Chapter 12, Problem 1P

(a)

To determine

Plot the sketch of given function $f\left(t\right)$ over the interval $0\text{s}\le t\le \infty$.

Explanation of Solution

Given data:

$f\left(t\right)=\{\begin{array}{l}0,t\le 0\\ 5t,0\text{s}\le t\le 10\text{s}\\ -5t+100,10\text{s}\le t\le 30\text{s}\\ -50,30\text{s}\le t\le 40\text{s}\\ 2.5t-150,40\text{s}\le t\le 60\text{s}\\ 0,60\text{s}\le t<\infty \end{array}$

Calculation:

Construct a single continuous definition for the function using step function as follows:

$\begin{array}{l}0,\text{at}t\le 0;\\ 5t,\text{on}\text{at}t=0,\text{off}\text{at}t=10;\\ -5t+100,\text{on}\text{at}t=10,\text{off}\text{at}t=30;\\ -50,\text{on}\text{at}t=30,\text{off}\text{at}t=40;\\ 2.5t-150,\text{on}\text{at}t=40,\text{off}\text{at}t=60;\\ 0,\text{at}60\le t<\infty \end{array}$

Use the values and draw the sketch of expression $f\left(t\right)$ as shown in Figure 1.

Conclusion:

Thus, the sketch of given expression $f\left(t\right)$ is shown in Figure 1.

(b)

To determine

Write the expression for the given function $f\left(t\right)$ using the concept of window function.

Answer to Problem 1P

The expression for the given function $f\left(t\right)$ is $\left\{\begin{array}{l}5t\left[u\left(t\right)-u\left(t-10\right)\right]+\left(100-5t\right)\left[u\left(t-10\right)-u\left(t-30\right)\right]\\ -50\left[u\left(t-30\right)-u\left(t-40\right)\right]+\left(2.5t-150\right)\left[u\left(t-40\right)-u\left(t-60\right)\right]\end{array}\right\}$.

Explanation of Solution

Calculation:

Construct a single continuous definition for the function using step function as follows:

$\begin{array}{l}0,\text{at}t\le 0;\\ 5t,\text{on}\text{at}t=0,\text{off}\text{at}t=10;\\ -5t+100,\text{on}\text{at}t=10,\text{off}\text{at}t=30;\\ -50,\text{on}\text{at}t=30,\text{off}\text{at}t=40;\\ 2.5t-150,\text{on}\text{at}t=40,\text{off}\text{at}t=60;\\ 0,\text{at}60\le t<\infty \end{array}$

Use the values and write the expression for $f\left(t\right)$ as follows:

$f\left(t\right)=\left\{\begin{array}{l}5t\left[u\left(t\right)-u\left(t-10\right)\right]+\left(100-5t\right)\left[u\left(t-10\right)-u\left(t-30\right)\right]\\ -50\left[u\left(t-30\right)-u\left(t-40\right)\right]+\left(2.5t-150\right)\left[u\left(t-40\right)-u\left(t-60\right)\right]\end{array}\right\}$

Conclusion:

Thus, the expression for the given function $f\left(t\right)$ is $\left\{\begin{array}{l}5t\left[u\left(t\right)-u\left(t-10\right)\right]+\left(100-5t\right)\left[u\left(t-10\right)-u\left(t-30\right)\right]\\ -50\left[u\left(t-30\right)-u\left(t-40\right)\right]+\left(2.5t-150\right)\left[u\left(t-40\right)-u\left(t-60\right)\right]\end{array}\right\}$.