Properization of second-order cyclostationary random processes and its application to signal presence detection

In this paper, we show that a second-order cyclo-stationary (SOCS) random process, whether it is proper or improper, can be always converted to an equivalent proper-complex SOCS random process with twice the cycle period. A simple linear-conjugate linear periodically time-varying operator called a FREquency SHift (FRESH) properizer is proposed to perform this conversion. As an application, we consider the presence detection of an improper-complex SOCS random process, which well models the complex envelopes of digitally modulated signals such as pulse amplitude modulation (PAM), staggered quaternary phase-shift keying (SQPSK), Gaussian minimum shift keying (GMSK), etc. In particular, the optimal presence detector that utilizes the FRESH properizer is derived for improper-complex SOCS Gaussian random processes, which provides the lower bound on the detection error probabilities. The derived optimal detector, which has the structural advantage in that it consists of a FRESH properizer followed by a single linear filter, achieves the same performance as the conventional detector that consists of parallel-connected linear and conjugate-linear filters. Numerical results are also provided.