Introductory Circuit Analysis (13th Edition)
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ISBN: 9780133923605
Author: Robert L. Boylestad
Publisher: PEARSON
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![Consider a causal LTI system. When the input is \( x_0(t) = u(t-4) \), the Laplace transform of the output is
\[
Y_0(s) = \frac{e^{-5s} - e^{-8s}}{s^2}.
\]
**Determine and sketch** the output of the system \( y(t) \) when the input is changed to
\[
x(t) = u(t) - u(t-6).
\]
Try both of the following two methods:
- **Method 1:** Find the system transfer function \( H(s) = Y_0(s)/X_0(s) \) from the input-output pair \((x_0(t), y_0(t))\), and then determine the Laplace transform \( Y(s) = H(s)X(s) \) of the output \( y(t) \).
- **Method 2:** Express \( x(t) \) as a linear combination of some time-shifted copies of \( x_0(t) \), and then use the LTI properties to find \( Y(s) \) or \( y(t) \).
Which one of the above two methods do you prefer in solving this problem?](https://content.bartleby.com/qna-images/question/13afa382-1a86-4382-a05a-e40d6b5430cc/b1294fd0-09fe-46b1-9ede-c022b3972627/3cspfh2_processed.png)
Transcribed Image Text:Consider a causal LTI system. When the input is \( x_0(t) = u(t-4) \), the Laplace transform of the output is
\[
Y_0(s) = \frac{e^{-5s} - e^{-8s}}{s^2}.
\]
**Determine and sketch** the output of the system \( y(t) \) when the input is changed to
\[
x(t) = u(t) - u(t-6).
\]
Try both of the following two methods:
- **Method 1:** Find the system transfer function \( H(s) = Y_0(s)/X_0(s) \) from the input-output pair \((x_0(t), y_0(t))\), and then determine the Laplace transform \( Y(s) = H(s)X(s) \) of the output \( y(t) \).
- **Method 2:** Express \( x(t) \) as a linear combination of some time-shifted copies of \( x_0(t) \), and then use the LTI properties to find \( Y(s) \) or \( y(t) \).
Which one of the above two methods do you prefer in solving this problem?
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