Math Problem Statement
A unit step signal, u(t) is defined as: u(t < 0) = 0 and u(t > 0)=1.
If x(t) = u(10-2t), which one of the following is true?
A. x(t>5)=1,x(t<5)=0 B. x(t<-5)=1,x(t>-5)=0 C. x(t<5)=1,x(t>5)=0 D. x(t<-5)=1,x(t>-5)=0
Solution
The given signal is , where is the unit step function. The unit step function is defined as:
\begin{cases} 0 & \text{if } t < 0 \\ 1 & \text{if } t \geq 0 \end{cases}$$ We need to determine the values of $$t$$ where $$x(t)$$ transitions from 0 to 1 by solving the inequality $$10 - 2t = 0$$. ### Step 1: Find the critical point Set the argument of the step function to zero to find the critical time where the step occurs: $$10 - 2t = 0$$ $$2t = 10$$ $$t = 5$$ ### Step 2: Analyze behavior of $$x(t)$$ - For $$t < 5$$: $$10 - 2t > 0 \quad \Rightarrow \quad u(10 - 2t) = 1$$ So, $$x(t) = 1$$ when $$t < 5$$. - For $$t > 5$$: $$10 - 2t < 0 \quad \Rightarrow \quad u(10 - 2t) = 0$$ So, $$x(t) = 0$$ when $$t > 5$$. ### Step 3: Conclusion The correct description is: $$x(t) = \begin{cases} 1 & \text{if } t < 5 \\ 0 & \text{if } t > 5 \end{cases}$$ This matches option **C**. ### Final Answer: **C** $$x(t<5)=1, x(t>5)=0$$ --- Would you like more details or have any questions? Here are 5 related questions to expand your understanding: 1. How is the unit step function $$u(t)$$ used in signal processing? 2. How would the function $$x(t)$$ change if the coefficient of $$t$$ were different in $$u(10 - 2t)$$? 3. What is the graphical representation of a unit step function? 4. How do unit step functions combine in Laplace transforms? 5. Can you describe how $$u(t)$$ shifts and scales based on the argument inside the function? **Tip**: In signal processing, the unit step function is often used to model signals that turn on or off at a specific point in time.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Signal Processing
Unit Step Function
Inequalities
Formulas
u(t) = {0 if t < 0, 1 if t >= 0}
Theorems
Step function transformation
Suitable Grade Level
Undergraduate Level (Signal Processing or Electrical Engineering)