  In this paper, the block processing of a discrete-time (DT) improper-complex
second-order cyclostationary (SOCS) random process is considered. In
particular, it is of interest to find a pre-processing operation that enables
computationally efficient near-optimal post-processing. An invertible
linear-conjugate linear (LCL) operator named the DT FREquency Shift (FRESH)
properizer is first proposed. It is shown that the DT FRESH properizer converts
a DT improper-complex SOCS random process input to an equivalent DT
proper-complex SOCS random process output by utilizing the information only
about the cycle period of the input. An invertible LCL block processing
operator named the asymptotic FRESH properizer is then proposed that mimics the
operation of the DT FRESH properizer but processes a finite number of
consecutive samples of a DT improper-complex SOCS random process. It is shown
that the output of the asymptotic FRESH properizer is not proper but
asymptotically proper and that its frequency-domain covariance matrix converges
to a highly-structured block matrix with diagonal blocks as the block size
tends to infinity. Two representative estimation and detection problems are
presented to demonstrate that asymptotically optimal low-complexity
post-processors can be easily designed by exploiting these asymptotic
second-order properties of the output of the asymptotic FRESH properizer.
