📂Fourier Analysis

Fourier Transform of Characteristic Functions

Formulas

The Fourier transform of the characteristic function is as follows:

$$\mathcal{F} \left[ \chi_{[-a,a]}(x) \right] = \dfrac{2 \sin(a\xi) }{\xi}$$

Proof

$$\begin{align*} \mathcal{F} \left[ \chi_{[-a,a]}(x) \right] &= \int_{-\infty}^{\infty} \chi_{[-a,a]}(x)e^{-i \xi x } dx \\ &= \int_{-a}^{a}e^{-i \xi x} dx \\ &= \dfrac{1}{-i\xi} \left. e^{-i\xi x}\right]_{-a}^{a} \\ &= \dfrac{1}{-i\xi} \left( e^{-i a \xi} - e^{i a \xi} \right) \\ &= \dfrac{2}{\xi} \dfrac{e^{ia\xi} -e^{-ia\xi}}{2i} \\ &= \dfrac{2}{\xi} \sin (a\xi) \end{align*}$$

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