Examples of Digital Filters

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Index: Introduction to Digital Filters with Audio Applications

Introduction to Digital Filters with Audio Applications

Linear Time-Invariant Filters

Definition of a Filter

Linear Filters

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

Noise (in signal processing) is a random signal. In the language of statistical signal processing, a noise signal is typically modeled as 'stochastic process', which is in turn defined as a sequence of random variables. — Click for https://ccrma.stanford.edu/~jos/sasp/What_Noise.html

Any function that is not linear is nonlinear. There are no general methods for solving nonlinear equations or nonlinear differential equations. This is why it is usually easier, although generally less accurate, to approximate a nonlinear function by a linear function if possible. — Click for http://en.wikipedia.org/wiki/Nonlinear

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

A causal filter is any filter whose impulse response is zero prior to time zero. — Click for https://ccrma.stanford.edu/~jos/filters/Causal_Recursive_Filters.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

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

The set of real numbers may be expressed informally as the set of all numbers that may be expressed as a decimal number with possibly an infinite number of digits. More formally, the set of real numbers is the field of rational and irrational numbers. — Click for http://mathworld.wolfram.com/RealNumber.html

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/

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

Linear Time-Invariant Filters

Definition of a Filter

Linear Filters

Examples of Digital Filters

While any mapping from signals to real numbers can be called a filter, we normally work with filters which have more structure than that. Some of the main structural features are illustrated in the following examples.

The filter analyzed in Chapter 1 was specified by

$\displaystyle y(n)=x(n) + x(n-1).$

Such a specification is known as a difference equation. This simple filter is a special case of an important class of filters called linear time-invariant (LTI) filters. LTI filters are important in audio engineering because they are the only filters that preserve signal frequencies.

The above example remains a real LTI filter if we scale the input samples by any real coefficients:

$\displaystyle y(n)=2\, x(n) - 3.1\, x(n-1)$

If we use complex coefficients, the filter remains LTI, but it becomes a complex filter:

$\displaystyle y(n)=(2+j)\,x(n) + 5 j \,x(n-1)$

The filter also remains LTI if we use more input samples in a shift-invariant way:

$\displaystyle y(n)=x(n) + x(n-1) + x(n+1) + \cdots$

The use of ``future'' samples, such as

in this difference equation, makes this a non-causal filter example. Causal filters may compute

using only present and/or past input samples

, and so on.

Another class of causal LTI filters involves using past output samples in addition to present and/or past input samples. The past-output terms are called feedback, and digital filters employing feedback are called recursive digital filters:

$\displaystyle y(n)=x(n) - x(n-1) + 0.1 \, y(n-1) + \cdots$

An example multi-input, multi-output (MIMO) digital filter is

$\displaystyle \left[\begin{array}{c} y_1(n) \\ [2pt] y_2(n) \end{array}\right] = \left[\begin{array}{cc} a & b \\ [2pt] c & d \end{array}\right] \left[\begin{array}{c} x_1(n) \\ [2pt] x_2(n) \end{array}\right] + \left[\begin{array}{cc} e & f \\ [2pt] g & h \end{array}\right]\left[\begin{array}{c} x_1(n-1) \\ [2pt] x_2(n-1) \end{array}\right],$

where we have introduced vectors and matrices inside square brackets. This is the 2D generalization of the SISO filter $y(n) = a \, x(n) + b\, x(n-1)$ .

The simplest nonlinear digital filter is

$\displaystyle y(n)=x^2(n),$

i.e., it squares each sample of the input signal to produce the output signal. This example is also a memoryless nonlinearity because the output at time

is not dependent on past inputs or outputs. The nonlinear filter

$\displaystyle y(n)=x(n)-y^2(n-1)$

is not memoryless.

Another nonlinear filter example is the median smoother of order which assigns the middle value of input samples centered about time to the output at time . It is useful for ``outlier'' elimination. For example, it will reject isolated noise spikes, and preserve steps.

An example of a linear time-varying filter is

$\displaystyle y(n)=x(n) + \cos(2\pi n /10)\, x(n-1).$

It is time-varying because the coefficient of

changes over time. It is linear because no coefficients depend on

These examples provide a kind of ``bottom up'' look at some of the major types of digital filters. We will now take a ``top down'' approach and characterize all linear, time-invariant filters mathematically. This characterization will enable us to specify frequency-domain analysis tools that work for any LTI digital filter.

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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

Examples of Digital Filters

``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