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 $\int_{-\infty}^{\infty} |x(t)|^2 dt = \int_{-\infty}^{\infty} |X(f)|^2 df$

## Alumni Liaison

Correspondence Chess Grandmaster and Purdue Alumni

Prof. Dan Fleetwood