Stable and Unstable Systems

A system is called “bounded input/bounded output” (BIBO) stable if and only if for every bounded input, the output is bounded.

Continuous time:

$$|x(t)|\le M_x<\infty \to |y(t)|\le M_y<\infty$$

Discrete time:

$$|x[n]|\le M_x<\infty \to |y[n]|\le M_y<\infty$$

where $M_x,M_y$ are positive, finite values.

Stable Example

Moving average system:

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

Assume bounded input

$$\begin{array}{rl}|x[n]|& \le M_x<\infty \text{for all }n\\ |y[n]|& =\frac{1}{3}|x[n]+x[n-1]+x[n-2]|\\ |y[n]|& \le \frac{1}{3}(x[n]+x[n-1]+x[n-2])\\ |y[n]|& \le \frac{1}{3}(M_x+M_x+M_x)\\ |y(n)|& \le \infty \\ & \therefore \text{BIBO stable}\end{array}$$

Unstable Example

System:

$$\begin{array}{r}y(t)=t^{2}x(t)\\ x(t)=u(t)\end{array}$$

Thus

$$\begin{array}{r}|y(t)|=|t^{2}x(t)|\\ \text{as }t\to \infty \\ |y(t)|\to \infty \\ \therefore \text{BIBO unstable}\end{array}$$