Introduction to Signals and Systems

• Signals are functions, typically of time e.g. $x(t)$
• Systems provide an abstraction for how an input signal is converted into an output signal e.g. $x(t) \rightarrow \text{System} \rightarrow y(t)$
• Signals can be discrete (time) or continuous (time)

Example 1

Suppose water flows into a leaky tank. Water comes in with rate $r_i(t)$ and flows out with rate $r_o(t)$. Let $h(t)$ be the height of water in the tank. If we assume $r_o(t) \propto h(t)$ and $h(t)$ obeys water conservation i.e. $\partial_t h(t) \propto r_i(t) - r_o(t)$, then

$$\partial_t r_o(t) \propto \partial_t h(t) \propto r_i(t) - r_o(t)$$

The constant of proportionality must have units $1/\text{time}$. One over the constant is frequently called the time constant $\tau$:

$$\tau \partial_t r_1(t) = r_i(t) - r_o(t)$$