Bilinear Transformation

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Physical Audio Signal Processing

Digitization of Lumped Models

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FDA vs. Bilinear Transform

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/

A frequency warp is a non-uniform scaling of frequency. Frequency warps are commonly used in audio signal processing in order to better mimic the non-uniform frequency resolution of human hearing. — Click for https://ccrma.stanford.edu/~jos/bbt/Filter_Design_Example.html

The frequency response is defined for LTI filters as the Fourier transform of the filter output signal divided by the Fourier transform of the filter input signal — Click for https://ccrma.stanford.edu/~jos/filters/Frequency_Response_I.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

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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

A filter is said to be stable if its impulse response decays to zero as time goes to infinity — Click for https://ccrma.stanford.edu/~jos/filters/Stability_Revisited.html

Sampling is the process of converting a continuous-time signal into a discrete-time signal. — Click for http://ccrma.stanford.edu/~jos/mdft/Sampling_Theory.html

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dc stands for direct current in electrical engineering and signal processing (as opposed to 'alternating current'). The mean (average value) of a signal may be called its dc component. Similarly, frequency zero is called the dc frequency, because a sinusoid with zero frequency is a constant signal. — Click for https://ccrma.stanford.edu/~jos/filters/Mathematical_Sine_Wave_Analysis.html

The transfer function is defined for LTI filters as the z transform of the filter output signal, divided by the z transform of the filter input signal — Click for https://ccrma.stanford.edu/~jos/filters/Transfer_Function_Analysis.html

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Index: Physical Audio Signal Processing

Physical Audio Signal Processing

Digitization of Lumped Models

Delay Operator Notation

FDA vs. Bilinear Transform

Bilinear Transformation

The bilinear transform is defined by the substitution

$\displaystyle s$	$\displaystyle =$	$\displaystyle c\frac{1-z^{-1}}{1+z^{-1}}, \quad c>0, \; c=2/T\;$ (typically)	(8.6)
$\displaystyle \,\,\Rightarrow\,\, z$	$\displaystyle =$	$\displaystyle \frac{1+s/c}{1-s/c} \eqsp 1 + 2(s/c) + 2(s/c)^2 + 2(s/c)^3 + \cdots$	(8.7)

where

is some positive constant [83,329]. That is, given a continuous-time transfer function

, we apply the bilinear transform by defining

$\displaystyle H_d(z) = H_a\left(c\frac{1-z^{-1}}{1+z^{-1}}\right) \protect$

(8.8)

where the ``

'' subscript denotes ``digital,'' and ``

'' denotes ``analog.''

It can be seen that analog dc ( ) maps to digital dc ( ) and the highest analog frequency ( $s=\infty$ ) maps to the highest digital frequency ( ). It is easy to show that the entire $j\omega$ axis in the plane (where $s\isdeftext \sigma+j\omega$ ) is mapped exactly once around the unit circle in the plane (rather than summing around it infinitely many times, or ``aliasing'' as it does in ordinary sampling). With real and positive, the left-half plane maps to the interior of the unit circle, and the right-half plane maps outside the unit circle. This means stability is preserved when mapping a continuous-time transfer function to discrete time.

Another valuable property of the bilinear transform is that order is preserved. That is, an th-order -plane transfer function carries over to an th-order -plane transfer function. (Order in both cases equals the maximum of the degrees of the numerator and denominator polynomials [452]).^8.6

The constant provides one remaining degree of freedom which can be used to map any particular finite frequency from the $j\omega_a$ axis in the plane to a particular desired location on the unit circle $e^{j\omega_d}$ in the plane. All other frequencies will be warped. In particular, approaching half the sampling rate, the frequency axis compresses more and more. Note that at most one resonant frequency can be preserved under the bilinear transformation of a mass-spring-dashpot system. On the other hand, filters having a single transition frequency, such as lowpass or highpass filters, map beautifully under the bilinear transform; one simply uses to map the cut-off frequency where it belongs, and the response looks great. In particular, ``equal ripple'' is preserved for optimal filters of the elliptic and Chebyshev types because the values taken on by the frequency response are identical in both cases; only the frequency axis is warped.

The frequency-warping of the bilinear transform is readily found by looking at the frequency-axis mapping in Eq.(7.7), i.e., by setting $s=j\omega_a$ and $z = e^{j\omega_d T}$ in the bilinear-transform definition:

$\displaystyle j\,\omega_a \;=\; c \, \frac{1-e^{-j\omega_d T}}{1+e^{-j\omega_d T}} \;=\; j \, c \, \tan\left(\frac{\omega_dT}{2}\right).$

Thus, we may interpret

as a frequency-scaling constant. At low frequencies, $\tan(x)\approx x$ , so that $\omega_a\approx c \omega_d T / 2$ at low frequencies, leading to the typical choice of

, where

denotes the sampling rate in Hz. However,

can be chosen to map exactly any particular interior frequency $\omega_a\in(0,f_s/2)$ .

The bilinear transform is often used to design digital filters from analog prototype filters [346]. An on-line introduction is given in [452].

Subsections

FDA vs. Bilinear Transform

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``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

Bilinear Transformation

``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