Complex and Trigonometric Identities

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Index: Introduction to Digital Filters with Audio Applications

Introduction to Digital Filters with Audio Applications

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Complex Numbers

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

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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/

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Index: Introduction to Digital Filters with Audio Applications

Introduction to Digital Filters with Audio Applications

Background Fundamentals

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Complex Numbers

Complex and Trigonometric Identities

This section gives a summary of some of the more useful mathematical identities for complex numbers and trigonometry in the context of digital filter analysis. For many more, see handbooks of mathematical functions such as Abramowitz and Stegun [2].

The symbol $\isdef$ means ``is defined as''; stands for a complex number; and , $\theta$ , , and stand for real numbers. The quantity is used below to denote $\tan(\theta/2)$ .

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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

Complex and Trigonometric Identities

``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