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Let's break down and explain each of the given signals step by step. Each signal involves the unit step function \( u(t) \), which is defined as:

\[
u(t) = \begin{cases} 
0 & \text{if } t < 0, \\
1 & \text{if } t \geq 0.
\end{cases}
\]

### a. \( u(t - 5) + u(t - 7) \)

- \( u(t - 5) \) is a unit step function that shifts to the right by 5 units. It starts at \( t = 5 \) with a value of 1 for \( t \geq 5 \).
- \( u(t - 7) \) is another unit step function that shifts to the right by 7 units, starting at \( t = 7 \) with a value of 1 for \( t \geq 7 \).
- The signal will have a value of 1 from \( t = 5 \), and at \( t = 7 \), the sum becomes 2 for \( t \geq 7 \).

### b. \( t^2 [u(t - 1) - u(t - 2)] \)

- The term \( u(t - 1) - u(t - 2) \) creates a pulse (or window) from \( t = 1 \) to \( t = 2 \). 
  - It is 1 for \( 1 \leq t < 2 \) and 0 otherwise.
- \( t^2 \) is applied within this interval, so the signal is \( t^2 \) only in the interval \( 1 \leq t < 2 \), and 0 elsewhere.

### c. \( (t - 4)[u(t - 2) - u(t - 4)] \)

- The term \( u(t - 2) - u(t - 4) \) creates a pulse (or window) from \( t = 2 \) to \( t = 4 \). 
  - It is 1 for \( 2 \leq t < 4 \) and 0 otherwise.
- \( (t - 4) \) is applied within this window, so the signal is \( (t - 4) \) for \( 2 \leq t < 4 \). Since \( (t - 4) \) is negative in this interval, the signal will be negative within this range and 0 elsewhere.

#### Next Step:
I can create sketches of these signals to help visualize them. Would you like me to proceed with generating the sketches, or do you have any specific questions about the analysis?

### Additional Questions:
1. Can you explain how the unit step function \( u(t) \) behaves at different time shifts?
2. What is the effect of multiplying a function by a window like \( u(t - a) - u(t - b) \)?
3. How does the quadratic term \( t^2 \) affect the shape of the signal in part (b)?
4. What would happen if the range of the pulse in part (c) was extended beyond \( t = 4 \)?
5. Could you analyze how combining two step functions like in part (a) affects the output?

### Tip:
When dealing with the unit step function, remember that it acts like a switch that turns a signal on or off at specific time instances.

Sketch and label the following signals: 
a. u(t - 5) + u(t - 7) 
b. t^2 [u(t - 1) - u(t - 2)] 
c. (t - 4)[u(t - 2) - u(t - 4)]

Learn how to sketch signals involving the unit step function, including shifts, quadratic modifications, and pulse generation. This solution covers three distinct problems: u(t - 5) + u(t - 7), t^2 [u(t - 1) - u(t - 2)], and (t - 4)[u(t - 2) - u(t - 4)], with detailed explanations.

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