3.2 How to check system is linear?

Do experiment 1: Feed the system with the sum of 2 scaled signals \(x_{1}\left ( t\right ) \) and \(x_{2}\left ( t\right ) \) and obtain the output.

Do experiment 2: feed the system with \(x_{1}\left ( t\right ) \,\), and scale the output. feed the system with \(x_{2}\left ( t\right ) \,,\) and scale the output (use same scales that were used in experiment 1).  Now, add the above 2 scaled outputs. Check if we get the same result as we did in experiment 1. If the same result, then system is linear, else not linear. Here are 2 examples to illustrate. In these, I used \(z_{1}\) as scale for the signal \(x_{1}\left ( t\right ) \) and used \(z_{2}\) as scale for \(x_{2}\left ( t\right ) \). These scales are some numerical values. This has to be valid for any set of scales used.

This below is an algebraic method to check for linearity, which might be easier to use in exams.

1.

Let \(u_{1}\rightarrow y_{1}\), and \(u_{2}\rightarrow y_{2}\), where \(u_{i}\) is input to system, and \(y_{i}\) is the output. Find \(y_{1}\) and \(y_{2}\) by using the system definition given

2.

Find \(u_{3}=\alpha u_{1}+\beta u_{2}\), i.e. scaled version of \(u_{1}\) and \(u_{2}\)

3.

Find \(y_{3}\) when input is \(u_{3}\) using system definition

4.

Let \(\tilde{y}_{3}=\alpha y_{1}+\beta y_{2}\), i.e. scaled version of the \(y_{1}\) and \(y_{2}\) found in step 1 above.

5.

Compare \(\tilde{y}_{3}\) and \(y_{3}\), if they are the same, then system is linear, else not.

Here is an example, let system be given as \(u\rightarrow au+b\), and apply the above 5 steps

1.

Let \(u_{1}\rightarrow au_{1}+b\), and \(u_{2}\rightarrow au_{2}+b.\) Hence \(y_{1}=au_{2}+b\) and \(y_{2}=au_{2}+b\)

2.

\(u_{3}=\alpha u_{1}+\beta u_{2}\)

3.

\(y_{3}=\) \(au_{3}+b=a(\alpha u_{1}+\beta u_{2})+b\)

4.

Let \(\tilde{y}_{3}=\alpha y_{1}+\beta y_{2}\), hence \(\tilde{y}_{3}=\alpha \left ( au_{1}+b\right ) +\beta \left ( au_{2}+b\right ) \), hence \(\tilde{y}_{3}=a(\alpha u_{1}+\beta u_{2})+b\left ( \alpha +b\right ) \)

5.

Compare \(\tilde{y}_{3}\) and \(y_{3}\), we see they are not the same, hence non-linear