Collective Table of Formulas

Continuous-time Fourier Transform Pairs and Properties

as a function of frequency f in hertz

(used in ECE438)



CT Fourier Transform and its Inverse
CT Fourier Transform $ X(f)=\mathcal{F}(x(t))=\int_{-\infty}^{\infty} x(t) e^{-i2\pi ft} dt $
Inverse DT Fourier Transform $ \, x(t)=\mathcal{F}^{-1}(X(f))=\int_{-\infty}^{\infty}X(f)e^{i2\pi ft} df \, $
CT Fourier Transform Pairs
signal (function of t) $ \longrightarrow $ Fourier transform (function of f)
CTFT of a unit impulse $ \delta (t)\ $ $ 1 \ $
CTFT of a shifted unit impulse $ \delta (t-t_0)\ $ $ e^{-i2\pi ft_0} $
CTFT of a complex exponential $ e^{iw_0t} $ $ \delta (f - \frac{\omega_0}{2\pi}) \ $
$ e^{-at}u(t), \ \text{ where } a\in {\mathbb R}, a>0 $ $ \frac{1}{a+i2\pi f} $
$ te^{-at}u(t), \ \text{ where } a\in {\mathbb R}, a>0 $ $ \left( \frac{1}{a+i2\pi f}\right)^2 $
CTFT of a cosine $ \cos(\omega_0 t) \ $ $ \frac{1}{2} \left[\delta (f - \frac{\omega_0}{2\pi}) + \delta (f + \frac{\omega_0}{2\pi})\right] \ $
CTFT of a sine $ sin(\omega_0 t) \ $ $ \frac{1}{2i} \left[\delta (f - \frac{\omega_0}{2\pi}) - \delta (f + \frac{\omega_0}{2\pi})\right] $
CTFT of a rect $ \left\{\begin{array}{ll}1, & \text{ if }|t|<T,\\ 0, & \text{else.}\end{array} \right. \ $ $ \frac{\sin \left(2\pi Tf \right)}{\pi f} \ $
CTFT of a sinc $ \frac{ \sin \left( W t \right)}{\pi t } \ $ $ \left\{\begin{array}{ll}1, & \text{ if }|f| <\frac{W}{2\pi},\\ 0, & \text{else.}\end{array} \right. \ $
CTFT of a periodic function $ \sum^{\infty}_{k=-\infty} a_{k}e^{ikw_{0}t} $ $ \sum^{\infty}_{k=-\infty}a_{k}\delta(f-\frac{kw_{0}}{2\pi}) \ $
CTFT of an impulse train $ \sum^{\infty}_{n=-\infty} \delta(t-nT) \ $ $ \frac{1}{T}\sum^{\infty}_{k=-\infty}\delta(f-\frac{k}{T}) \ $
CT Fourier Transform Properties
$ x(t) \ $ $ \longrightarrow $ $ X(f) \ $
multiplication property $ x(t)y(t) \ $ $ X(f)*Y(f) =\int_{-\infty}^{\infty} X(\theta)Y(f-\theta)d\theta $
time shifting property $ x(t-t_0) \ $ $ X(f)e^{-j 2 \pi f t_0} \ $
frequency shifting (also called "modulation") property $ x(t) e^{j 2 \pi f_0 t} \ $ $ X(f-f_0) \ $
scaling and shifting property $ x\left( \frac{ t- t_0}{a} \right) \ $ $ |a| X(af) e^{-j 2 \pi f t_0} \ $
convolution property $ x(t)*y(t) \ $ $ X(f)Y(f) \ $
time reversal $ \ x(-t) $ $ \ X(-f) $
Other CT Fourier Transform Properties
Parseval's relation $ \int_{-\infty}^{\infty} |x(t)|^2 dt = \int_{-\infty}^{\infty} |X(f)|^2 df $

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