### Application of the Bilinear Transform

The impedance of a mass in the frequency domain is

*i.e.*, maps to which implies . Then the impedance relation maps across as

*difference equation*

This difference equation is diagrammed in Fig. 7.16.
We recognize this recursive digital filter as the *direct form I*
structure. The direct-form II structure is obtained by commuting the
feedforward and feedback portions and noting that the two delay
elements contain the same value and can therefore be shared [449].
The two other major
filter-section forms are obtained by *transposing* the two direct
forms by exchanging the input and output, and reversing all
arrows. (This is a special case of Mason's Gain Formula which works
for the single-input, single-output case.) When a filter structure is
transposed, its summers become branching nodes and vice versa.
Further discussion of the four basic filter section forms can be found
in [333].

#### 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 (). (2) There is a delay-free path from input to output.

The first problem can easily be circumvented by introducing a loss factor , moving the pole from to , where and . 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 taken up in §F.1.

**Next Section:**

Limitations of Lumped Element Digitization

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Bilinear Transformation