An analog LTI system with impulse response function h(t) = u(3 − t) + u(t − 3) is

presented with an input x(t) = (t-1)[u(t − 1) − u(t − 5)].

a) Determine and plot the system output y(t) = x(t) ∗ h(t)

The plots show the input signal \( x(t) \), the impulse response \( h(t) \), and the output signal \( y(t) = x(t) \ast h(t) \).

1. **Input Signal \( x(t) \)**: A ramp that starts at \( t = 1 \) and stops at \( t = 5 \).
2. **Impulse Response \( h(t) \)**: Defined by two shifted step functions centered at \( t = 3 \), creating a window between \( t = 0 \) and \( t = 3 \).
3. **Output Signal \( y(t) \)**: This is the result of convolving \( x(t) \) and \( h(t) \). The output represents the combined effect of the input signal passing through the system characterized by \( h(t) \).

Would you like more details or have any questions?

Here are some related questions for deeper exploration:
1. How would the output change if \( h(t) \) were a different step function?
2. What happens if the input signal is extended beyond \( t = 5 \)?
3. Can we derive an analytical expression for \( y(t) \)?
4. How does the duration of the impulse response affect the output?
5. What is the physical interpretation of the convolution result in this system?

**Tip**: When convolving signals, always consider the duration and overlap of the signals to predict the result's shape.

An analog LTI system with impulse response function h(t) = u(3 − t) + u(t − 3) is presented with an input x(t) = (t-1)[u(t − 1) − u(t − 5)].

a) Determine and plot the system output y(t) = x(t) ∗ h(t).

Explore how to determine and plot the output y(t) of an analog LTI system with the given impulse response function h(t) and input signal x(t). The problem demonstrates the convolution process, crucial in signal processing, with step functions and system theory concepts.

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