(New page: Proving the property of Linearity in Fourier Transform: <math>ax(t)+by(t)</math> is linearly related to <math>aX(j\omega)+bY(j\omega)</math>) |
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<math>aX(j\omega)+bY(j\omega)</math> | <math>aX(j\omega)+bY(j\omega)</math> | ||
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| + | Proof: | ||
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| + | <math>\mathcal{F}(ax(t))=aX(j\omega) </math> and <math> \mathcal{F}(by(t))=bY(j\omega)</math> | ||
| + | for <math>ax(t)=2\delta(t)</math> and <math>bx(t)=3\delta(t-1)</math> we find that | ||
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| + | <math>\mathcal{F}(2\delta(t)=2 </math> and <math>\mathcal{F}(3\delta(t-1)= 3e^{-j\omega}</math> | ||
| + | and that <math>aX+bY = 2+3e^{-j \omega}</math> | ||
| + | Summing the two prior to transform we derive | ||
| + | <math>\mathcal{F}(2\delta(t) + 3\delta(t-1)) = \int_{-\infty}^{+\infty}2\delta(t)+3\delta(t-1)e^{j\omega t}d\omega = 2+3e^{-j \omega} </math> | ||
Latest revision as of 15:17, 8 July 2009
Proving the property of Linearity in Fourier Transform:
$ ax(t)+by(t) $
is linearly related to
$ aX(j\omega)+bY(j\omega) $
Proof:
$ \mathcal{F}(ax(t))=aX(j\omega) $ and $ \mathcal{F}(by(t))=bY(j\omega) $ for $ ax(t)=2\delta(t) $ and $ bx(t)=3\delta(t-1) $ we find that
$ \mathcal{F}(2\delta(t)=2 $ and $ \mathcal{F}(3\delta(t-1)= 3e^{-j\omega} $ and that $ aX+bY = 2+3e^{-j \omega} $ Summing the two prior to transform we derive $ \mathcal{F}(2\delta(t) + 3\delta(t-1)) = \int_{-\infty}^{+\infty}2\delta(t)+3\delta(t-1)e^{j\omega t}d\omega = 2+3e^{-j \omega} $
