FDA in the Frequency Domain

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Index: Physical Audio Signal Processing

Physical Audio Signal Processing

Finite Difference Approximation

Delay Operator Notation

In the context of linear time-invariant filter analysis, a zero is a point in the complex s or z plane at which the transfer function goes to zero. A pole is a point in the s or z plane at which the transfer function goes to infinity. The s plane is used to analyze analog filters while the z plane is used for digital filters. — Click for https://ccrma.stanford.edu/~jos/filters/Pole_Zero_Analysis_I.html

A conformal map is a complex-valued function of a complex variable that is an angle-preserving mapping from one complex plane to another. That is, the mapping preserves the angles between intersecting curves in the complex plane. — Click for https://mathworld.wolfram.com/ConformalMapping.html

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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

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

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Physical Audio Signal Processing

Finite Difference Approximation

Delay Operator Notation

FDA in the Frequency Domain

Viewing Eq.(7.2) in the frequency domain, the ideal differentiator transfer-function is , which can be viewed as the Laplace transform of the operator (left-hand side of Eq.(7.2)). Moving to the right-hand side, the z transform of the first-order difference operator is $(1-z^{-1})/T$ . Thus, in the frequency domain, the finite-difference approximation may be performed by making the substitution

$\displaystyle s \eqsp \frac{1-z^{-1}}{T} \protect$

(8.3)

in any continuous-time transfer function (Laplace transform of an integro-differential operator) to obtain a discrete-time transfer function (z transform of a finite-difference operator).

The inverse of substitution Eq.(7.3) is

$\displaystyle z \eqsp \frac{1}{1 - sT} \eqsp 1 + sT+ (sT)^2 + \cdots \, .$

As discussed in §8.3.1, the FDA is a special case of the matched transformation applied to the point .

Note that the FDA does not alias, since the conformal mapping $s = {1 - z^{-1}}$ is one to one. However, it does warp the poles and zeros in a way which may not be desirable, as discussed further on p. below.

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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

FDA in the Frequency Domain

``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