Signal Notation

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Index: Spectral Audio Signal Processing

Spectral Audio Signal Processing

Notation

Frequency and Time

Fourier Transform Notation

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/

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Index: Spectral Audio Signal Processing

Spectral Audio Signal Processing

Notation

Frequency and Time

Fourier Transform Notation

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Signal Notation

$x \in \mathbb{C}^N$ means is a length complex sequence
$x = x(\cdot)$
$X = \hbox{\sc DFT}(x) \in \mathbb{C}^N$
$X(k) = \hbox{\sc DFT}_k(x)$
$X(k) = \hbox{\sc DFT}_{N,k}(x)$ denotes the th bin of a length DFT
$X(k) \in \mathbb{C}$
$x(n) = \hbox{\sc IDFT}_n(X)$
$x(n)\in\mathbb{R}$ or $\mathbb{C}$
$n,k \in \mathbb{Z}$ or $n,k \in \mathbb{Z}_N$ (integers modulo )
For $x\in \mathbb{C}^\infty$ , $X = \hbox{\sc DTFT}(x) \in \mathbb{C}_{2\pi}^{\infty}$
$\overline{x} =$ conjugate of
$\angle x =$ phase of

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``Spectral Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2011, ISBN 978-0-9745607-3-1.
Copyright © 2022-02-28 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA), Stanford University