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| <math>e^{-j\omega t_0}X(\omega)</math>  
 
| <math>e^{-j\omega t_0}X(\omega)</math>  
 
| <math>\mathfrak{F}(x(t - t_{o})) = \int_{-\infty}^{\infty}x(t-t_{0})e^{-j\omega t} dt</math><br />
 
| <math>\mathfrak{F}(x(t - t_{o})) = \int_{-\infty}^{\infty}x(t-t_{0})e^{-j\omega t} dt</math><br />
let <math> /tau = t - t_{o} </math><br />
+
let <math> t' = t - t_{o} </math><br />
<math>\int_{-\infty}^{\infty}x(/tau)e^{-j\omega /tau - t_{o}} d/tau </math><br />
+
<math>\int_{-\infty}^{\infty}x(t')e^{-j\omega t'- t_{o}} dt' </math><br />
<math>= e^{-j\omega t_{o}}\int_{-\infty}^{\infty}x(/tau))e^{-j\omega /tau) d/tau </math><br />
+
<math>= e^{-j\omega t_{o}}\int_{-\infty}^{\infty}x(t')e^{-j\omega t') dt' </math><br />
 
<math>= e^{-j\omega t_{o}}\chi(\omega)</math> <br />
 
<math>= e^{-j\omega t_{o}}\chi(\omega)</math> <br />
 
|-
 
|-

Revision as of 23:46, 14 November 2018

CTFT of periodic signals with properties

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}) \ $
$ \sum^{\infty}_{n=-\infty} \delta(t-nT) \ $ $ \frac{2\pi}{T}\sum^{\infty}_{k=-\infty}\delta(w-\frac{2\pi k}{T}) $
Name $ x(t) \longrightarrow \ $ $ \mathcal{X}(\omega) $ Proof
Linearity $ ax(t) + by(t) \ $ $ a \mathcal{X}(\omega) + b \mathcal{Y} (\omega) $ $ \mathfrak{F}(ax(t) + by(t)) = \int_{-\infty}^{\infty}[ax(t) + by(t)]e^{-j\omega t} dt $

$ \int_{-\infty}^{\infty}ax(t)e^{-j\omega t} dt + \int_{-\infty}^{\infty}by(t)e^{-j\omega t} dt $
$ =a\mathcal{X}(\omega) + b\mathcal{Y}(\omega) $

Time Shifting $ x(t-t_0) \ $ $ e^{-j\omega t_0}X(\omega) $ $ \mathfrak{F}(x(t - t_{o})) = \int_{-\infty}^{\infty}x(t-t_{0})e^{-j\omega t} dt $

let $ t' = t - t_{o} $
$ \int_{-\infty}^{\infty}x(t')e^{-j\omega t'- t_{o}} dt' $
$ = e^{-j\omega t_{o}}\int_{-\infty}^{\infty}x(t')e^{-j\omega t') dt' $
$ = e^{-j\omega t_{o}}\chi(\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) $
Duality $ \mathcal{X} (-t) $ $ 2 \pi 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

Abstract algebra continues the conceptual developments of linear algebra, on an even grander scale.

Dr. Paul Garrett