For $a,b \in \omega$ with $a > 0$, let $f_{a,b}: \omega\to\omega$ be defined by $n \mapsto an+b$. What is an example of an infinite binary string $b:\omega\to\{0,1\}$ with the following property?

Whenever $(a,b), (a_1,b_1)\in (\omega\setminus\{0\})\times \omega$ with $(a,b)\neq (a_1,b_1)$, then $b\circ f_{a,b} \neq b\circ f_{a_1,b_1}$.