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Gaussian Characteristic Function

Since the Gaussian PDF is

$\displaystyle p(t) \isdef \frac{1}{\sqrt{2\pi\sigma^2}} e^{-\frac{(t-\mu)^2}{2\sigma^2}}$ (D.50)

and since the Fourier transform of $ p(t)$ is

$\displaystyle P(\omega) = e^{-j\mu \omega} e^{-\frac{1}{2}\sigma^2\omega^2}$ (D.51)

It follows that the Gaussian characteristic function is

$\displaystyle \Phi(\omega) = \overline{P(\omega)} = e^{j\mu \omega} e^{-\frac{1}{2}\sigma^2\omega^2}.$ (D.52)

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Moment Theorem