  Many turbulent flows exhibit time-periodic statistics. These include
turbomachinery flows, flows with external harmonic forcing, and the wakes of
bluff bodies. Many existing techniques for identifying turbulent coherent
structures, however, assume the statistics are statistically stationary. In
this paper, we leverage cyclostationary analysis, an extension of the
statistically stationary framework to processes with periodically varying
statistics, to generalize the spectral proper orthogonal decomposition (SPOD)
to the cyclostationary case. The resulting properties of the cyclostationary
SPOD (CS-SPOD for short) are explored, a theoretical connection between CS-SPOD
and the harmonic resolvent analysis is provided, simplifications for the low
and high forcing frequency limits are discussed, and an efficient algorithm to
compute CS-SPOD with SPOD-like cost is presented. We illustrate the utility of
CS-SPOD using two example problems: a modified complex linearized
Ginzburg-Landau model and a high-Reynolds-number turbulent jet.
