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Embedded SystemsFPGAElectronics

Spectral Audio Signal Processing
    Multirate Filter Banks
       Wavelet Filter Banks
          Geometric Signal Theory
             Normalized Fourier Transform Basis

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Normalized Fourier Transform Basis

The Fourier transform projects a continuous-time signal $ x(t)$ onto an infinite set of continuous-time complex sinusoids $ \exp(j\omega t)$, for $ t,\omega\in(-\infty,\infty)$. These sinusoids all have infinite $ L2$ norm, but a simple normalization by $ 1/\sqrt{2\pi}$ can be chosen so that the inverse Fourier transform has the desired form of a superposition of projections:

$\displaystyle \varphi _\omega (t) \isdef e^{j\omega t}/\sqrt{2\pi},\quad \omega , t\in (-\infty,\infty) $


$\displaystyle \displaystyle \implies\quad \left<\varphi _\omega ,x\right>$ $\displaystyle =$ $\displaystyle \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty} x(t) e^{-j\omega t} dt$  
  $\displaystyle \isdef$ $\displaystyle \hbox{FT}_\omega (x)/\sqrt{2\pi} \isdef X(\omega )/\sqrt{2\pi}$  
       
$\displaystyle x(t)$ $\displaystyle =$ $\displaystyle \int_{-\infty}^{\infty} \left<\varphi _\omega ,x\right> \varphi _\omega (t) d\omega$  
  $\displaystyle =$ $\displaystyle \frac{1}{2\pi} \int_{-\infty}^{\infty} X(\omega )e^{j\omega t} d\omega$  
  $\displaystyle \isdef$ $\displaystyle \hbox{FT}_t^{-1}(X)$  

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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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