How do I convert this equation from Iverson brackets to make use of the Heaviside function?

Question

I have the equation

$\sum_{i=0}^{\infty} 2^{i}[0 \leq x - 2^{i}][x - 2^{i + 1} < 0]$

and I would like to convert the Iverson brackets to the Heaviside function.

I've read this post but I'm still a bit confused as to how this would work inside the summation.

EDIT:

If it helps, I believe I've reduced it to

$u_1(x) + \sum_{i=1}^\infty 2^{i-1}u_{2^i}(x)$