Fourier Transform

Definition

There are multiple ways to define the Fourier Transform, but this is the convention commonly used in electrical engineering. The correction factor in the inverse transform arises due to the change of variables from f to ω.

$$\begin{aligned} \mathcal F[ f(t) ] &= F(\omega) = \int_{-\infty}^{\infty} f(t) e^{-j\omega t} \; dt \\ \\ \mathcal F^{-1}[ F(j\omega) ] &= f(t) = \frac{1}{2\pi} \int_{-\infty}^{\infty} F(\omega) e^{-j\omega t} \; d\omega \end{aligned}$$

Functions that depend on t are time domain representations, while functions that use ω are frequency domain representations.

Simple Functions

Time domain	Frequency domain
$$1$$	$$2\pi\delta(\omega)$$
$$\delta(t)$$	$$1$$
$$u(t)$$	$$\pi \delta(\omega) + \frac{1}{j\omega} $$
$$e^{j\omega_0 t}$$	$$2\pi \delta(\omega - \omega_0)$$
$$u(t) e^{-\alpha t}$$	$$\frac{1}{\alpha + j \omega}$$
$$\cos(\omega_0 t)$$	$$\pi (\delta(\omega - \omega_0) + \delta(\omega + \omega_0))$$
$$\sin(\omega_0 t)$$	$$\frac{\pi}{j} (\delta(\omega - \omega_0) - \delta(\omega + \omega_0))$$

Properties

Linearity

$$\begin{aligned} \mathcal F[ c_1 f(t) + c_2 g(t) ] &= c_1 \mathcal F[ f(t) ] + c_2 \mathcal F[ g(t) ] \end{aligned}$$

Shifting

$$\begin{aligned} \mathcal F[ f(t - t_0) ] &= e^{-j\omega t_0} \mathcal F[ f(t) ] \end{aligned}$$

Scaling

$$\begin{aligned} \mathcal F[ f(at) ] &= \frac{1}{|a|} F(\frac{\omega}{a}) \end{aligned}$$

Reversal

$$\begin{aligned} \mathcal F[ f(-t) ] &= F(-\omega) \end{aligned}$$

Conjugation

$$\begin{aligned} \mathcal F[ f^{*}(t) ] &= F^{*}(-\omega) \end{aligned}$$

Differentiation

$$\begin{aligned} \mathcal F[ \frac{d}{dt} f(t) ] &= j\omega \mathcal F[ f(t) ] \\ \\ \mathcal F^{-1}[ j \frac{d}{d\omega} F(\omega) ] &= t \mathcal F^{-1}[ F(\omega) ] \end{aligned}$$

Convolution Theorem

$$\begin{aligned} \mathcal F[ f(t) \ast g(t) ] &= \mathcal F[ f(t) ] \mathcal F[ g(t) ] \\ \\ \mathcal F[ f(t) g(t) ] &= \mathcal F[ f(t) ] \ast \mathcal F[ g(t) ] \end{aligned}$$