Linear and non-linear systems

Additive property: If:

• $x_1(t)\to y_1(t)$
• $x_2(t)\to y_2(t)$

Then:

$$x_1(t)+x_2(t)\to y_1(t)+y_2(t)$$

Scaling property: If $x_1(t)\to y_1(t)$, then $k{x}_1(t)\to k{y}_2(t)$

Additive + Scaling = Linear: If:

• $x_1(t)\to y_1(t)$
• $x_2(t)\to y_2(t)$

Then:

$$k_1{x}_1(t)+k_2{x}_2(t)\to k_1{y}_1(t)+k_2{y}_2(t)$$

Linear Example

Given $y(t)=tx(t)$, we have:

$$\begin{array}{r}{x}_1(t)\to y_1(t)=t{x}_1(t)\\ x_2(t)\to y_2(t)=t{x}_2(t)\end{array}$$

Then:

$$\begin{array}{rl}{x}_3(t)=a{x}_1(t)+b{x}_2(t)\to y_3(t)& =t{x}_3(t)\\ & =t[a{x}_1(t)+b(x_2)t]\\ & =ta{x}_1(t)+tb{x}_2(t)\\ & =a{y}_1(t)+b{y}_2(t)\\ & \therefore \text{ System is linear}\end{array}$$

Non-linear Example 1

Given a system of $y(t)=x^3(t)$, we have

$$\begin{array}{r}{x}_1(t)\to y_1(t)=x_1^3(t)\end{array}$$

Thus:

$$\begin{array}{rl}{x}_2(t)=k{x}_1(t)\to y_2(t)& =x_2^3(t)=(k{x}_1(t){)}^3\\ & =k^3{x}_1^3(t)\\ & \ne k{y}_1(t)\\ & \therefore \text{ Scaling doesn’t hold, non-linear}\end{array}$$

Non-linear Example 2

Given a system of $y[n]=\mathrm{Re}\{x[n]\}$, such that:

$$\begin{array}{rl}{x}_1[n]={\delta }_1[n]+j{\omega }_1[n]& \to y_1[n]={\delta }_1[n]\end{array}$$

However, let’s say that $x_2[n]$ is defined to be $j{x}_1[n]$, we would have:

$$\begin{array}{rl}{x}_2[n]=j{x}_1[n]=j{\delta }_1[n]-{\omega }_1[n]& \to y_2[n]=-{\omega }_1[n]\end{array}$$

so $y_2[n]\ne j{y}_1[n]$, failing the scaling property, making the system nonlinear.