Correlation-based testing for the convergence of decision feedback equalizers

It has been thought that convergence tests developed for linear decision-directed equalizers based on the correlation matching criterion may not be directly applicable to decision feedback equalizers (DFE's). Indeed, the correlation-based convergence criteria proved for linear equalizers have so far been accepted as conjectural for DFE's. In this correspondence we show that identical channel input and DFE output autocorrelations is a necessary and sufficient condition for the channel eye to be open, if the channel input is binary phase-shift keyed. A threshold test based on the correlation matching criterion and assuming a priori knowledge of the channel input autocorrelation is also constructed and analyzed.