Radians versus Cycles

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Radians versus Cycles

Our usual frequency variable is $\omega$ in radians per second. However, certain Fourier theorems are undeniably simpler and more elegant when the frequency variable is chosen to be in cycles per second. The two are of course related by

$\displaystyle \omega = 2\pi f.$

As an example, $e^{j\omega t}$ is more compact than $e^{j2\pi f t}$ . On the other hand, it is nice to get rid of all normalization constants in the Fourier transform and its inverse:

$\begin{eqnarray*} X(f) &=& \ensuremath{\int_{-\infty}^{\infty}}x(t)e^{-j2\pi f t... ... &=& \ensuremath{\int_{-\infty}^{\infty}}X(f)e^{j2\pi f t} df\\ \end{eqnarray*}$

The ``editorial policy'' for this book is this: Generally, $\omega$ is preferred, but is used when considerable simplification results. With a bit of thought, it is not hard to convert back and forth as needed.

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written by 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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Spectral Audio Signal Processing
    Fourier Transforms for Continuous/Discrete Time/Frequency
       Selected Continuous Fourier Theorems
          Radians versus Cycles