Force-Pulse Filter Design

The design of the filters whose impulse response approximates a correct force pulse will be addressed in this section.

Assume we are given a desired hammer-string interaction force for each key and for each key velocity which looks qualitatively as in Fig. 5.1. Such a curve can be computed, for example, using a nonlinear model of the hammer-string interaction [531].

There are two basic types of filter design problems we consider: (1) Design a single filter whose input is a series of interaction impulses (three of them in the example figure) and whose output is the desired force curve, or (2) somehow separate the individual force pulses in the force curve and design a separate filter whose impulse response gives each desired pulse shape.

Consider the dual of this problem. That is, suppose the desired force curve were a desired spectrum rather than a desired time trace. In that case, the ``interaction impulses'' would be sinusoidal peaks, and their parameters would be amplitude and frequency instead of amplitude and arrival time. The ``spreading'' of peaks in spectrum analysis is caused by some kind of modulation which causes it to deviate from a pure sinusoid. Examples include (1) a window function (tapering) in the time domain, and (2) exponential decay of each sinusoid. The window function in case (1) can be obtained by performing an inverse