# Summary of the Properties of the Fourier Transform

 Property $f(t)$ $F(\omega)$ Linearity $a_{1} f_{1}(t)+a_{2} f_{2}(t)$ $a_{1} F_{1}(\omega)+a_{2} F_{2}(\omega)$ Scaling $f(a t)$ $\frac{1}{|a|} F\left(\frac{\omega}{a}\right)$ Time shift $f(t-a) u(t-a)$ $e^{-j \omega a} F(\omega)$ Frequency shift $e^{j \omega_{0} t} f(t)$ $F\left(\omega-\omega_{0}\right)$ Modulation $\cos \left(\omega_{0} t\right) f(t)$ $\frac{1}{2}\left[F\left(\omega+\omega_{0}\right)+F\left(\omega-\omega_{0}\right)\right]$ Time differentiation $\frac{d f}{d t}$ $j \omega F(\omega)$ $\frac{d^{n} f}{d t^{n}}$ $(j \omega)^{n} F(\omega)$ Time integration $\int_{-\infty}^{t} f(t) d t$ $\frac{F(\omega)}{j \omega}+\pi F(0) \delta(\omega)$ Frequency differentiation $t^{n} f(t)$ $(j)^{n} \frac{d^{n}}{d \omega^{n}} F(\omega)$ Reversal $f(-t)$ $F(-\omega)$ or $F^{*}(\omega)$ Duality $F(t)$ $2 \pi f(-\omega)$ Convolution in $t$ $f_{1}(t) * f_{1}(t)$ $F_{1}(\omega) F_{2}(\omega)$ Convolution in $\omega$ $f_{1}(t) f_{1}(t)$ $\frac{1}{2 \pi} F_{1}(\omega) * F_{2}(\omega)$
 $f (t)$ $F (ω)$ $δ(t)$ $1$ $1$ $2πδ(ω)$ $u(t)$ $πδ(ω) + { 1 \over jω}$ $u(t + τ) − u(t − τ)$ $2 {sin ωτ \over ω}$ $|t|$ ${ -2 \over ω^2}$ $sgn(t)$ ${ 2 \over jω}$ $e^{−at}u(t)$ ${ 1 \over a + jω }$ $e^{at}u(-t)$ ${ 1 \over a - jω }$ $t^ne^{−at}u(t)$ ${ n! \over (a + jω)^{n+1} }$ $e^{−a|t|}$ ${ 2a \over a^2 + ω^2}$ $e^{jω_0t}$ $2πδ(ω − ω_0)$ $\sin ω_0t$ $jπ[δ(ω + ω_0) − δ(ω − ω_0)]$ $\cos ω_0t$ $π[δ(ω + ω_0) + δ(ω − ω_0)]$ $e^{−at}\sin ω_0t u(t)$ ${ω_0 \over (a + jω)^2 + ω_0^2}$ $e^{−at}\cos ω_0t u(t)$ ${a + jω \over (a + jω)^2 + ω_0^2}$