LTI properties; is y[n] = e^(x[n]) linear

Properties of LTI systems

An LTI system is linear and time-invariant; its output is $y[n]=x[n]*h[n]$. Its main properties:

1. Commutative: $x*h=h*x$.
2. Associative: $(x*h_1)*h_2=x*(h_1*h_2)$; a cascade equals one system $h_1*h_2$, in any order.
3. Distributive: $x*(h_1+h_2)=x*h_1+x*h_2$; parallel systems add their impulse responses.
4. Memoryless only if $h[n]=K\delta[n]$.
5. Causal if and only if $h[n]=0$ for $n<0$.
6. BIBO stable if and only if $\sum_n|h[n]|<\infty$.
7. Invertible if some $h_i[n]$ satisfies $h*h_i=\delta[n]$.
8. Step response $s[n]=\sum_{k\le n}h[k]$, and $h[n]=s[n]-s[n-1]$.
9. Eigenfunction: $e^{j\omega n}\rightarrow H(e^{j\omega})e^{j\omega n}$.

Is $y[n]=e^{x[n]}$ linear?

A system is linear if $a_1x_1+a_2x_2\rightarrow a_1y_1+a_2y_2$.

Let $y_1[n]=e^{x_1[n]}$ and $y_2[n]=e^{x_2[n]}$. For $x_3=a_1x_1+a_2x_2$:

$$y_3[n]=e^{a_1x_1[n]+a_2x_2[n]}=\left(e^{x_1[n]}\right)^{a_1}\left(e^{x_2[n]}\right)^{a_2}$$

while

$$a_1y_1[n]+a_2y_2[n]=a_1e^{x_1[n]}+a_2e^{x_2[n]}$$

These are not equal in general (one is a product, the other a sum).

Also, zero input gives $y[n]=e^0=1\ne0$, which a linear system cannot do.

Numerical check: $x=1\rightarrow y=e\approx2.718$; $x=2\rightarrow y=e^2\approx7.389\ne2(2.718)=5.437$.

Answer: $y[n]=e^{x[n]}$ is non-linear. (It is, however, time-invariant, memoryless, causal, and BIBO stable, since $|x|\le B$ gives $|y|\le e^{B}$.)