Contents
A CT-LTI System
$ y(t) = 10x(t)\, $
Unit Impulse Response
$ 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)\, $
