Some more rigor and time is spent developing LTI concepts, convolution, and system properties in the Discrete Systems page. This page will consist of a “fast-motion” adaptation of this information for continuous time systems.
Convolution and LTI Systems
Imagine an impulse centered at $t=t_0$ represented by $\delta(t – t_0)$. Assume the system’s response as this impulse is fed through the system is then represented by
$$
\begin{align}
\delta(t – t_0) \rightarrow h_{t_0}(t) \\
\end{align}
$$
If the system is linear, then scaling the impulse at the input yields a correspondingly scaled version of the output.
$$
\begin{align}
a \delta(t – t_0) \rightarrow a h_{t_0}(t) && \text{Linear System}\\
\end{align}
$$
Additionally, if the system is time invariant, the impulse response remains the same regardless of the time, that is
$$
\begin{align}
\delta(t – t_0) \rightarrow h(t – t_0) &&\text{Time Invariant System} \\
\end{align}
$$
From the Sifting Property of continuous time signals, we can represent any signal $x(t)$ as a sum of scaled impulses.
\begin{align}
x(t) &= \int_{-\infty}^{\infty} x(\tau) \delta(t – \tau) d\tau \\
\end{align}
Therefore, for Linear Time Invariant systems, a sum of impulses that comprise an input signal $x(t)$ would be transformed to a sum of scaled impulse responses as an output signal $y(t)$.
$$
\begin{align}
\delta(t) &\rightarrow h(t) \\
x(0) \delta(t) &\rightarrow x(0) h(t) \\
x(t_0) \delta(t-t_0) &\rightarrow x(t_0) h(t – t_0) \\
x(t) = \int_{-\infty}^{\infty} x(\tau) \delta(t – \tau) d\tau &\rightarrow y(t) = \int_{-\infty}^{\infty} x(\tau) h(t – \tau) d\tau \\
\end{align}
$$
This is called the convolution sum which is also represented as
$$
y(t) = x(t) * h(t)
$$