Math Problem Statement

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)]

Solution

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

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.

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Math Problem Analysis

Mathematical Concepts

Unit Step Function
Signal Shifting
Pulse Function
Piecewise Function

Formulas

u(t - a): Shifted Unit Step Function
Pulse Window: u(t - a) - u(t - b)

Theorems

Unit Step Function Definition

Suitable Grade Level

University Level (Signal Processing, Electrical Engineering)