In this paper, we define a new Joint Sparsity Model (JSM) and use Principal Component Analysis followed by Minimum Description Length and Compressive Sensing to reconstruct spatially and temporally correlated signals in a sensor network. The proposed model decomposes each sparse signal into two sparse components. The first component has a common support across all sensed signals. The second component is an innovation part that is specific to each sensor and might have a support that is different from the support of the other innovation signals. We use the fact that the common component generates a common subspace that can be found using the principal component analysis and the minimum description length. We show that with this general model, we can reconstruct the signal with smaller samples that are needed by the direct application of the compressive sensing on each sensor.