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Index: Introduction to Digital Filters with Audio Applications

Introduction to Digital Filters with Audio Applications

Matrix Filter Representations

General Causal Linear Filter Matrix

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The Discrete Fourier Transform (DFT) computes a discrete-frequency spectrum from a discrete-time signal of finite length. — Click for https://ccrma.stanford.edu/~jos/mdft/

The Fourier Transform (FT) gives the (complex) spectrum of a continuous-time signal of any length. — Click for https://ccrma.stanford.edu/~jos/mdft/Fourier_Transform_FT_Inverse.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 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

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

Matrix Filter Representations

General Causal Linear Filter Matrix

Introduction

It is illuminating to look at matrix representations of digital filters.^F.1Every linear digital filter can be expressed as a constant matrix $\mathbf{h}$ multiplying the input signal ${\underline{x}}$ (the input vector) to produce the output signal (vector) $\underline{y}$ , i.e.,

$\displaystyle \underline{y}= \mathbf{h}{\underline{x}}.$

For simplicity (in this appendix only), we will restrict attention to finite-length inputs ${\underline{x}}^T = [x_0,\ldots,x_{N-1}]$ (to avoid infinite matrices), and the output signal will also be length

. Thus, the filter matrix $\mathbf{h}$ is a square $N\times N$ matrix, and the input/output signal vectors are $N\times 1$ column vectors.

More generally, any finite-order linear operator can be expressed as a matrix multiply. For example, the Discrete Fourier Transform (DFT) can be represented by the ``DFT matrix'' $[e^{-j2\pi kn/N}]$ , where the column index and row index range from 0 to [84, p. 111].^F.2Even infinite-order linear operators are often thought of as matrices having infinite extent. In summary, if a digital filter is linear, it can be represented by a matrix.

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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

Introduction

``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