We introduce discrete symplectic systems, which contain as special cases Sturm-Liouville difference equations and linear Hamiltonian difference systems. A special kind of a symplectic system, namely so-called trigonometric systems, are introduced. These systems are called trigonometric, since its solutions (which are matrix-valued) satisfy identities that are well-known trigonometric identities in the scalar case (e.g., S times the transpose of S + C times the transpose of C is equal to the identity matrix etc.). A transformation is given which transforms any symplectic system into a trigonometric system without changing the oscillatory behavior of the original system.