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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:

$\displaystyle \varphi _\omega (t) \isdef e^{j\omega t}/\sqrt{2\pi},\quad \omega , t\in (-\infty,\infty)$ (12.113)

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

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