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b) Compute the response of your system to the signal you defined in Question 1 using H(s) and the Fourier series coefficients of your signal. | b) Compute the response of your system to the signal you defined in Question 1 using H(s) and the Fourier series coefficients of your signal. | ||
| − | CT signal : <math> sin ( | + | CT signal : |
| + | <math>x(t) = 6\cos(2\pi t) + 8\sin(4\pi t)\,</math> | ||
| − | <math> | + | <math>x(t) = \sum^{\infty}_{k = -\infty} a_k e^{jk\pi t}\,</math> |
| + | |||
| + | <math>y(t) = \sum^{\infty}_{k = -\infty} a_k H(s) e^{jk\pi t}\,</math> | ||
| + | |||
| + | From Question 1: | ||
| + | |||
| + | <math>x(t) = 3e^{j2\pi t}+3e^{-j2\pi t} + 4e^{j4\pi t}-4e^{-j4\pi t}\,</math> | ||
| + | |||
| + | <math>a_1 = 3\,</math> <math>a_{-1} = 3\,</math> | ||
| + | |||
| + | <math>a_2 = 4\,</math> <math>a_{-2} = -4\,</math> | ||
| + | |||
| + | <math>x(t) = 3H(s)e^{j2\pi t}+3H(s)e^{-j2\pi t} + 4H(s)e^{j4\pi t}-4H(s)e^{-j4\pi t}\,</math> | ||
| + | |||
| + | <math>x(t) = 3j\omega_0e^{j2\pi t}+3j\omega_0e^{-j2\pi t} + 4j\omega_0e^{j4\pi t}-4j\omega_0e^{-j4\pi t}\,</math> | ||
| + | |||
| + | <math>\omega_0\,</math> f=2 | ||
| + | |||
| + | <math>x(t) = 6je^{j2\pi t}+6je^{-j2\pi t} + 8je^{j4\pi t}-8je^{-j4\pi t}\,</math> | ||
Latest revision as of 18:42, 26 September 2008
3. Define a CT LTI system.
System:
$ y(t)=x(t-1) $
a) Obtain the unit impulse response h(t) and the system function H(s) of your system.
$ d(t) --> System --> d(t-1)\ $
$ h(t)= d(t-1) $
$ H(s)= e^{- s} $
b) Compute the response of your system to the signal you defined in Question 1 using H(s) and the Fourier series coefficients of your signal.
CT signal : $ x(t) = 6\cos(2\pi t) + 8\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}\, $
From Question 1:
$ x(t) = 3e^{j2\pi t}+3e^{-j2\pi t} + 4e^{j4\pi t}-4e^{-j4\pi t}\, $
$ a_1 = 3\, $ $ a_{-1} = 3\, $
$ a_2 = 4\, $ $ a_{-2} = -4\, $
$ x(t) = 3H(s)e^{j2\pi t}+3H(s)e^{-j2\pi t} + 4H(s)e^{j4\pi t}-4H(s)e^{-j4\pi t}\, $
$ x(t) = 3j\omega_0e^{j2\pi t}+3j\omega_0e^{-j2\pi t} + 4j\omega_0e^{j4\pi t}-4j\omega_0e^{-j4\pi t}\, $
$ \omega_0\, $ f=2
$ x(t) = 6je^{j2\pi t}+6je^{-j2\pi t} + 8je^{j4\pi t}-8je^{-j4\pi t}\, $
