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I will use my signal from Question 1. | I will use my signal from Question 1. | ||
| − | <math>x(t)=7\sin(2t)+(1+j)\cos(3t)=\frac{1+j}{2}e^{- | + | <math>x(t)=7\sin(2t)+(1+j)\cos(3t)=\frac{1+j}{2}e^{-3jt}-\frac{7}{2j}e^{-2jt}+\frac{7}{2j}e^{2jt}+\frac{1+j}{2}e^{3jt}</math> |
| + | |||
| + | Since we have represented the original signal as a sum of complex exponentials, we simply have to multiply each term of the original input signal by <math>H(j\omega)</math> to obtain the corresponding term in the output. | ||
Revision as of 05:25, 25 September 2008
Contents
A Continuous Time, Linear, Time-Invariant System
Consider the system $ y(t)=2x(t)-x(t-2) $.
Unit Impulse Response
Let $ x(t)=\delta(t) $. Then $ h(t)=2\delta(t)-\delta(t-2) $.
System Function
As discussed in class, the system function is
$ H(j\omega)=\int_{-\infty}^\infty h(\tau)e^{-j\omega\tau}d\tau $
In this case, we can apply the sifting property to arrive at the system function quite easily. After applying the property, we arrive at $ H(j\omega)=2-e^{-2j\omega} $. One might recognize this is the Laplace transform of the impulse response.
Response to a Signal from Question 1
I will use my signal from Question 1.
$ x(t)=7\sin(2t)+(1+j)\cos(3t)=\frac{1+j}{2}e^{-3jt}-\frac{7}{2j}e^{-2jt}+\frac{7}{2j}e^{2jt}+\frac{1+j}{2}e^{3jt} $
Since we have represented the original signal as a sum of complex exponentials, we simply have to multiply each term of the original input signal by $ H(j\omega) $ to obtain the corresponding term in the output.
