  The impact of control sequences on the environmental coupling of a quantum
system can be described in terms of a filter. Here we analyze how the coherent
evolution of two interacting spins subject to periodic control pulses, at the
example of a nitrogen vacancy center coupled to a nuclear spin, can be
described in the filter framework in both the weak and the strong coupling
limit. A universal functional dependence around the filter resonances then
allows for tuning the coupling type and strength. Originally limited to small
rotation angles, we show how the validity range of the filter description can
be extended to the long time limit by time-sliced evolution sequences. Based on
that insight, the construction of tunable, noise decoupled, conditional gates
composed of alternating pulse sequences is proposed. In particular such an
approach can lead to a significant improvement in fidelity as compared to a
strictly periodic control sequence. Moreover we analyze the decoherence impact,
the relation to the filter for classical noise known from dynamical decoupling
sequences, and we outline how an alternating sequence can improve spin sensing
protocols.
