Linear Systems

Despite being the most marvelous creation in the universe, our brains are simple structures that find it difficult to handle complexity beyond a certain level. In DSP, our main emphasis is on a class of systems knowns as linear systems. But what does linearity mean?

Linearity implies that the response of a system to a sum of scaled input signals is the sum of responses to those scaled individual signals. This makes our lives much easier as we will see next.

Scaling

If the response of a system to a particular input $x[n]$ is $y[n]$,

$$x[n]\qquad \rightarrow \qquad y[n]$$

then a scalar product simply appears at the output as:

$$a x[n] \qquad \rightarrow \qquad a y[n]$$

This looks like a straightforward relation, but many systems do not obey this property. For example, an object falling under the influence of gravity covers a distance equal to:

$$s= v_0t + \frac{1}{2}gt^2$$