The six basic properties of Systems

Memory

A system with memory has outputs that depend on previous (or future) inputs.

• Example of a system with memory:

$ y(t) = x(t - \pi) $

• Example of a system without memory:

$ y(t) = x(t) $

Invertibility

An invertible system is one in which there is a one-to-one correlation between inputs and outputs.

• Example of an invertible system:

$ y(t) = x(t) $

• Example of a non-invertible system:

$ y(t) = |x(t)| $

In the second example, both x(t) = -3 and x(t) = 3 yield the same result.

Causality

A causal system has outputs that only depend on current and/or previous inputs.

• Example of a causal system:

$ y(t) = x(t) + x(t - 1) $

• Example of a non-causal system:

$ y(t) = x(t) + x(t + 1) $

Stability

There are many types of stability, for this course, we first consider BIBO (Bounded Input Bounded Output) stability.

A system is BIBO stable if, for all bounded inputs ($ \exist B \epsilon \Re, |x(t)| < B $), the output is also bounded ($ |y(t)| < \infty $)

Time Invariance

A system is time invariant if a shift in the time domain corresponds to the same shift in the output.

• Example of a time invariant system:

$ y_1(t) = x_1(t) \mapsto y_2(t - t_0) = x_2(t - t_0) $

• Example of a time variant system:

$ y_1(t) = \sin(t) x_1(t) \mapsto y_2(t - t_0) = \sin(t) x_2(t - t_0) $

In the first example, $ y_2 $ is the shifted version of $ y_1 $. This is not true of the second example.

Linearity

A system is linear if the superposition property holds, that is, that linear combinations of inputs lead to the same linear combinations of the outputs.

A system with inputs $ x_1 $ and $ x_2 $ and corresponding outputs $ y_1 $ and $ y_2 $ is linear if: $ ax_1 + bx_2 = ay_1 + by_2 $ for any constants a and b.

• Example of a linear system:

$ y(t) = 10x(t) $

• Example of a nonlinear system:

$ y(t) = x(t)^2 $