# Linearity of the given system

I am given the following system and I am checking the additive property:

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

where $$y(t)$$ is the output and $$x(t)$$ is the input given to the system.

Now this is what I did so far:

\begin{align} x_1(t)&\rightarrow \text{system} \rightarrow& y_1(t)=&x_1(e^t)\\ x_2(t)&\rightarrow \text{system} \rightarrow& y_2(t)=&x_2(e^t)\\ \end{align}

$$y_1(t) + y_2(t) = x_1(e^t) + x_2(e^t)$$ How will system respond to this input?

$$x_1(t) + x_2(t)\rightarrow system \rightarrow =$$

What I think is that this is what will happen:

$$x_1(t) + x_2(t)\rightarrow system \rightarrow = x_1(e^t)+x_2(e^t)$$

Am I right?

• What do $y$ and $x$ represent in the first equation you wrote ($y(t)=x(e^t)$)? – Tendero Oct 10 '18 at 13:24
• I think you are right. A system is additive if $S\{ x_1(t) + x_2(t) \} = S\{ x_1(t) \} + S\{ x_2(t) \}$, where $S\{ \cdot \}$ is the system. – Maxtron Oct 10 '18 at 19:21
• $y(t)=x(t)$ describes a perfect system where signals enter and leave undistorted. Do you think you could clarify the question a little bit? – A_A Oct 14 '18 at 9:52

Additivity requires a little more than a direct addition of some $$x_1$$ to some $$x_2$$. It should involve any linear combination of $$x_1$$ and $$x_2$$, $$a_1 x_1 + a_2 x_2$$ where $$a_1$$ and $$a_2$$ are "kind of scalars" (this could make us dive into complicated stuff, like in Does scaling property imply superposition?)
In your case, however, linearity works, because you are touching the "time base" ($$t$$ vs. $$e^t$$) without touching the amplitudes. Your operation is a sort of time warping.
$$\mathcal{S}(a_1 x_1(t) + a_2 x_2(t)) = a_1 x_1(e^t) + a_2 x_2(e^t) = \mathcal{S}(a_1 x_1(t)) + \mathcal{S}(a_2 x_2(t))$$
• Afaik it sufficies for $a_1$ and $a_2$ to be complex? What other complications did you imply? (or did you?) I read from the other link that you mention about irrational and rational number fields... – Fat32 Oct 10 '18 at 22:11