fourier decomposition of white noise

Question

I found in some lecture notes that Brownian motion is defined by its Fourier series:

$$ B(t) = \sum_{m \in \mathbb{Z},\; m \neq 0} \frac{a_m}{m} e^{2\pi i \,mt} $$

Then I would get that its derivative is white noise:

$$ W(t) = \sum_{m \in \mathbb{Z},\; m \neq 0} a_m e^{2\pi i \,mt} $$

What is the random Fourier series if we truncate it to the positive half? Is there a name for that random process?

$$ f(t) = \sum_{ m \geq 0} a_m e^{2\pi i \,mt} $$

Here $a_m$ are standard identical, independently distributed Gaussian random variables with mean 0 and variance 1.