Find LTI system output Impulse response of impulse step signal

For a Linear Time-Invariant (LTI) system, let's consider the system output in response to an input step signal.

### Impulse Response from Step Response:

Given an LTI system with an impulse response \( h(t) \), the output of the system due to a unit impulse input \( \delta(t) \) is defined as \( h(t) \). If the input to the LTI system is a **unit step signal** \( u(t) \), the system output \( y(t) \) can be expressed as a convolution of the impulse response \( h(t) \) with the step signal \( u(t) \):

\[
y(t) = h(t) * u(t)
\]

### Relation Between Step and Impulse Responses:

1. **Step Response**: The output due to the step input \( u(t) \) is often denoted as \( s(t) \). Therefore:

   \[
   s(t) = h(t) * u(t)
   \]

2. The relationship between the impulse response \( h(t) \) and the step response \( s(t) \) is given by the derivative:

   \[
   h(t) = \frac{d}{dt} s(t)
   \]

### Explanation:

- **Impulse response** \( h(t) \) is the derivative of the **step response** \( s(t) \).
- If you are given the step response \( s(t) \), you can find the impulse response by differentiating \( s(t) \) with respect to time \( t \).

### Summary:
The key takeaway is that the impulse response \( h(t) \) of an LTI system can be obtained by differentiating the system's output when a step signal \( u(t) \) is applied. Mathematically, the impulse response is:

\[
h(t) = \frac{d}{dt} s(t)
\]

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**Relative Questions:**
1. What is the definition of a unit impulse signal \( \delta(t) \) in an LTI system?
2. How does the convolution property play a role in determining system output?
3. What are some physical interpretations of the impulse and step response in systems?
4. How can the impulse response be used to find the output for other input signals?
5. What are some practical examples of LTI systems where the impulse response is essential?

**Tip**: In LTI systems, understanding the impulse response is crucial as it characterizes the entire system's behavior. Once known, it allows you to determine the output for any arbitrary input through convolution.

Explore how to find the impulse response of a Linear Time-Invariant (LTI) system when given a unit step signal input. Understand the relationship between the impulse and step responses, and learn the mathematical derivation and convolution properties involved. This problem is suited for college-level students studying signal processing or control systems.

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