• I suggest putting a link next to each formula explaining how to obtain it from the formula in terms of $\omega$. Also, I would not use the "mathcal" (curly) font for the transform variable, just a capital letter instead. --Mboutin 08:52, 3 September 2010 (UTC).
• The explanation for each formula still needs to be added! In particular, some students said it was not clear how to get the convolution property in terms of f. So this needs to be explained clearly. --Mboutin 09:04, 7 September 2010 (UTC)
• Provided explanation for each formula. -Zhao
• Modified explanation for each formula. -Zhao 17:20, 15 September 2010 (UTC).
CT Fourier Transform Pairs and Properties (frequency f in hertz per time unit) (info)
(Click title to see explanation on how to obtain the formula in terms of f in hertz)
Definition 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 CT Fourier Transform $\, x(t)=\mathcal{F}^{-1}(X(f))=\int_{-\infty}^{\infty}X(f)e^{i2\pi ft} df \,$
CT Fourier Transform Pairs
x(t) $\longrightarrow$ $X(f)$
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)\$, where $a\in {\mathbb R}, a>0$ $\frac{1}{a+i2\pi f}$
$te^{-at}u(t)\$, 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{2 \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) \$ $X(f)*Y(f) =\int_{-\infty}^{\infty} X(\theta)Y(f-\theta)d\theta$
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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