The Discrete Fourier Transform (DFT) has many applications to signal processing. One application leverages the uncertainty principle associated with the DFT, allowing a message to be reconstructed exactly from incomplete data. We generalize this uncertainty principle to an arbitrary invertible linear transformation, and show that the DFT is in some sense the optimal transform for this purpose. Stronger recovery properties are possible for sparse signals via the DFT, and we will give an overview of this theory, and provide examples of non-DFT transforms that satisfy this stronger recovery property.