D Fourier Tables

D.1 Continuous-Time Fourier Transforms

\[ G(f) = \int_{-\infty}^\infty g(t) e^{-j2\pi f t}\,dt,\ \ -\infty < f < \infty \]

Time-Domain \(g(t)\)	Frequency-Domain \(G(f) = \mathscr{F}[g(t)]\)
\(1\)	\(\delta(f)\)
\(u(t) \overset{\text{def}}{=} \begin{cases} 1 & t \geq 0 \\ 0 & t < 0 \end{cases}\)	\(\displaystyle\frac{1}{2}\delta(f) + \frac{1}{j2\pi f}\)
\(\cos(2\pi f_0 t)\)	\(\frac{1}{2}(\delta(f - f_0) + \delta(f + f_0))\)
\(\sin(2\pi f_0 t)\)	\(\frac{1}{2j}(\delta(f - f_0) - \delta(f + f_0))\)
\(e^{-t}u(t)\)	\(\displaystyle\frac{1}{1 + j2\pi f}\)
\(e^{-\|t\|}\)	\(\displaystyle\frac{2}{1 + (2\pi f)^2}\)
\(e^{-\pi t^2}\)	\(e^{-\pi f^2}\)
\(\text{rect}(t) \overset{\text{def}}{=} \begin{cases} 1 & \|t\| \leq 0.5 \\ 0 & \|t\| > 0.5 \end{cases}\)	\(\displaystyle\text{sinc}(f) \overset{\text{def}}{=} \frac{\sin(\pi f)}{\pi f}\)
\(\text{tri}(t) \overset{\text{def}}{=} \begin{cases} 1 - \|t\| & \|t\| \leq 1 \\ 0 & \|t\| > 1 \end{cases}\)	\(\text{sinc}^2(f)\)

Note that \(f\) here denotes normalized frequency (cycles/sample). \[ G(f) = \sum_{n=-\infty}^\infty g[n] e^{-j2\pi f n},\ \ -0.5 < f < 0.5 \]

Time-Domain \(g[n]\)	Frequency-Domain \(G(f) = \mathscr{F}[g[n]]\)	Restrictions
\(1\)	\(\delta(f)\)
\(\delta[n]\)	\(1\)
\(\cos(2\pi f_0 n)\)	\(\frac{1}{2}(\delta(f - f_0) + \delta(f + f_0))\)	\(-0.5 < f < 0.5\)
\(\sin(2\pi f_0 n)\)	\(\frac{1}{2j}(\delta(f - f_0) - \delta(f + f_0))\)	\(-0.5 < f < 0.5\)
\(\alpha^{\|n\|}\)	\(\displaystyle\frac{1 - \alpha^{2}}{1 + \alpha^{2} - 2\alpha \cos(2\pi f)}\)	\(\|\alpha\| < 1\)
\(\alpha^{n} u[n]\)	\(\displaystyle\frac{1}{1 - \alpha e^{-j2\pi f}}\)	\(\|\alpha\| < 1\)

Suppose \(g\), \(g_1\), and \(g_2\) are time-domain signals with Fourier transforms \(G\), \(G_1\), and \(G_2\), respectively.

Property	When It Applies	Time-Domain		Frequency-Domain
Linearity	continuous-time, discrete-time	\(a g_1(t) + b g_2(t)\)	\(\overset{\mathscr{F}}{\longleftrightarrow}\)	\(a G_1(f) + b G_2(f)\)
Scaling	continuous-time only	\(g(at)\)	\(\overset{\mathscr{F}}{\longleftrightarrow}\)	\(\frac{1}{a} G\left(\frac{f}{a}\right)\)
Shifting	continuous-time, discrete-time	\(g(t + b)\)	\(\overset{\mathscr{F}}{\longleftrightarrow}\)	\(G(f) e^{j2\pi b f}\)
Convolution	continuous-time, discrete-time	\((g_1 * g_2)(t)\)	\(\overset{\mathscr{F}}{\longleftrightarrow}\)	\(G_1(f) \cdot G_2(f)\)
Reversal	continuous-time, discrete-time	\(g(-t)\)	\(\overset{\mathscr{F}}{\longleftrightarrow}\)	\(G(-f)\)
DC Offset	continuous-time, discrete-time	\(\int_{-\infty}^\infty g(t)\,dt\)	\(\overset{\mathscr{F}}{\longleftrightarrow}\)	\(G(0)\)