Ethan's Notes

    Fourier Transform

    #Math

    $\displaystyle f(x)=\frac{1}{\sqrt{ 2\pi }}\int_{-\infty}^{\infty} \tilde{f}(k)e^{ikx} , \mathrm{d}k$

    $\displaystyle \mathcal{F}[f(x)]=\tilde{f}(k)=\frac{1}{\sqrt{ 2\pi }}\int_{-\infty}^{\infty} f(x)e^{-ikx} , \mathrm{d}x$

    • This is the Fourier transform of $\displaystyle f(x)$ that takes inputs from $\displaystyle k$ space

    $\displaystyle \int_{-\infty}^{\infty} f(t) , \mathrm{d}t=\tilde{f}(0)$

    $\displaystyle \mathcal{F}[f(t)\cdot g(t)]=\mathcal{F}[f(t)]*\mathcal{F}[g(t)]$

    Quantum Physics

    Topics

    $\displaystyle \Psi(x)=\frac{1}{\sqrt{ 2\pi }}\int_{-\infty}^{\infty} \tilde{\Psi}(k)e^{ikx} , \mathrm{d}k$

    $\displaystyle \tilde{\Psi}(k)=\frac{1}{\sqrt{ 2\pi }}\int_{-\infty}^{\infty} \Psi(x)e^{-ikx} , \mathrm{d}x$

    Signal Processing

    $\displaystyle F(j\omega)=\int_{-\infty}^{\infty} f(t)e^{-j\omega t} , \mathrm{d}t$

    • The $\displaystyle F(j\omega)$ is effectively the same as $\displaystyle F(\omega)$

    $\displaystyle f(t)=\frac{1}{2\pi}\int_{-\infty}^{\infty} F(j\omega)e^{j\omega t} , \mathrm{d}\omega$

    Topics