The Shannon-Whittaker sampling theorem provides an exact digitization of an analog signal (a scalar function of a single variable). It is not immediately obvious that this digitization is robust against error in the sampled signal values. We will present an algorithm which shows that for certain types of sample error, the Shannon-Whittaker sampling formula can be modified to recover a signal exactly even in the presence of error. We will also present several extensions of the Shannon-Whittaker sampling formula which make error recovery more stable in practice. Along the way, we will also discuss aspects of frame theory, which extend the idea of a basis for a vector space. Frames allow for information redundancy, making them much more robust against noise than an ordinary basis.