Exercise6.3.1: Let𝑓(𝑡)=𝑡2for𝑡≥0,and𝑔(𝑡)=𝑢(𝑡−1).Compute𝑓∗𝑔.

To compute the convolution \( f * g \) of two functions \( f(t) = t^2 \) for \( t \geq 0 \) and \( g(t) = u(t - 1) \), we proceed as follows:

1. **Identify the convolution formula**: The convolution of two functions \( f(t) \) and \( g(t) \) is defined as:
   \[
   (f * g)(t) = \int_{0}^{t} f(\tau) g(t - \tau) \, d\tau
   \]

2. **Substitute \( f(t) \) and \( g(t) \)**: Given \( f(t) = t^2 \) and \( g(t) = u(t - 1) \), where \( u(t - 1) \) is the unit step function that "turns on" at \( t = 1 \). This means that \( g(t - \tau) = u(t - \tau - 1) \), which is 1 when \( t - \tau \geq 1 \) or \( \tau \leq t - 1 \), and 0 otherwise.

3. **Set up the integral**: The step function implies that the convolution integral will only have a nonzero contribution when \( \tau \leq t - 1 \). Thus,
   \[
   (f * g)(t) = \int_{0}^{t-1} \tau^2 \cdot 1 \, d\tau = \int_{0}^{t-1} \tau^2 \, d\tau
   \]

4. **Evaluate the integral**:
   \[
   (f * g)(t) = \int_{0}^{t-1} \tau^2 \, d\tau
   \]
   Using the power rule for integration, \(\int \tau^2 \, d\tau = \frac{\tau^3}{3}\), we get:
   \[
   (f * g)(t) = \left[ \frac{\tau^3}{3} \right]_{0}^{t-1} = \frac{(t-1)^3}{3} - 0 = \frac{(t-1)^3}{3}
   \]

Therefore, the convolution \( f * g \) is:
\[
(f * g)(t) = \frac{(t - 1)^3}{3} \quad \text{for } t \geq 1
\]

For \( t < 1 \), \( (f * g)(t) = 0 \) because the unit step function \( g(t) = u(t - 1) \) does not contribute.

### Final Answer
\[
(f * g)(t) = 
\begin{cases} 
0, & t < 1 \\ 
\frac{(t - 1)^3}{3}, & t \geq 1 
\end{cases}
\]

Would you like further clarification on the steps, or have any additional questions?

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### Related Questions
1. How would the convolution change if \( g(t) = u(t - a) \) for a different constant \( a \)?
2. What would be the result of convolving \( f(t) = t^2 \) with a delta function \( \delta(t - 1) \)?
3. How does the unit step function impact the limits of integration in convolution problems?
4. Can we compute \( f * g \) when \( f \) and \( g \) are defined on \( t < 0 \)?
5. What is the physical interpretation of convolution in signal processing?

### Tip
When working with convolutions involving step functions, always check where each function is nonzero to simplify the integration limits.

Exercise 6.3.1: Let f(t) = t^2 for t ≥ 0, and g(t) = u(t - 1). Compute f ∗ g.

This page provides a detailed solution to the convolution of the functions f(t) = t^2 for t ≥ 0 and g(t) = u(t - 1). It covers the convolution formula, properties of the unit step function, and integration techniques. Suitable for undergraduate students.

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