Transfer Function

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Index: Introduction to Digital Filters with Audio Applications

Introduction to Digital Filters with Audio Applications

Analysis of a Digital Comb Filter

Impulse Response

Frequency Response

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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

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

Matlab is a high-level scripting language for scientific computing — Click for https://ccrma.stanford.edu/~jos/matlab/

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

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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A signal is typically a real-valued function of time. A discrete-time signal is typically a real-valued function of discrete time, and is therefore a time-ordered sequence of real numbers. — Click for http://ccrma.stanford.edu/~jos/filters/Definition_Signal.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 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

Search JOS Website

JOS Home Page

JOS Online Publications

Index of terms in JOS Website

Index: Introduction to Digital Filters with Audio Applications

Introduction to Digital Filters with Audio Applications

Analysis of a Digital Comb Filter

Impulse Response

Frequency Response

Transfer Function

As we cover in Chapter 6, the transfer function of a digital filter is defined as where is the z transform of the input signal , and is the z transform of the output signal . We may find from Eq.(3.1) by taking the z transform of both sides and solving for :

$\begin{eqnarray*} \boldmath {{\cal Z}}\{y(n)\} &=& \boldmath {{\cal Z}}\{x(n) + g_1 x(n-M_1) - g_2 y(n-M_2)\}\\ [5pt] &=& \boldmath {{\cal Z}}\{x(n)\} + g_1 \boldmath {{\cal Z}}\{x(n-M_1)\} \\ && \qquad\qquad \mathrel{-} g_2 \boldmath {{\cal Z}}\{y(n-M_2)\}\\ [5pt] \implies\quad Y(z) &=& X(z) + g_1 z^{-M_1} X(z) - g_2 z^{-M_2} Y(z)\\ [5pt] \implies\quad Y(z) + g_2 z^{-M_2} Y(z) &=& X(z) + g_1 z^{-M_1} X(z) \\ [5pt] \implies\quad H(z) \isdef \frac{Y(z)}{X(z)} &=& \frac{1 + g_1 z^{-M_1}}{1 + g_2 z^{-M_2}} \qquad\hbox{(Transfer Function)} \protect \end{eqnarray*}$

Some principles of this analysis are as follows:

The z transform ${\cal Z}\{\cdot\}$ is a linear operator which means, by definition,

$\displaystyle \boldmath {{\cal Z}}\{\alpha x_1(n) + \beta x_2(n)\} = \alpha \boldmath {{\cal Z}}\{x_1(n)\} + \beta \boldmath {{\cal Z}}\{x_2(n)\} \isdef \alpha X_1(z) + \beta X_2(z).$
${\cal Z}\{x(n-M)\} = z^{-M}X(z)$ . That is, the z transform of a signal delayed by samples, $x(n-M),\,n=0,1,2,\ldots$ , is $z^{-M}X(z)$ . This is the shift theorem for z transforms, which can be immediately derived from the definition of the z transform, as shown in §6.3.

Note that these two properties of the z transform are all we really need to find the transfer function of any linear, time-invariant digital filter from its difference equation (its implementation formula in the time domain).

In matlab, difference-equation coefficients are specified as transfer-function coefficients (vectors B and A in Fig.3.9). This is why a minus sign is needed in Eq.(3.3).

Ok, so finding the transfer function is not too much work. Now, what can we do with it? There are two main avenues of analysis from here: (1) finding the frequency response by setting $z=e^{j\omega T}$ , and (2) factoring the transfer function to find the poles and zeros of the filter. One also uses the transfer function to generate different implementation forms such as cascade or parallel combinations of smaller filters to achieve the same overall filter. The following sections will illustrate these uses of the transfer function on the example filter of this chapter.

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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

Transfer Function

``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