How to identify a process via its Karhunen-Loeve expansion?

Question

Suppose that you are given the following Karhunen-Loève expansion of a real-valued continuous Gaussian stochastic process, $x$. $$x(t) = \sum_{k=1}^{\infty}z_{k}\cdot \frac{\sqrt{2}\sin((k-0.5)\pi t)}{(k-0.5)\pi}\ ,\quad t \in[0,1]\ , $$ where $z_{k}$ denote independent standard normal distributed random variables. How do you infer from this that process $x$ is Brownian motion?