Gaussian Pulse MATLAB

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Gaussian Pulse - FFT & PSD in Matlab & Python - GaussianWaves

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$g(t) = \displaystyle{\frac{1}{\sqrt{2 \pi } \sigma} e^{- \frac{t^2}{2 \sigma^2}}}$

Gaussian Pulse - FFT & PSD in Matlab & Python - GaussianWaves

$\begin{aligned} G(f) &=F[g(t)]\\ &= \int_{-\infty }^{\infty } g(t)e^{-j2\pi ft}\, dt\\ &= \frac{1}{\sigma \sqrt{2 \pi } } \int_{-\infty }^{\infty } e^{- \frac{t^2}{2 \sigma^2}}e^{-j2\pi ft}\, dt\\ &=\frac{1}{\sigma \sqrt{2 \pi } } \int_{-\infty }^{\infty } e^{- \frac{1}{2 \sigma^2}\left[t^2+j4 \pi \sigma^2 ft \right]}\, dt\\ &=\frac{1}{\sigma \sqrt{2 \pi } } \int_{-\infty }^{\infty } e^{- \frac{1}{2 \sigma^2}\left[t^2+j4 \pi \sigma^2 ft + (j 2 \pi \sigma^2 f)^2 - (j 2 \pi \sigma^2 f)^2\right]}\, dt\\ &=e^{ \frac{1}{2 \sigma^2}(j 2 \pi \sigma^2 f)^2}\frac{1}{\sigma \sqrt{2 \pi } } \int_{-\infty }^{\infty } e^{- \frac{1}{2 \sigma^2}\left[t+j 2 \pi \sigma^2 f \right]^2}\, dt\\ &=e^{ \frac{1}{2 \sigma^2}(j 2 \pi \sigma^2 f)^2}=e^{ - \frac{1}{2}( 2 \pi \sigma f)^2}\end{aligned}$

Gaussian Pulse - FFT & PSD in Matlab & Python - GaussianWaves

Solved Question 7. Use Matlab to plot the Gaussian pulse | Chegg.com

$\displaystyle{\frac{1}{\sigma \sqrt{2 \pi} }\int_{-\infty }^{\infty }e^{- \frac{1}{2 \sigma^2}\left[t+j 2 \pi \sigma^2 f \right]^2}\, dt =\frac{1}{\sigma \sqrt{2 \pi } }\int_{-\infty }^{\infty }e^{- \frac{1}{2 \sigma^2} u^2}\, du =1}$

Gaussian Pulse - FFT & PSD in Matlab & Python - GaussianWaves

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