Math Problem Statement
Given Function x(t) and Period T=4 , the given function is a periodic function with period x(t) = cases2 - t & for 0 t < 2 t - 4 & for 2 t < 4cases
Solution
Let's break down the given periodic function with a period . The function seems to be defined piecewise over the interval and repeats for every interval of length 4.
We have:
\begin{cases} 2 - t & \text{for } 0 \leq t < 2 \\ t - 4 & \text{for } 2 \leq t < 4 \end{cases}$$ Since this function is periodic with a period $$T = 4$$, it repeats for all intervals of length 4. This means that for any $$t$$, the function can be expressed as: $$x(t) = x(t + nT) \quad \text{for any integer } n.$$ ### Explanation of the Piecewise Function: 1. For $$0 \leq t < 2$$: The function is $$x(t) = 2 - t$$. This is a line with a slope of $$-1$$ and a y-intercept of 2, which decreases from $$x(0) = 2$$ to $$x(2) = 0$$. 2. For $$2 \leq t < 4$$: The function is $$x(t) = t - 4$$. This is a line with a slope of $$1$$ and a y-intercept of $$-4$$, which increases from $$x(2) = -2$$ to $$x(4) = 0$$ (but the interval ends just before 4). Since the function is periodic with $$T = 4$$, it will repeat this pattern every 4 units of time. Thus, for any value of $$t$$, we can reduce $$t$$ modulo 4 to fit it within the interval $$[0, 4)$$ and then apply the corresponding piecewise rule. ### Graphical Representation If you graph $$x(t)$$, it will look like a zig-zag pattern that repeats every 4 units. Specifically: - From $$t = 0$$ to $$t = 2$$, the function decreases linearly. - From $$t = 2$$ to $$t = 4$$, the function increases linearly. Would you like me to go deeper into the function's properties, like Fourier series, or any other aspect of the periodic function? --- Here are five questions to deepen understanding: 1. How do you find the Fourier series representation of this periodic function? 2. What is the behavior of the function outside the interval $$[0, 4)$$? 3. How would you sketch the function for multiple periods? 4. Can you express the function in terms of a shifted Heaviside function? 5. What is the average (DC component) of this periodic function? **Tip:** Always analyze periodic functions in terms of their fundamental period, as this simplifies calculations and visualizations across the entire time domain.Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Piecewise Functions
Periodic Functions
Linear Functions
Modular Arithmetic
Formulas
x(t) = { 2 - t for 0 ≤ t < 2; t - 4 for 2 ≤ t < 4 }
x(t) = x(t + nT) for any integer n
Theorems
Properties of Periodic Functions
Suitable Grade Level
Undergraduate Mathematics or Advanced High School (Grades 11-12)