  In this paper we study the use of cross-correlations between multiple
gravitational wave (GW) data streams for detecting long-lived periodic signals.
Cross-correlation searches between data from multiple detectors have
traditionally been used to search for stochastic GW signals, but recently they
have also been used in directed searches for periodic GWs. Here we further
adapt the cross-correlation statistic for periodic GW searches by taking into
account both the non-stationarity and the long term-phase coherence of the
signal. We study the statistical properties and sensitivity of this search, its
relation to existing periodic wave searches, and describe the precise way in
which the cross-correlation statistic interpolates between semi-coherent and
fully-coherent methods. Depending on the maximum duration over we wish to
preserve phase coherence, the cross-correlation statistic can be tuned to go
from a standard cross-correlation statistic using data from distinct detectors,
to the semi-coherent time-frequency methods with increasing coherent time
baselines, and all the way to a full coherent search. This leads to a unified
framework for studying periodic wave searches and can be used to make informed
trade-offs between computational cost, sensitivity, and robustness against
signal uncertainties.
