A novel hybrid observer that estimates the states of an oscillating system composed of a linear chain structure and an intrinsic pulse-modulated feedback is considered. This particular type of plant model appears in e.g. endocrine systems with pulsatile hormone secretion. The observer reconstructs the continuous states of the model as well as the firing times and weights of the feedback impulses. Since the pulse-modulated feedback is intrinsic, no measurements of the discrete part of the plant are available to the observer. For a periodical plant solution, to reconstruct the hybrid state, the impulses in the observer have to be synchronized with those in the plant. The observer is equipped with two feedback loops driven by the output estimation error. One of these is utilized to correct the estimates of the continuous states. In contrast with previous observer designs, the estimate of the next impulse firing time is implemented by means of a finite-memory convolution operator. A pointwise mapping capturing the propagation of the continuous plant and observer states through the discrete cumulative sequence of the feedback firing instants is derived. Local stability properties of the synchronous mode are related to the spectral radius of the Jacobian of the pointwise mapping. The observer design is based on assigning a guaranteed convergence rate to the local dynamics of a synchronous mode through the output error feedback gains to the continuous and discrete part of the observer. The observation of a stable m-cycle in the plant is treated to establish a general scenario, whereas the special case of 1-cycle is worked out in detail as the most common one. A numerical example illustrates the observer performance in the case of periodic modes of low multiplicity in an impulsive model of testosterone regulation in the male. Despite the local nature of the design approach, convergence to a synchronous mode is observed for a wide range of initial conditions for the discrete state estimate in the observer.