Linearity of the given system

Question

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?

Answer 1

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.

Accordingly:

$$\mathcal{S}(a_1 x_1(t) + a_2 x_2(t)) = a_1 x_1(e^t) + a_2 x_2(e^t) = \mathcal{S}(a_1 x_1(t)) + \mathcal{S}(a_2 x_2(t)) $$