Convex Cost Functions in Blind Equalization

Existing blind adaptive equalizers that use nonconvex cost functions and stochastic gradient descent suffer from lack of global convergence to an equalizer setup that removes sufficient ISI when an FIR equalizer is used. In this paper, we impose convexity on the cost function and anchoring of the equalizer away from the all-zero setup. We establish that there exists a globally convergent blind equalization strategy for 1-D pulse amplitude modulation (PAM) systems with bounded input data (discrete or continuous) even when the equalizer is truncated. The resulting cost function is a constrained l₁ norm of the joint impulse response of the channel and the equalizer. Our results apply to arbitrary linear channels (provided there are no unit circle zeros) and apply regardless of the initial ISI (that is whether the eye is initially open or closed). We also show a globally convergent stochastic gradient scheme based on an implementable approximation of the l₁ cost function.