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,

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

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 [452].
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 [336].

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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University