Time series with missing data are signals encountered in important settings for machine learning. Some of the most successful prior approaches for modeling such time series are based on recurrent neural networks that transform the input and previous state to account for the missing observations, and then treat the transformed signal in a standard manner. In this paper, we introduce a single unifying framework, Recursive Input and State Estimation (RISE), for this general approach and reformulate existing models as specific instances of this framework. We then explore additional novel variations within the RISE framework to improve the performance of any instance. We exploit representation learning techniques to learn latent representations of the signals used by RISE instances. We discuss and develop various encoding techniques to learn latent signal representations. We benchmark instances of the framework with various encoding functions on three data imputation datasets, observing that RISE instances always benefit from encoders that learn representations for numerical values from the digits into which they can be decomposed.