In this paper, an analytical solution for the differential equation of the simple but nonlinear pendulum is derived. This solution is valid for any time and is not limited to any special initial instance or initial values. Moreover, this solution holds if the pendulum swings over or not. The method of approach is based on JACOBI elliptic functions and starts with the solution of a pendulum that swings over. Due to a meticulous sign correction term, this solution is also valid if the pendulum does not swing over.

The degrees of freedom (DoF) of the MIMO Ychannel, a multi-way communication network consisting of 3 users and a relay, are characterized for arbitrary number of antennas. The converse is provided by cut-set bounds and novel genie-aided bounds. The achievability is shown by a scheme that uses beamforming to establish network coding on-the-fly at the relay in the uplink, and zero-forcing pre-coding in the downlink. It is shown that the network has min{2M2+2M3, M1+ M 2 + M3, 2N } DoF, where Mj and N represent the number of antennas at user j and the relay, respectively. Thus, in the extreme case where M1 +M2 +M3 dominates the DoF expression and is smaller than N , the network has the same DoF as the MAC between the 3 users and the relay. In this case, a decode and forward strategy is optimal. In the other extreme where 2N dominates, the DoF of the network is twice that of the aforementioned MAC, and hence network coding is necessary. As a byproduct of this work, it is shown that channel output feedback from the relay to the users has no impact on the DoF of this channel.

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SUMMARYThis paper deals with wave digital modeling of passive state-space models. The set of differential equations must be of linear state-space form, but all parameters can be time-variant and/or nonlinear. For such statespace models, a canonical internally passive reference circuit is presented and used for deriving wave digital structures. In order to show the usability, special solutions for important basic linear time-variant models are compared with wave digital simulation results. Moreover, the wave digital modeling of a nonlinear and time-variant oscillator is discussed. Especially for a lossless oscillator an implementation is proposed, which preserves energy under finite-arithmetic conditions. This is verified by comparing simulation results with the analytical solution of a gravity pendulum.

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