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In this paper, the block processing of a discrete-time (DT) improper-complexsecond-order cyclostationary (SOCS) random process is considered. Inparticular, it is of interest to find a pre-processing operation that enablescomputationally efficient near-optimal post-processing. An invertiblelinear-conjugate linear (LCL) operator named the DT FREquency Shift (FRESH)properizer is first proposed. It is shown that the DT FRESH properizer convertsa DT improper-complex SOCS random process input to an equivalent DTproper-complex SOCS random process output by utilizing the information onlyabout the cycle period of the input. An invertible LCL block processingoperator named the asymptotic FRESH properizer is then proposed that mimics theoperation of the DT FRESH properizer but processes a finite number ofconsecutive samples of a DT improper-complex SOCS random process. It is shownthat the output of the asymptotic FRESH properizer is not proper butasymptotically proper and that its frequency-domain covariance matrix convergesto a highly-structured block matrix with diagonal blocks as the block sizetends to infinity. Two representative estimation and detection problems arepresented to demonstrate that asymptotically optimal low-complexitypost-processors can be easily designed by exploiting these asymptoticsecond-order properties of the output of the asymptotic FRESH properizer.

Asymptotic FRESH Properizer for Block Processing of Improper-Complex Second-Order Cyclostationary Random Processes