Filter Order = Transfer Function Order

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Index: Introduction to Digital Filters with Audio Applications

Introduction to Digital Filters with Audio Applications

Pole-Zero Analysis

Graphical Amplitude Response

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

The topic of linear systems theory is primarily about linear, time-invariant (LTI) filters. A linear filter is characterized by the property that its output-signal amplitude is linearly proportional to its input-signal amplitude. A time-invariant filter, or 'constant-coefficient' filter, performs the same filtering operation at all times. Only LTI filters can be characeterized by their impulse response, or their frequency response, or their poles and zeros plus a gain, or the like. Also, only LTI filters have the superposition property for audio signals, in which the filtering of an audio mix can be obtained alternatively by filtering each channel separately and adding the filtered channels together. Equivalently, only LTI filters exhibit no intermodulation distortion for audio signals. — Click for https://ccrma.stanford.edu/~jos/filters/Linear_Time_Invariant_Digital_Filters.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

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Search JOS Website

JOS Home Page

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Index of terms in JOS Website

Index: Introduction to Digital Filters with Audio Applications

Introduction to Digital Filters with Audio Applications

Pole-Zero Analysis

Graphical Amplitude Response

Filter Order = Transfer Function Order

Recall that the order of a polynomial is defined as the highest power of the polynomial variable. For example, the order of the polynomial is 2. From Eq.(8.1), we see that is the order of the transfer-function numerator polynomial in $z^{-1}$ . Similarly, is the order of the denominator polynomial in $z^{-1}$ .

A rational function is any ratio of polynomials. That is, is a rational function if it can be written as

$\displaystyle R(z)\eqsp \frac{P(z)}{Q(z)}$

for finite-order polynomials

and

. The order of a rational function is defined as the maximum of its numerator and denominator polynomial orders. As a result, we have the following simple rule:

$\textstyle \parbox{0.9\textwidth}{\emph{The order of an LTI filter is the order of its transfer function.}}$

It turns out the transfer function can be viewed as a rational function of either $z^{-1}$ or without affecting order. Let $K=\max\{M,N\}$ denote the order of a general LTI filter with transfer function expressible as in Eq.(8.1). Then multiplying by gives a rational function of (as opposed to $z^{-1}$ ) that is also order when viewed as a ratio of polynomials in . Another way to reach this conclusion is to consider that replacing by $z^{-1}$ is a conformal map [57] that inverts the -plane with respect to the unit circle. Such a transformation clearly preserves the number of poles and zeros, provided poles and zeros at $z=\infty$ and are either both counted or both not counted.

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``Introduction to Digital Filters with Audio Applications'', by Julius O. Smith III, (September 2007 Edition)
Copyright © 2024-09-03 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

Filter Order = Transfer Function Order

``Introduction to Digital Filters with Audio Applications'', by Julius O. Smith III, (September 2007 Edition) Copyright © 2024-09-03 by Julius O. Smith III Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

``Introduction to Digital Filters with Audio Applications'', by Julius O. Smith III, (September 2007 Edition)
Copyright © 2024-09-03 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA), Stanford University