Math Problem Statement
Find y(t) = x(t) * h(t) where x(t) = u(t) and h(t) = u(t).
Solution
The problem involves determining the convolution , where both and are unit step functions, . The convolution operation is defined as:
Here, and . Let us calculate the convolution step by step:
Step 1: Represent the unit step functions
The unit step function is defined as:
\begin{cases} 1, & t \geq 0 \\ 0, & t < 0 \end{cases}$$ ### Step 2: Substitute into the convolution integral The convolution becomes: $$y(t) = \int_{-\infty}^\infty u(\tau) u(t - \tau) \, d\tau$$ Since $$u(\tau)$$ and $$u(t - \tau)$$ are zero for negative arguments, the limits of integration can be modified: $$y(t) = \int_0^t 1 \cdot 1 \, d\tau$$ ### Step 3: Evaluate the integral $$y(t) = \int_0^t 1 \, d\tau = [\tau]_0^t = t$$ ### Step 4: Define the result Since the convolution only occurs for $$t \geq 0$$, we write: $$y(t) = \begin{cases} t, & t \geq 0 \\ 0, & t < 0 \end{cases}$$ Or simply: $$y(t) = t \cdot u(t)$$ This is the result of the convolution. Would you like further clarifications or an expansion on this explanation? --- Here are 5 related questions for practice: 1. What is the convolution of $$x(t) = u(t)$$ and $$h(t) = e^{-t}u(t)$$? 2. How does the convolution integral change for functions defined over a finite interval? 3. What are the properties of the unit step function $$u(t)$$? 4. How does the convolution operation relate to systems in the time domain? 5. What happens if $$x(t)$$ and $$h(t)$$ are both shifted unit step functions? **Tip:** The convolution of two unit step functions results in a ramp function, $$t \cdot u(t)$$, which grows linearly with $$t$$.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Convolution
Unit Step Function
Formulas
y(t) = ∫[x(τ) * h(t - τ)] dτ
Theorems
Properties of Convolution
Suitable Grade Level
Grades 11-12 or Undergraduate
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