### Application of the Bilinear Transform

The impedance of a mass in the frequency domain is

In the plane, we have

where the a'' subscript denotes analog''. For simplicity, let's choose the free constant in the bilinear transform such that rad/sec maps to one fourth the sampling rate, i.e., maps to which implies . Then the impedance relation maps across as

where the d'' subscript denotes digital. Multiplying through by the denominator and applying the shift theorem for transforms gives the corresponding 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.
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Bilinear Transformation