Contents
The six basic properties of Systems
(Basic concept covered in ECE301: Signals and 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 $
