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

## D.2 Discrete-Time Fourier Transforms

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
$$1$$ $$\delta(f)$$
$$\delta[n]$$ $$1$$
$$\cos(2\pi f_0 n)$$ $$\frac{1}{2}(\delta(f - f_0) + \delta(f + f_0))$$ $$-0.5 < f < 0.5$$
$$\sin(2\pi f_0 n)$$ $$\frac{1}{2j}(\delta(f - f_0) - \delta(f + f_0))$$ $$-0.5 < f < 0.5$$
$$\alpha^{|n|}$$ $$\displaystyle\frac{1 - \alpha^{2}}{1 + \alpha^{2} - 2\alpha \cos(2\pi f)}$$ $$|\alpha| < 1$$
$$\alpha^{n} u[n]$$ $$\displaystyle\frac{1}{1 - \alpha e^{-j2\pi f}}$$ $$|\alpha| < 1$$

## D.3 Fourier Properties

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 $$\int_{-\infty}^\infty g(t)\,dt$$ $$\overset{\mathscr{F}}{\longleftrightarrow}$$ $$G(0)$$