In compressed sensing, one takes Graphic samples of an N-dimensional vector Graphic using an Graphic matrix A, obtaining undersampled measurements Graphic. For random matrices with independent standard Gaussian entries, it is known that, when Graphic is k-sparse, there is a precisely determined phase transition: for a certain region in the (Graphic,Graphic)-phase diagram, convex optimization Graphic typically finds the sparsest solution, whereas outside that region, it typically fails. It has been shown empirically that the same property—with the same phase transition location—holds for a wide range of non-Gaussian random matrix ensembles. We report extensive experiments showing that the Gaussian phase transition also describes numerous deterministic matrices, including Spikes and Sines, Spikes and Noiselets, Paley Frames, Delsarte-Goethals Frames, Chirp Sensing Matrices, and Grassmannian Frames. Namely, for each of these deterministic matrices in turn, for a typical k-sparse object, we observe that convex optimization is successful over a region of the phase diagram that coincides with the region known for Gaussian random matrices. Our experiments considered coefficients constrained to Graphic for four different sets Graphic, and the results establish our finding for each of the four associated phase transitions.