7.	Mention the convolution integral with all combinations of integral limits( if the system is casual)?

The convolution integral is a fundamental concept in signal processing and systems theory, particularly when analyzing linear time-invariant (LTI) systems. For a system to be **causal**, the output at any time \( t \) depends only on the current and past inputs, not on future inputs. This causality condition affects the limits of the convolution integral.

Given two functions \( x(t) \) (input signal) and \( h(t) \) (impulse response of the system), the convolution \( y(t) \) of these two functions is defined as:

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

However, for a **causal system**, the impulse response \( h(t) \) is zero for \( t < 0 \). This causality simplifies the convolution integral. Depending on the time \( t \), the limits of the convolution integral vary:

### 1. **For \( t < 0 \):**
\[
y(t) = \int_{-\infty}^{t} x(\tau) h(t - \tau) \, d\tau
\]
Since \( h(t - \tau) = 0 \) for \( \tau > t \), the convolution integral only includes values where \( t - \tau \geq 0 \), i.e., \( \tau \leq t \).

### 2. **For \( t = 0 \):**
\[
y(0) = \int_{-\infty}^{0} x(\tau) h(-\tau) \, d\tau
\]
At \( t = 0 \), the integral is over all past times since \( h(-\tau) \) is defined for \( \tau \leq 0 \).

### 3. **For \( t > 0 \):**
\[
y(t) = \int_{0}^{t} x(\tau) h(t - \tau) \, d\tau
\]
Here, the limits of integration are from 0 to \( t \), because for a causal system \( h(t - \tau) \) is non-zero only when \( \tau \) is within the interval \( [0, t] \).

### Summary of Integral Limits for a Causal System:
- **For \( t < 0 \):** The limits are from \( -\infty \) to \( t \).
- **For \( t = 0 \):** The limits are from \( -\infty \) to 0.
- **For \( t > 0 \):** The limits are from 0 to \( t \).

These conditions ensure that the output \( y(t) \) only depends on the present and past values of the input \( x(t) \), adhering to the causality of the system.

Would you like more details or have any specific questions?

Here are five related questions to deepen your understanding:
1. How does the convolution integral change if the system is non-causal?
2. What are the properties of the convolution operation in signal processing?
3. Can you provide an example of computing the convolution integral for a given \( x(t) \) and \( h(t) \)?
4. How does the convolution integral relate to the Laplace transform in analyzing LTI systems?
5. What is the significance of the convolution theorem in Fourier analysis?

**Tip:** Always check the causality of a system before determining the limits of the convolution integral, as it affects the integral's evaluation and the system's behavior.

Explore the convolution integral in causal systems, focusing on how integral limits vary with time for causal systems. Includes detailed examples and explanations suitable for undergraduate students studying signal processing and system theory.

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