An asymptotic GLRT for the detection of cyclostationary signals

We derive the generalized likelihood ratio test (GLRT) for detecting cyclostationarity in scalar-valued time series. The main idea behind our approach is Gladyshev´s relationship, which states that when the scalar-valued cyclostationary signal is blocked at the known cycle period it produces a vector-valued wide-sense stationary process. This result amounts to saying that the covariance matrix of the vector obtained by stacking all observations of the time series is block-Toeplitz if the signal is cyclostationary, and Toeplitz if the signal is wide-sense stationary. The derivation of the GLRT requires the maximum likelihood estimates of Toeplitz and block-Toeplitz matrices. This can be managed asymptotically (for large number of samples) exploiting Szegö´s theorem and its generalization for vector-valued processes. Simulation results show the good performance of the proposed GLRT.