There are two properties that make a system linear: \emph{homogeneity} and \emph{additivity}.


{\bf Scaling (homogeneity) property}

Given $\alpha \in \C$, or $\alpha \in \R$, If you scale the input signal by $\alpha$, then the output signal will also be scaled by $\alpha$.

$$\alpha x \to \fbox{F} \to \alpha y$$

{\bf Additivity property}

Given two input signals and their respective outputs after passing them through a system $F$, the signal representing the sum of the input signals corresponds to the output signal representing the sum of the original output signals.

\begin{align*}
x_1 \to &\fbox{F} \to y_1 \\
x_2 \to &\fbox{F} \to y_2 \\
\Rightarrow x_1 + x_2 \to &\fbox{F} \to y_1 + y_2
\end{align*}


\emph{Superposition} combines these two properties and characterizes LTI systems:

$$ \alpha_1 x_1 + \alpha_2 x_2 \to \fbox{F} \to \alpha_1 y_1 + \alpha_2 y_2 $$

\begin{example}
In discrete-time (DT), consider a signal representing a 3-day moving average of a stock quote:

$$ y(n) = \frac{x(n) + x(n-1) + x(n-2)}{3} $$

where $x(n)$ is he closing price of a stock, and $y(n)$ is the 3-day moving average. It can be shown that this signal is linear.
\end{example}

\begin{example}
In continuous-time (CT), consider a circuit where the voltage across the circuit is given by Ohm's Law, $V=IR$, where $I$ is the current, and $R$ is the resistance.

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%\includegraphics{images/circuits/resistor.ps}
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Let $\hat{x} = \alpha x$., then $\hat{y} = R\hat{x} = \alpha R x = \alpha y$. Let $\tilde{x} = x_1 +  x_2$, then $\tilde{y}=R\tilde{x}=Rx_1+Rx_2=y_1+y_2$. Therefore the system is linear. Since we never used the constancy of $R$, we can observe that in both cases we have linearity.
\end{example}


Linearity