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.