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 y(t)=x(t)h(t)y(t) = x(t) * h(t), where both x(t)x(t) and h(t)h(t) are unit step functions, u(t)u(t). The convolution operation is defined as:

y(t)=x(τ)h(tτ)dτy(t) = \int_{-\infty}^\infty x(\tau) h(t - \tau) \, d\tau

Here, x(t)=u(t)x(t) = u(t) and h(t)=u(t)h(t) = u(t). Let us calculate the convolution step by step:

Step 1: Represent the unit step functions

The unit step function u(t)u(t) 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$$.

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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