Difference between revisions of "CT Fourier Transform (frequency in radians per time unit)" - Rhea

Revision as of 12:16, 10 April 2011

CT Fourier Transform Pairs and Properties (frequency $ω$ in radians per time unit) (info)
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 DT 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
		$x (t)$	$\longrightarrow$	$\mathcal{X}(\omega)$
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{2 \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		$2πδ(ω)$
13	CTFT of a Periodic Square Wave	$\left\{\begin{array}{ll}1, & \|t\|<T_1,\\ 0, & T_1<\|t\|<=\frac{T}{2}\end{array} \right.$ and $x (t + T) = x (t)$		$\sum^{\infty}_{k=-\infty}\frac{2 \sin(k\omega_0T_1}{k}\delta(\omega-k\omega_0)$
14	CTFT of a Step Function	$u (t)$		$\frac{1}{j\omega}+\pi\delta(\omega)$

			CT Fourier Transform Properties
	$x (t)$	$\longrightarrow$	$\mathcal{X}(\omega)$
(info) multiplication property	$x(t)y(t) \$		$\frac{1}{2\pi} \mathcal{X}(\omega)*\mathcal{Y}(\omega) =\frac{1}{2\pi} \int_{-\infty}^{\infty} \mathcal{X}(\theta)\mathcal{Y}(\omega-\theta)d\theta$
convolution property	$x(t)*y(t) \!$		$\mathcal{X}(\omega)\mathcal{Y}(\omega) \!$
time reversal	$\ x(-t)$		$\ \mathcal{X}(-\omega)$
Linearity	$$ ax(t)+by(t) $$		$aX(\omega)+bY(\omega)$
Time Shifting	$$ x(t-t_0) $$		$e^{-j\omega t_0}X(\omega)$

Other CT Fourier Transform Properties
Parseval's relation	$\int_{-\infty}^{\infty} \|x(t)\|^2 dt = \frac{1}{2\pi} \int_{-\infty}^{\infty} \|\mathcal{X}(w)\|^2 dw$

Sources:

@@ Line 19: / Line 19: @@
 | align="right" style="padding-right: 1em;" | <br>
 | align="right" style="padding-right: 1em;" |
-| <span class="texhtml">''1x''(''t'')</span>
+| <span class="texhtml">''x''(''t'')</span>
 | <math>\longrightarrow</math>
 | <math> \mathcal{X}(\omega) </math>
@@ Line 115: / Line 115: @@
 |
 | <span class="texhtml">2πδ(ω)
-</span>
+</span>
 |
 |
@@ Line 134: / Line 137: @@
 |
 |-
-| align="right" style="padding-right: 1em;" | 14
+| align="right" style="padding-right: 1em;" |
-| align="right" style="padding-right: 1em;" | CTFT of a Step Function
-| <math>u(t)</math>
+| align="right" style="padding-right: 1em;" | CTFT of a Step Function
+| <span class="texhtml">''u''(''t'')</span>
 |
 | <math>\frac{1}{j\omega}+\pi\delta(\omega)</math>
 |
 |
@@ Line 173: / Line 178: @@
 |-
 | align="right" style="padding-right: 1em;" |
-| align="right" style="padding-right: 1em;" | Lin
+| align="right" style="padding-right: 1em;" | Linearity
-|
+| <math>ax(t)+by(t)</math>
 |
+| <math>aX(\omega)+bY(\omega)</math>
+|-
+| align="right" style="padding-right: 1em;" |
+| align="right" style="padding-right: 1em;" | Time Shifting
+| <math>x(t-t_0)</math>
 |
+| <math>e^{-j\omega t_0}X(\omega)</math>
 |}