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Why y(t) is increasing from -1 to 1 and decreasing from 6 to 8

The image you've provided shows the convolution of an input signal \( X(t) \) with an impulse response \( h(t) \), which gives the output \( Y(t) = X(t) * h(t) \). 

The output \( y(t) \) behavior between \( t = -1 \) to \( t = 1 \) and \( t = 6 \) to \( t = 8 \) can be explained by the properties of convolution.

### Understanding Convolution in Time Domain:

1. **From \( t = -1 \) to \( t = 1 \):**
   The graph of \( h(t) \), the impulse response, shows a spike at \( t = 3 \), and the shape of \( X(t) \) is increasing before \( t = 4 \). When you convolve the two, the increasing region in \( y(t) \) from \( t = -1 \) to \( t = 1 \) happens because the right edge of the signal \( X(t) \) is encountering the impulse response. The convolution integrates the overlapping areas of \( X(t) \) and \( h(t) \), which leads to an increase in the signal at the start.

2. **From \( t = 6 \) to \( t = 8 \):**
   This part shows a decreasing trend in \( y(t) \). Here, \( X(t) \) is at its maximum and then starts to sharply decrease after \( t = 6 \), leading to the corresponding output response in \( y(t) \). Since the impulse response is centered around \( t = 3 \), the convolution shifts \( X(t) \), so the tail end of the input signal aligns with the impulse response, creating this decreasing shape in the output.

### Summary:
- **Increasing from \( t = -1 \) to \( t = 1 \):** This is due to the early phase of the convolution where the overlapping area between the input and impulse response begins to grow.
- **Decreasing from \( t = 6 \) to \( t = 8 \):** This happens as the high part of the input signal finishes interacting with the impulse response, leading to a decrease in the output.

### Would you like more details or have any questions on this?

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Here are five related questions that expand upon this explanation:

1. What is the general mathematical definition of convolution?
2. How does the shape of the impulse response \( h(t) \) affect the output \( y(t) \)?
3. What would happen if \( h(t) \) were a different shape, such as a rectangular function?
4. How can we interpret the properties of \( X(t) \) in the frequency domain for this scenario?
5. How does the duration of \( X(t) \) affect the duration of \( Y(t) \)?

**Tip:** When analyzing convolution, always look at how the input function aligns with the impulse response at various points, as this affects the shape of the output.

Why y(t) is increasing from -1 to 1 and decreasing from 6 to 8?

Explore the behavior of convolution in signal processing. Understand why y(t) increases between t = -1 to 1 and decreases from t = 6 to 8 through a detailed explanation of convolution with an input signal and impulse response.

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