Definition

Linear Frequency \(f\)

Fourier Transform:

\[X(f) = \mathcal{F}\{x(t)\} = \int_{-\infty}^{\infty} x(t) e^{-j2\pi tf} dt\]

Inverse Fourier Transform:

\[x(t) = \mathcal{F}^{-1}\{X(f)\} = \int_{-\infty}^{\infty} X(f) e^{j2\pi tf} dt\]

Where

• \(f = \frac{1}{T}\) is frequency

The physical unit is [Hz] \(= \frac{1}{s}\)

Angular frequency \(\omega = 2\pi f\)

Fourier Transform:

\[X(\omega) = \mathcal{F}\{x(t)\} = \int_{-\infty}^{\infty} x(t) e^{-j\omega t} dt\]

Inverse Fourier Transform:

\[x(t) = \mathcal{F}^{-1}\{X(\omega)\} = \frac{1}{2\pi} \int_{-\infty}^{\infty} X(\omega) e^{j\omega t} dt\]

Where

• \(\omega = 2\pi f\) is angular frequency

The physical unit is [rad/s]

If the integral goes to infinity, then the function \(x(t)\) does not have a fourier transform.

Basic Pairs

\(x(t)\)	\(X(\omega)\)
\(\delta(t-t_0)\)	\(e^{-j\omega t_0}\)
\(1\)	\(2\pi \delta(\omega)\)
\(u(t)\)	\(\frac{1}{j\omega} + \pi \delta(\omega)\)
\(e^{j\omega_0 t}\)	\(2\pi \delta(\omega - \omega_0)\)
\(\sin{\omega_0 t}\)	\(\frac{\pi}{j} (\delta(\omega - \omega_0) - \delta(\omega + \omega_0))\)
\(\cos{\omega_0 t}\)	\(\frac{\pi}{j} (\delta(\omega - \omega_0) + \delta(\omega + \omega_0))\)
\(1,\ \mid \omega \mid \ < T_1\)	\(\frac{2\sin{\omega T_1}}{\omega}\)
\(\frac{\sin{Wt}}{\pi t}\)	\(1,\ \mid \omega \mid \ < W\)

Properties

Linearity

\[\mathcal{F}\{ax_1 (t) + bx_2 (t)\} = aX_1 (\omega) + bX_2 (\omega)\]

Time shifting

\[\mathcal{F}\{x(t - t_0)\} = X(\omega) e^{-j\omega t_0}\]

Scaling

\[\mathcal{F}\{x(at)\} = \frac{1}{\mid a \mid} X \left( \frac{\omega}{a} \right)\]

Frequency shifting

\[\mathcal{F}\{x(t)\ e^{j\omega_0 t} \} = X(\omega - \omega_0)\]

Differentiation

\[\mathcal{F}\left \{ \frac{d^n}{dt^n} x(t) \right \} = (j\omega)^n X(\omega)\]

Integration

\[\mathcal{F}\left \{ \int_{-\infty}^{t} x(\tau) d\tau \right \} = \frac{1}{j\omega} X(\omega) + \pi X(0) \delta(\omega)\]

Convolution

\[\mathcal{F}\{x_1(t) * x_2(t) \} = X_1(\omega)\ X_2(\omega)\]

Multiplication

\[\mathcal{F}\{x_1(t)\ x_2(t) \} = \frac{1}{2\pi} X_1(\omega) * X_2(\omega)\]