Additivity (system property)

A system is additive if the response to a sum of inputs equals the sum of the individual responses. In symbols: if $g(t)\to y_1(t)$ and $h(t)\to y_2(t)$, then $g(t)+h(t)\to y_1(t)+y_2(t)$, for any $g$ and $h$.

How to test

1. Pick $x_1=g$. Find $y_1$.
2. Pick $x_2=h$. Find $y_2$.
3. Pick $x_3=g+h$. Find $y_3$.
4. Check whether $y_3=y_1+y_2$ for every $g,h$.

A common gotcha

$y(t)=u(x(t))$ — apply the unit step to the input. Let $x_1=g$, $y_1=u(g)$. Let $x_2=h$, $y_2=u(h)$. Let $x_3=g+h$, $y_3=u(g+h)$.

Pick a counterexample: $t=3$, $g(3)=4$, $h(3)=2$. Then $y_1(3)+y_2(3)=u(4)+u(2)=1+1=2$. But $y_3(3)=u(4+2)=u(6)=1$. So $2\ne 1$, and the system is not additive.

The reason: the unit step is a nonlinear function. $u(a+b)$ is not generally $u(a)+u(b)$. Any nonlinear transformation of the input fails additivity.

Where additivity sits

Additivity is one of the two ingredients of linearity; the other is homogeneity. Linearity is just both at once.

A single property combines both:

$\text{If }g\to y_1,h\to y_2,\text{ then }\alpha g+\beta h\to \alpha y_1+\beta y_2$

for any constants $\alpha ,\beta$. Setting $\alpha =\beta =1$ recovers additivity; setting $\beta =0$ recovers homogeneity. So linearity, additivity, and homogeneity are tightly coupled.

There are weird systems that are homogeneous but not additive, or additive but not homogeneous, but in practice you almost never encounter them outside of contrived examples. For real engineering systems, the two travel together as linearity.