Collective Table of Formulas

Table of Continuous-time (CT) Fourier Transform Pairs and Properties

as a function of $ \omega $ in radians per time unit

(used in ECE301)


Definition CT Fourier Transform and its Inverse
(info) CT Fourier Transform $ \mathcal{X}(\omega)=\mathcal{F}(x(t))=\int_{-\infty}^{\infty} x(t) e^{-i\omega t} dt $
(info) Inverse CT Fourier Transform $ \, x(t)=\mathcal{F}^{-1}(\mathcal{X}(\omega))=\frac{1}{2\pi} \int_{-\infty}^{\infty}\mathcal{X}(\omega)e^{i\omega t} d \omega\, $
CT Fourier Transform Pairs

signal (function of t) $ \longrightarrow $ Fourier transform (function of $ \omega $)
1 CTFT of a unit impulse $ \delta (t)\ $ $ 1 \ $
2 CTFT of a shifted unit impulse $ \delta (t-t_0)\ $ $ e^{-iwt_0} $
3 CTFT of a complex exponential $ e^{iw_0t} $ $ 2\pi \delta (\omega - \omega_0) \ $
4 $ e^{-at}u(t),\ $ $ a\in {\mathbb R}, a>0 $ $ \frac{1}{a+i\omega} $
5 $ te^{-at}u(t),\ $ $ a\in {\mathbb R}, a>0 $ $ \left( \frac{1}{a+i\omega}\right)^2 $
6 CTFT of a cosine $ \cos(\omega_0 t) \ $ $ \pi \left[\delta (\omega - \omega_0) + \delta (\omega + \omega_0)\right] \ $
7 CTFT of a sine $ sin(\omega_0 t) \ $ $ \frac{\pi}{i} \left[\delta (\omega - \omega_0) - \delta (\omega + \omega_0)\right] $
8 CTFT of a rect $ \left\{\begin{array}{ll}1, & \text{ if }|t|<T,\\ 0, & \text{else.}\end{array} \right. \ $ $ \frac{2 \sin \left( T \omega \right)}{\omega} \ $
9 CTFT of a sinc $ \frac{\sin \left( W t \right)}{\pi t } \ $ $ \left\{\begin{array}{ll}1, & \text{ if }|\omega| <W,\\ 0, & \text{else.}\end{array} \right. \ $
10 CTFT of a periodic function $ \sum^{\infty}_{k=-\infty} a_{k}e^{ikw_{0}t} $ $ 2\pi\sum^{\infty}_{k=-\infty}a_{k}\delta(w-kw_{0}) \ $
11 CTFT of an impulse train $ \sum^{\infty}_{n=-\infty} \delta(t-nT) \ $ $ \frac{2\pi}{T}\sum^{\infty}_{k=-\infty}\delta(w-\frac{2\pi k}{T}) $
12 $ 1 \ $ $ 2\pi \delta (\omega) \ $
13 CTFT of a Periodic Square Wave $ x(t+T)=x(t)=\left\{\begin{array}{ll}1, & |t|\leq T_1,\\ 0, & T_1<|t|\leq T/2 \end{array} \right. $ $ \sum^{\infty}_{k=-\infty}\frac{2 \sin(k\frac{2\pi}{T}T_1)}{k}\delta(\omega-k\frac{2\pi}{T}) $
14 CTFT of a Step Function $ u(t) \ $ $ \frac{1}{j\omega}+\pi\delta(\omega) $
15 $ e^{-\alpha |t|} \ $ $ \frac{2\alpha}{\alpha^{2}+\omega^{2}} $
CT Fourier Transform Properties
$ x(t) \ $ $ \longrightarrow $ $ \mathcal{X}(\omega) $
16 (info) multiplication property $ x(t)y(t) \ $ $ \frac{1}{2\pi} \mathcal{X}(\omega)*\mathcal{Y}(\omega) =\frac{1}{2\pi} \int_{-\infty}^{\infty} \mathcal{X}(\theta)\mathcal{Y}(\omega-\theta)d\theta $
17 convolution property $ x(t)*y(t) \ $ $ \mathcal{X}(\omega)\mathcal{Y}(\omega) \! $
18 time reversal $ \ x(-t) $ $ \ \mathcal{X}(-\omega) $
19 Frequency Shifting $ e^{j\omega_0 t}x(t) $ $ \mathcal{X} (\omega - \omega_0) $
20 Conjugation $ x^{*}(t) \ $ $ \mathcal{X}^{*} (-\omega) $
21 Time and Frequency Scaling $ x(at) \ $ $ \frac{1}{|a|} \mathcal{X} (\frac{\omega}{a}) $
23 Differentiation in Frequency $ tx(t) \ $ $ j\frac{d}{d\omega} \mathcal{X} (\omega) $
24 Symmetry $ x(t)\ \text{ real and even} $ $ \mathcal{X} (\omega) \ \text{ real and even} $
25 $ x(t) \ \text{ real and odd} $ $ \mathcal{X} (\omega) \ \text{ purely imaginary and odd} $
26 Duality $ \mathcal{X} (-t) $ $ 2 \pi x (\omega) \ $
27 Differentiation $ \frac{d^{n}x(t)}{dt^{n}} $ $ (j \omega)^{n} \mathcal{X} (\omega) $
28 Linearity $ ax(t) + by(t) \ $ $ a \mathcal{X}(\omega) + b \mathcal{Y} (\omega) $
29 Time Shifting $ x(t-t_0) \ $ $ e^{-j\omega t_0}X(\omega) $
Other CT Fourier Transform Properties
Parseval's relation $ \int_{-\infty}^{\infty} |x(t)|^2 dt = \frac{1}{2\pi} \int_{-\infty}^{\infty} |\mathcal{X}(w)|^2 dw $



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Questions/answers with a recent ECE grad

Ryne Rayburn