To find the output $y(t)$ of a Linear Time-Invariant (LTI) system given an input signal $x(t)$ and an impulse response $h(t)$, we can use the convolution integral:

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

From the given figure:

1. $x(t)$ is a rectangular pulse that exists from $t = 0$ to $t = 2$ with an amplitude of $1$. 1 & 0 \leq t \leq 2 \\ 0 & \text{otherwise} \end{cases}$$
2. $h(t)$ is another rectangular pulse that exists from $t = -2$ to $t = 2$ with an amplitude of $2$. 2 & -2 \leq t \leq 2 \\ 0 & \text{otherwise} \end{cases}$$

Steps for Convolution

To compute $y(t) = x(t) * h(t)$, follow these steps:

1. Define the limits: Convolution will only have a non-zero result when the two signals overlap. Determine the range of $t$ values for which $x(\tau)$ and $h(t - \tau)$ overlap.

2. Set up the integral: For each interval where $x(\tau)$ and $h(t - \tau)$ overlap, set up and evaluate the integral.

3. Combine results for all intervals: After calculating the integral for each interval, combine them to obtain $y(t)$.

Would you like me to go through each of these steps in detail and solve for $y(t)$ explicitly?

Relative Questions

1. How is the convolution process influenced by the duration of the signals $x(t)$ and $h(t)$?
2. What would happen to the convolution result if the impulse response $h(t)$ had a different amplitude?
3. How does shifting the position of $h(t)$ affect the convolution outcome?
4. Why is the convolution integral important in determining the system's response?
5. Can the convolution integral be solved using Laplace or Fourier transforms for simplicity?

Tip

When performing convolution, visualize the overlap between $x(t)$ and $h(t - \tau)$ as you shift one function across the other to understand the regions where they contribute to the output $y(t)$.