In this thesis we study the relation between analytic signals and a variety of pendulum systems. The representation of a signal as a pair of time varying amplitude and phase has been well studied and often related to linear mass spring systems. The differential equations describing pendulum systems are nonlinear and we provide analytical and numerical results regarding interpretation about the amplitude and the phase of signals in different pendulum settings. We report an explicit solution of the Elastic Pendulum problem in the case of linear phase. We develop an experimental procedure to piece-wise approximate bounded functions on a partition of a finite interval. On each sub-interval the function is approximated by a solution of a Pendulum system. The parameters of the corresponding differential equations are determined by optimization on each sub-interval. The smoothness of the approximation is controlled by the initial conditions provided by the given function
