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Kernel Papers

- 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)

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 \(\tau\):

\[\tau \partial_t r_1(t) = r_i(t) - r_o(t)\]