  The filter function formalism from quantum control theory is typically used
to determine the noise susceptibility of pulse sequences by looking at the
overlap between the filter function of the sequence and the noise power
spectral density. Importantly, the square modulus of the filter function is
used for this method, hence directional and phase information is lost. In this
work, we take advantage of the full filter function including directional and
phase information. By decomposing the filter function with phase preservation
before taking the modulus, we are able to consider the contributions to $x$-,
$y$- and $z$-rotation separately. Continuously driven systems provide noise
protection in the form of dynamical decoupling by cancelling low-frequency
noise, however, generating control pulses synchronously with an arbitrary
driving field is not trivial. Using the decomposed filter function we look at
the controllability of a system under arbitrary driving fields, as well as the
noise susceptibility, and also relate the filter function to the geometric
formalism.
