Revision as of 15:34, 14 November 2018

CTFT of periodic signals and some properties with proofs

Function	CTFT
$sin(\omega_0t)$	$\frac{\pi}{j}(\delta(\omega - \omega_0) - \delta(\omega+\omega_0))$
$cos(\omega_0t)$	$\pi(\delta(\omega - \omega_0) + \delta(\omega+\omega_0))$
$e^{j\omega_0t}$	$2\pi\delta(\omega - \omega_0)$
$\sum^{\infty}_{k=-\infty} a_{k}e^{ikw_{0}t}$	$2\pi\sum^{\infty}_{k=-\infty}a_{k}\delta(w-kw_{0}) \$

Name	$x(t) \longrightarrow \$	$\mathcal{X}(\omega)$
Linearity	$ax(t) + by(t) \$	$a \mathcal{X}(\omega) + b \mathcal{Y} (\omega)$
Time Shifting	$x(t-t_0) \$	$e^{-j\omega t_0}X(\omega)$
Frequency Shifting	$e^{j\omega_0 t}x(t)$	$\mathcal{X} (\omega - \omega_0)$
Conjugation	$x^{*}(t) \$	$\mathcal{X}^{*} (-\omega)$
Scaling	$x(at) \$	$\frac{1}{\|a\|} \mathcal{X} (\frac{\omega}{a})$
Multiplication	$x(t)y(t) \$	$\frac{1}{2\pi} \mathcal{X}(\omega)*\mathcal{Y}(\omega)$
Convolution	$x(t)*y(t) \$	$\mathcal{X}(\omega)\mathcal{Y}(\omega) \!$
Differentiation	$tx(t) \$	$j\frac{d}{d\omega} \mathcal{X} (\omega)$
Parseval's Relation	$\int_{-\infty}^{\infty} \|x(t)\|^2 dt =$	$\frac{1}{2\pi} \int_{-\infty}^{\infty} \|\mathcal{X}(w)\|^2 dw$

@@ Line 5: / Line 5: @@
 ! Function
 ! CTFT
-! Proof
 |-
 |<math>sin(\omega_0t) </math>
 |<math>\frac{\pi}{j}(\delta(\omega - \omega_0) - \delta(\omega+\omega_0))</math>
-|<math> </math>
 |-
 |<math>cos(\omega_0t) </math>
 |<math>\pi(\delta(\omega - \omega_0) + \delta(\omega+\omega_0))</math>
-|
 |-
 |<math>e^{j\omega_0t} </math>
 |<math>2\pi\delta(\omega - \omega_0) </math>
-|
 |-
-|<math> \sum_{k=-\infty}^{\infty}u(t+5k) - u(t-1+5k) </math>
+| <math>\sum^{\infty}_{k=-\infty} a_{k}e^{ikw_{0}t}</math>
-|
+| <math>2\pi\sum^{\infty}_{k=-\infty}a_{k}\delta(w-kw_{0}) \ </math>
-|
 |-}