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Chapters

Spectral Audio Signal Processing
    Fourier Transforms for Continuous/Discrete Time/Frequency
       Continuous-Time Fourier Theorems
          Scaling Theorem

Chapter Contents:

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Scaling Theorem

The scaling theorem (or similarity theorem) says that if you horizontally ``stretch'' a signal by the factor $\alpha$ in the time domain, you ``squeeze'' and amplify its Fourier transform by the same factor in the frequency domain. This is an important general Fourier duality relationship:

Theorem: For all continuous-time functions possessing a Fourier transform,

$\displaystyle \zbox {\hbox{\sc Stretch}_\alpha(x) \;\longleftrightarrow\;\left\vert\alpha\right\vert\hbox{\sc Stretch}_{(1/\alpha)}(X)}$

where

$\displaystyle \hbox{\sc Stretch}_{\alpha,t}(x) \isdefs x\left(\frac{t}{\alpha}\right)$

and $\alpha$ is any nonzero real number (the abscissa stretch factor). A more commonly used notation is the following:

$\displaystyle \zbox {x\left(\frac{t}{\alpha}\right) \;\longleftrightarrow\; \left\vert\alpha\right\vert\cdot X(\alpha\omega)}$

Proof: Taking the Fourier transform of the stretched signal gives

$\begin{eqnarray*} \hbox{\sc FT}_{\omega}(\hbox{\sc Stretch}_\alpha(x)) &\isdef ... ...\tau \\ &\isdef & \left\vert\alpha\right\vert X(\alpha\omega). \end{eqnarray*}$

The absolute value appears above because, when $\alpha<0$ , $d (\alpha\tau) < 0$ , which brings out a minus sign in front of the integral from $-\infty$ to $\infty$ .

The scaling theorem is fundamentally restricted to the continuous-time, continuous-frequency (Fourier transform) case.

For this and other continuous-time Fourier theorems, see §B.1.

The closest we come to the scaling theorem among the DTFT theorems (§2.3) is the stretch (repeat) theorem (page ).

Previous: Continuous-Time Fourier Theorems
Next: Spectral Roll-Off

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About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.

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