Causal Wiener filter banks for periodically correlated time series

A causal filter bank implementation of the cyclic Wiener filter for periodically correlated (PC) time series is developed. By converting a PC time series into a vector-valued wide-sense stationary (WSS) time series, the existing literature on factorization of spectral density matrices may be utilized. However, because PC analytic and equivalent baseband signals are generally complex improper, spectral factorization algorithms must be adapted to the improper case. Based on the factorization of the spectral density matrix for the equivalent WSS vector process, causal synthesis and whitening filters for PC time series can be built. These techniques are exploited to implement a causal cyclic Wiener filter as a multirate filter bank or an equivalent polyphase structure. This filter bank is shown to be an efficient equivalent implementation of a frequency shift (FRESH) filter. Therefore, the results derived in this paper also show how to build a causal FRESH cyclic Wiener filter.