We also believe that they form a better model of perception than dynamic estimators.