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! Function  
 
! Function  
 
! CTFT  
 
! CTFT  
! Proof
 
 
|-
 
|-
  
 
|<math>sin(\omega_0t) </math>
 
|<math>sin(\omega_0t) </math>
 
|<math>\frac{\pi}{j}(\delta(\omega - \omega_0) - \delta(\omega+\omega_0))</math>
 
|<math>\frac{\pi}{j}(\delta(\omega - \omega_0) - \delta(\omega+\omega_0))</math>
|<math> </math>
 
 
|-
 
|-
  
 
|<math>cos(\omega_0t) </math>
 
|<math>cos(\omega_0t) </math>
 
|<math>\pi(\delta(\omega - \omega_0) + \delta(\omega+\omega_0))</math>
 
|<math>\pi(\delta(\omega - \omega_0) + \delta(\omega+\omega_0))</math>
|
 
 
|-
 
|-
  
 
|<math>e^{j\omega_0t} </math>
 
|<math>e^{j\omega_0t} </math>
 
|<math>2\pi\delta(\omega - \omega_0) </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>  
|
+
 
|-}
 
|-}
  

Revision as of 16:34, 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 $

Alumni Liaison

Ph.D. on Applied Mathematics in Aug 2007. Involved on applications of image super-resolution to electron microscopy

Francisco Blanco-Silva