Analysis of decision-directed equalizer convergence

Digital data signals are usually equalized by passing samples of the received signal through an adaptive equalizer consisting of a tapped delay line having adjustable coefficients (tap weights). The equalizer tap weights are adjusted by starting the transmission with a short training sequence of digital data known in advance by the receiver. This paper analyzes the situation when the known training sequence is replaced by a sequence of data symbols estimated from the equalizer output and treated as known data. Such procedures are called “decision-directed” startup. With a known training sequence, the “least-mean-square” adjustment algorithm corresponds mathematically to searching for the unique minimum of a quadratic “error” surface whose unimodal nature assures convergence. In decision-directed startup, by contrast, the use of estimated and unreliable data changes the error surface into a multimodal one so that complex behavior may result. We describe the nature of the error surfaces for binary and four-level transmission, thereby gaining insight into convergence problems. The most significant conclusion is that a poor choice for the initial tap settings may result in the taps converging to an undesirable setting. We show that, because of finite step-size effects, fluctuations are significant at the undesired settings and cause the spurious capture to have a long, but finite, duration. Finally we provide information on stability, convergence times, and lifetimes and their relation to the adaptation parameter (step size).