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==System Function== | ==System Function== | ||
| + | As discussed in class, the system function is | ||
| + | |||
| + | <math>H(s)=\int_{-\infty}^\infty h(\tau)e^{-s\tau}d\tau</math> | ||
| + | |||
| + | In this case, we can apply the sifting property to arrive at the system function quite easily. After applying the property, we arrive at <math>H(s)=2-e^{-2s}</math>. One might recognize this is the Laplace transform of the impulse response. | ||
==Response to a Signal from Question 1== | ==Response to a 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^{-3j}-\frac{7}{2j}e^{-2j}+\frac{7}{2j}e^{2j}+\frac{1+j}{2}e^{3j}</math> | ||
Revision as of 05:09, 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(s)=\int_{-\infty}^\infty h(\tau)e^{-s\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(s)=2-e^{-2s} $. 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^{-3j}-\frac{7}{2j}e^{-2j}+\frac{7}{2j}e^{2j}+\frac{1+j}{2}e^{3j} $
