Fourier Transform Reference Cheat Sheet

Key Facts

• Using the angular frequency convention, the Fourier transform is $F(\omega)=\mathcal{F}\{f(t)\}=\int_{-\infty}^{\infty} f(t)e^{-i\omega t}\,dt$.
• The inverse Fourier transform is $f(t)=\mathcal{F}^{-1}\{F(\omega)\}=\frac{1}{2\pi}\int_{-\infty}^{\infty} F(\omega)e^{i\omega t}\,d\omega$.
• Linearity means $\mathcal{F}\{af(t)+bg(t)\}=aF(\omega)+bG(\omega)$ for constants $a$ and $b$.
• Time shifting gives $\mathcal{F}\{f(t-t_0)\}=e^{-i\omega t_0}F(\omega)$, while frequency shifting gives $\mathcal{F}\{e^{i\omega_0 t}f(t)\}=F(\omega-\omega_0)$.
• Scaling gives $\mathcal{F}\{f(at)\}=\frac{1}{|a|}F\left(\frac{\omega}{a}\right)$ for $a\ne 0$.
• Differentiation in time becomes multiplication in frequency: $\mathcal{F}\{f'(t)\}=i\omega F(\omega)$ when boundary terms vanish.
• Convolution in time becomes multiplication in frequency: $\mathcal{F}\{(f*g)(t)\}=F(\omega)G(\omega)$, where $(f*g)(t)=\int_{-\infty}^{\infty} f(\tau)g(t-\tau)\,d\tau$.
• Parseval's theorem states $\int_{-\infty}^{\infty}|f(t)|^2\,dt=\frac{1}{2\pi}\int_{-\infty}^{\infty}|F(\omega)|^2\,d\omega$.

Vocabulary

Fourier transform

A transform that rewrites a function $f(t)$ as a frequency-domain function $F(\omega)$ using complex exponentials.

Inverse Fourier transform

The operation that reconstructs $f(t)$ from $F(\omega)$ using $f(t)=\frac{1}{2\pi}\int_{-\infty}^{\infty}F(\omega)e^{i\omega t}\,d\omega$.

Angular frequency

The frequency variable $\omega$ measured in radians per unit, related to ordinary frequency by $\omega=2\pi f$.

Convolution

An integral combination of two functions defined by $(f*g)(t)=\int_{-\infty}^{\infty}f(\tau)g(t-\tau)\,d\tau$.

Dirac delta function

A generalized function $\delta(t)$ with the sifting property $\int_{-\infty}^{\infty}f(t)\delta(t-a)\,dt=f(a)$.

Transform pair

A matched pair $f(t)\leftrightarrow F(\omega)$ showing a function and its Fourier transform under a chosen convention.