<aside> 🦈 Both the Fourier and Laplace transform can be applied to one sided or two sided inputs.

</aside>

• But usually working with two-sided inputs, we don’t require the input to be causal

CT systems

Any entity that interacts with its external environment through input and output signals

Property: Linearity

CT system is linear if for any two input signals, $x$, $!x$ and any two constants $\alpha, !\alpha \in C$

$$ T\{\alpha x + !\alpha x\} = \alpha T\{x\}+ !\alpha T\{!x\} $$

Property: Time Invariance

A CT system is time-invariant if or any input $x$ with output $y = T\{x\}$, it holds that $y_{\tau} = T\{x_{\tau}\}$ for all time shifts $\tau \in \R$

• To check if a system is time-invariant, we compute $y_{\tau}$, compute $T\{x_{\tau}\}$ and see if they are equal

Property: Causality

A CT system $T$ is causal if for all $t \in \R$, the output value $y(t)$ depends only on the past and present input values $\{ x(\tau)\}_{\tau <= t}$

• System does not “look into the future”
• Can often check causality by inspecting the formula for $y(t)$

Property: Memoryless

Ct system $T$ is memoryless if for all times $t \in \R$, the output value $y(t)$ depends only on the input value $x(t)$ at that same time

• Only the current time matters, not the past or future
• Special case of causality

Property: Invertibility