To solve this problem, we need to compute the output $y(t)$, which is the convolution of the input $x(t)$ with the impulse response $h(t)$. The output can be expressed as:

$y(t) = x(t) \ast h(t) = \int_{-\infty}^{\infty} x(\tau) h(t - \tau) d\tau$

Step 1: Analyze the input and impulse response functions

We start by analyzing both the input $x(t)$ and the impulse response $h(t)$:

• The impulse response is given as: $h(t) = u(2 - t) + u(t - 2)$ This is a combination of step functions $u(t)$. Specifically:

  • $u(2 - t)$ is a step function that equals 1 for $t \leq 2$ and 0 for $t > 2$.
  • $u(t - 2)$ is a step function that equals 0 for $t < 2$ and 1 for $t \geq 2$.

  So, $h(t)$ is 1 everywhere, as the sum of these two step functions results in: $h(t) = 1 \quad \text{for all} \, t.$

• The input $x(t)$ is given as: $x(t) = t[u(t - 2) - u(t - 4)]$ This is a piecewise function, nonzero only between $t = 2$ and $t = 4$. It can be expressed as:

  \begin{cases} t, & 2 \leq t \leq 4, \\ 0, & \text{otherwise}. \end{cases}$$

Step 2: Set up the convolution integral

We now compute the convolution $y(t) = x(t) \ast h(t)$, which simplifies due to $h(t) = 1$ for all $t$. The convolution becomes: $y(t) = \int_{-\infty}^{\infty} x(\tau) d\tau$

Since $x(\tau)$ is nonzero only in the interval $[2, 4]$, the integral reduces to: $y(t) = \int_2^4 \tau \, d\tau$

Step 3: Perform the integration

Let's now compute this integral: $y(t) = \int_2^4 \tau \, d\tau = \left[\frac{\tau^2}{2}\right]_2^4$ Substituting the limits: $y(t) = \frac{4^2}{2} - \frac{2^2}{2} = \frac{16}{2} - \frac{4}{2} = 8 - 2 = 6$

Step 4: Final Output

The output $y(t)$ will be a constant value $y(t) = 6$ for the time interval where $x(t)$ and $h(t)$ overlap. Since $x(t)$ is nonzero only between $t = 2$ and $t = 4$, the convolution will be nonzero in this region.

Plotting $y(t)$

The output $y(t)$ is a piecewise function:

\begin{cases} 6, & 4 \leq t \leq 6, \\ 0, & \text{otherwise}. \end{cases}$$ I can generate a plot for you as well if you'd like! Would you like me to provide more details or help you with a plot? --- Here are some related questions you might explore: 1. How does the convolution change if the impulse response is not constant? 2. What would happen if the input function $$x(t)$$ was shifted by a different amount? 3. How does the unit step function impact the behavior of linear time-invariant (LTI) systems? 4. What are the properties of the convolution integral? 5. How would the output change if the limits of the input $$x(t)$$ were altered? **Tip:** In convolution problems, identifying where the functions are nonzero simplifies the integration significantly.