Ethan's Notes

    Fourier Transform Properties

    #Math

    $\displaystyle \mathcal{F}[(f(t))]\equiv \tilde{f}(\omega)\equiv F(\omega)$

    • Generally, the Fourier transform a function is denoted by the uppercase letter version

    $\displaystyle \mathcal{F}[af_{1}(t)+bf_{2}(t)]=aF_{1}(\omega)+bF_{2}(\omega)$

    • Linearity

    $\displaystyle \mathcal{F}[f(at)]=\frac{1}{\lvert a\rvert}F\left( j \frac{\omega}{a} \right)$

    • Time scaling

    $\displaystyle \mathcal{F}[f(-t)]=F(-j\omega)$

    • Time reversal

    $\displaystyle \mathcal{F}[f^{}(t)]=F^{}(-j\omega)$

    • Complex conjugate

    $\displaystyle \mathcal{F}[F(t)]=2\pi f(-j\omega)$

    • Duality
    • Shown by taking $\displaystyle f(t)=\frac{1}{2\pi}\int_{-\infty}^{\infty} F(j\omega)e^{j\omega t} , \mathrm{d}\omega$
    • then rearranging to $\displaystyle 2\pi f(-t)=\int_{-\infty}^{\infty} F(j\omega)e^{-j\omega t} , \mathrm{d}\omega=\mathcal{F}[F(t)]$

    $\displaystyle \mathcal{F}[f(t-\tau)]=e^{-j\omega \tau}F(j\omega)$

    • Time-shifting

    $\displaystyle \mathcal{F}[f'(t)]=j\omega F(j\omega)$

    • Derivative

    $\displaystyle \mathcal{F}[(f_{1}*f_{2})(t)]=F_{1}(j\omega)F_{2}(j\omega)$

    • Convolution property

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

    • Parseval's Theorem

    $\displaystyle \mathcal{F}[f_{1}(t)f_{2}(t))]=\frac{1}{2\pi}(F_{1}*F_{2})(j\omega)$

    • Multiplication
    • Can be proved by using duality + convolution

    $\displaystyle \mathcal{F}[f(t)e^{j{\omega}{0}t}]=F(j(\omega-{\omega}{0}))$

    • Modulation

    $\displaystyle \int_{-\infty}^{t} f(\tau) , \mathrm{d}\tau <\Rightarrow \pi F(0)\delta(\omega)+\frac{F(j\omega)}{j\omega}$

    • Integral