PostprintThis work addresses the design of a state observer for systems whose outputs are measured through a communication network. The measurements from each sensor node are assumed to arrive randomly, scarcely and with a time-varying delay. The proposed model of the plant and the network measurement scenarios cover the cases of multiple sensors, out-of-sequence measurements, buffered measurements on a single packet and multi-rate sensor measurements. A jump observer is proposed, that selects a different gain depending on the number of periods elapsed between successfully received measurements and on the available data. A finite set of gains is precalculated off-line with a tractable optimization problem, where the complexity of the observer implementation is a design parameter. The computational cost of the observer implementation is much lower than in the Kalman filter, whilst the performance is similar. Several examples illustrate the observer design for different measurements scenarios and observer complexity and show the achievable performance.