Many time series of practical interest are periodic and digital in nature. A simple state space formulation of a general digital periodic time series is constructed. This allows us to design and propose a simple partially recurrent backpropagation neural network with adaptive error thresholding suitable for prediction and parameter estimation of periodic time series sequences. Such an approach is designed to be robust to corrupted data and discontinuous parameter changes. That this is the case is demonstrated with relevant examples. The method is ideally suited to problems requiring extremely rapid, recursive updates as each new time series value is encountered.