Here are the detailed answers to your questions regarding convolution in linear time-invariant (LTI) systems:

1. Convolution Integral with All Combinations of Integral Limits for a Causal System:

For a causal system, the convolution of the input signal $x(t)$ and the impulse response $h(t)$ is given by the convolution integral:

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

Since the system is causal (meaning the output at any time $t$ depends only on the values of the input at the current time and earlier times), we adjust the limits of integration accordingly:

• For causal systems:

  The convolution integral simplifies to:

  $y(t) = \int_{0}^{t} x(\tau) h(t - \tau) \, d\tau$

This is because for $t < 0$, the impulse response $h(t - \tau)$ is zero for $\tau > t$.

2. Relation Between $x(t)$, $h(t)$, and $y(t)$ for an LTI System:

For a Linear Time-Invariant (LTI) system, the output $y(t)$ is the convolution of the input signal $x(t)$ and the system’s impulse response $h(t)$:

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

This relation shows that the output is determined by the superposition of the scaled and shifted impulse responses.

3. Conditions for Stability of an LTI System Based on Impulse Response:

An LTI system is stable if its impulse response $h(t)$ is absolutely integrable. Mathematically, the condition for stability is:

$\int_{-\infty}^{\infty} |h(t)| \, dt < \infty$

This implies that the total area under the absolute value of the impulse response must be finite.

4. Relation Between Impulse Response and Step Response:

The step response $s(t)$ of a system is related to its impulse response $h(t)$ as follows:

• The step response is the integral of the impulse response:

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

This means that if you integrate the impulse response over time, you obtain the step response.

Conversely, the impulse response is the derivative of the step response:

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

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Would you like further details or have any specific questions? Here are some related questions to deepen your understanding:

1. What happens to the convolution integral limits when the system is not causal?
2. How does convolution relate to frequency domain analysis in LTI systems?
3. Can you explain the significance of the convolution theorem in signal processing?
4. How would you apply the convolution integral in discrete-time systems?
5. What is the physical interpretation of the step response in a practical system?

Tip: Understanding the integral bounds in convolution is crucial for applying these concepts to both continuous and discrete systems.