Fourier Transform Properties

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

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

$\mathcal{F}[a{f}_1(t)+b{f}_2(t)]=a{F}_1(\omega )+b{F}_2(\omega )$

• Linearity

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

• Time scaling

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

• Time reversal

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

• Complex conjugate

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

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

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

• Time-shifting

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

• Derivative

$\mathcal{F}[(f_1\ast f_2)(t)]=F_1(j\omega )F_2(j\omega )$

• Convolution property

${\int }_{-\infty }^{\infty }|f(t){|}^2\mathrm{d}t=\frac{1}{2\pi }{\int }_{-\infty }^{\infty }|F(j\omega ){|}^2\mathrm{d}\omega$

• Parseval’s Theorem

$\mathcal{F}[f_1(t)f_2(t))]=\frac{1}{2\pi }(F_1\ast F_2)(j\omega )$

• Multiplication
• Can be proved by using duality + convolution

$\mathcal{F}[f(t)e^{j{\omega }_{0}t}]=F(j(\omega -{\omega }_0))$

• Modulation

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

• Integral