Practical Considerations

While the digital mass simulator has the desirable properties of the bilinear transform, it is also not perfect from a practical point of view: (1) There is a pole at half the sampling rate ($z=-1$). (2) There is a delay-free path from input to output.

The first problem can easily be circumvented by introducing a loss factor $g$, moving the pole from $z=-1$ to $z=-g$, where $g\in[0,1)$ and $g\approx1$. This is sometimes called the ``leaky integrator''.

The second problem arises when making loops of elements (e.g., a mass-spring chain which forms a loop). Since the individual elements have no delay from input to output, a loop of elements is not computable using standard signal processing methods. The solution proposed by Alfred Fettweis is known as ``wave digital filters,'' a topic we take up in §N.1.