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Physical Audio Signal Processing
    Introduction to Lumped Models
       Bilinear Transformation
          Application of the Bilinear Transform

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Application of the Bilinear Transform

The impedance of a mass in the frequency domain is

$\displaystyle R_M(s) = Ms$

In the

plane, we have

$\displaystyle F_a(s) = (Ms) V_a(s)$

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

$\displaystyle F_d(z) = \left(M\frac{1-z^{-1}}{1+z^{-1}}\right) V_d(z)$

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

transforms gives the corresponding difference equation

$\begin{eqnarray*} (1+z^{-1})F_d(z) &=& M (1-z^{-1}) V_d(z) \\ \;\longleftrighta... ... \,\,\Rightarrow\,\,f_d(n) &=& M[v_d(n) - v_d(n-1)] - f_d(n-1) \end{eqnarray*}$

This difference equation is diagrammed in Fig. L.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. 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 [340].

**Figure L.16:** A direct-form-I digital filter simulating a mass created using the bilinear transform $s=(1-z^{-1})/(1+z^{-1})$ .
$\includegraphics[width=4in]{eps/lmassFilterDF1}$

Subsections

Practical Considerations

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written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See