Application of the Bilinear Transform

International Standard Book Number — Click for https://mathworld.wolfram.com/ISBN.html

Search JOS Website

JOS Home Page

JOS Online Publications

Index of terms in JOS Website

Index: Physical Audio Signal Processing

Physical Audio Signal Processing

Digitization of Lumped Models

FDA vs. Bilinear Transform

Practical Considerations

The nodes of a standing wave are the locations where the wave motion has zero amplitude. — Click for http://cnx.org/content/m12413/latest/

Click for https://ccrma.stanford.edu/~jos/filters/Time_Domain_Filter_Representations.html

A filter in the audio signal processing context is any operation that accepts a signal as an input and produces a signal as an output. Most practical audio filters are linear and time invariant, in which case they can be characterized by their impulse response or their frequency response. — Click for https://ccrma.stanford.edu/~jos/filters/What_Filter.html

Click for https://ccrma.stanford.edu/~jos/filters/Direct_Form_II.html

Click for https://ccrma.stanford.edu/~jos/filters/Direct_Form_I.html

A digital filter can be visualized as a 'black box' that accepts a sequence of numbers and emits a new sequence of numbers. In digital audio signal processing applications, such number sequences usually represent sounds. For example, digital filters are used to implement graphic equalizers and other digital audio effects. — Click for https://ccrma.stanford.edu/~jos/filters/

The difference equation is a recipe for computing samples of the output signal of a digital filter based on samples of the input signal and the filter coefficients. In an Infinite-Impulse-Response (IIR) digital filter, there are typically both feedforward and feedback coefficients in the difference equation. — Click for https://ccrma.stanford.edu/~jos/filters/Difference_Equation_I.html

Click for https://ccrma.stanford.edu/~jos/mdft/Shift_Theorem.html

The wave impedance in a propagation medium relates the force-related component of the wave motion to the displacement-related component of the motion. — Click for http://ccrma.stanford.edu/~jos/OnePorts/Impedance.html

The sampling rate of a discrete-time signal is defined as the number of samples per second. Its units are thus in Hertz (Hz). Shannon's Sampling Theorem states that the original continuous-time signal can be recovered exactly from the samples if and only if the sampling rate is higher than twice the highest frequency present in the original signal. Any higher frequencies will alias to frequencies below half the sampling rate. — Click for https://ccrma.stanford.edu/~jos/mdft/Sampling_Theory.html

Click for https://ccrma.stanford.edu/~jos/pasp/Bilinear_Transformation.html

The Fourier transform of a signal is a frequency-domain function — Click for https://ccrma.stanford.edu/~jos/mdft/

Click for http://hyperphysics.phy-astr.gsu.edu/hbase/mass.html#mas

The wave impedance in a propagation medium is the force variable (such as pressure for acoustic waves) divided by the velocity variable. In ideal plane waves and traveling waves, the wave impedance is a real, positive number. For spherical waves in air, the wave impedance is different at each frequency. — Click for https://ccrma.stanford.edu/~jos/OnePorts/Impedance.html

Search JOS Website

JOS Home Page

JOS Online Publications

Index of terms in JOS Website

Index: Physical Audio Signal Processing

Physical Audio Signal Processing

Digitization of Lumped Models

FDA vs. Bilinear Transform

Practical Considerations

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

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

This difference equation is diagrammed in Fig. 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].

**Figure 7.16:** 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

[How to cite this work] [Order a printed hardcopy] [Comment on this page via email]

``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4
Copyright © 2024-06-28 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

Application of the Bilinear Transform

``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4 Copyright © 2024-06-28 by Julius O. Smith III Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4
Copyright © 2024-06-28 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA), Stanford University