(New page: == A CT-LTI System == <math>y(t) = 10x(t)\,</math> == Unit Impulse == <math>x(t) = \delta(t)\,</math> <math>h(t) = 10 \delta(t)\,</math> == Frequency Response == <math>y(t) = \int^{\in...) |
(→Response of the CT system defiend in Q1) |
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CT Periodic Signal : <math>x(t) = \cos(3\pi t) + \sin(4\pi t)\,</math> | CT Periodic Signal : <math>x(t) = \cos(3\pi t) + \sin(4\pi t)\,</math> | ||
| − | <math>x(t) = \ | + | <math>x(t) = \sum^{\infty}_{k = -\infty} a_k e^{jk\pi t}\,</math> |
| − | <math> | + | <math>y(t) = \sum^{\infty}_{k = -\infty} a_k H(s) e^{jk\pi t}\,</math> |
| − | <math> | + | <math>y(t) = \sum^{\infty}_{k = -\infty} a_k (10) e^{jk\pi t}\,</math> |
| − | <math> | + | <math>y(t) = 10\sum^{\infty}_{k = -\infty} a_k e^{jk\pi t}\,</math> |
| − | <math> | + | <math>y(t) = 10(\cos(3\pi t) + \sin(4\pi t))\,</math> |
| + | |||
| + | <math>y(t) = 10\cos(3\pi t) + 10\sin(4\pi t)\,</math> | ||
Revision as of 17:54, 24 September 2008
Contents
A CT-LTI System
$ y(t) = 10x(t)\, $
Unit Impulse
$ x(t) = \delta(t)\, $
$ h(t) = 10 \delta(t)\, $
Frequency Response
$ y(t) = \int^{\infty}_{-\infty} h(t) * x(t) dt\, $ where $ x(t) = e^{jwt} \, $
$ y(t) = \int^{\infty}_{-\infty} 10 \delta(t) * e^{jwt} dt\, $
$ y(t) = \int^{\infty}_{-\infty} 10 \delta(r) e^{jw(t-r} dr\, $
$ y(t) = e^{jwt} \int^{\infty}_{-\infty} 10 \delta(r) e^{-jwr} dr\, $
$ H(s) = \int^{\infty}_{-\infty} 10 \delta(r) e^{-jwr} dr\, $
$ H(s) = 10 e^{-jw0}\, $
$ H(s) = 10\, $
Response of the CT system defiend in Q1
CT Periodic Signal : $ x(t) = \cos(3\pi t) + \sin(4\pi t)\, $
$ x(t) = \sum^{\infty}_{k = -\infty} a_k e^{jk\pi t}\, $
$ y(t) = \sum^{\infty}_{k = -\infty} a_k H(s) e^{jk\pi t}\, $
$ y(t) = \sum^{\infty}_{k = -\infty} a_k (10) e^{jk\pi t}\, $
$ y(t) = 10\sum^{\infty}_{k = -\infty} a_k e^{jk\pi t}\, $
$ y(t) = 10(\cos(3\pi t) + \sin(4\pi t))\, $
$ y(t) = 10\cos(3\pi t) + 10\sin(4\pi t)\, $
