When Has a Decision-Directed Equalizer Converged?

Adaptive decision-directed equalizers can become locked for long periods onto incorrect equilibria. This note provides a test involving data available at the equalizer output for determining whether an equilibrium is correct or not, up to a fixed overall delay. The key result is the following. If an independent sequence of random variables taking values ± 1 is the input to a finite impulse response filter, and the output of the filter is passed through a slicer, then the slicer output is uncorrected if and only if the slicer output is a delayed version of the filter input. Determination of the actual delay parameter is a separate issue and not addressed. An analogous result for Mary rather than binary data is outlined.