This paper considers the identification of time-varying linear channels using maximum likelihood estimation. The channel is modelled as a tapped-delay line filter with complex coefficients. Due to the complexity of the likelihood function and the large number of parameters to be estimated, an analytical maximization of the likelihood function is infeasible. Therefore, gradient algorithms are considered. We consider the structure of the channel tap covariances, and present a gradient algorithm for finding the constrained maximum likelihood estimates of the channel parameters.