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 $\ensuremath{L_2}$ 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:

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

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

$$\isdef \hbox{FT}_\omega (x)/\sqrt{2\pi} \isdef X(\omega )/\sqrt{2\pi}$$

$$x(t) = \int_{-\infty}^{\infty} \left<\varphi _\omega ,x\right> \varphi _\omega (t) d\omega$$

$$= \frac{1}{2\pi} \int_{-\infty}^{\infty} X(\omega )e^{j\omega t} d\omega$$

$$\isdef \hbox{FT}_t^{-1}(X)$$