Optimizing the design of aviation propulsion systems using computational fluid dynamics is essential to increase their efficiency and reduce pollutant as well as noise emissions. Nowadays, this design process is increasingly aided by computational fluid dynamic methods for which and with the adequate modeling approach it is possible to perform meaningful unsteady computations of the various components of a gas-turbine engine. However, these simulations are often carried out independently of each other and only share averaged quantities at the component interfaces minimizing the impact and interactions between components. The present work investigates the interactions between fan, compressor and annular combustion chamber at takeoff conditions by simulating a 360 azimuthal degrees large-eddy simulation of over 2100 million cells of the DGEN-380 demonstrator. In that case, the domain includes: 14 fan blades; 42 outlet-guide vanes (OGV); the impeller with 11 main blades and 11 splitter blades; a radial and an axial diffuser with 22 and 55 vanes, respectively; and the annular combustion chamber with a contouring casing and 13 swirlers on the back of the chamber. At take-off conditions it is found that the compressor operates in transonic conditions in the rotating frame of reference of the impeller and a shock is formed at the leading edge of the main blades which propagates upstream towards the fan and it is perceived at half the impeller blade-passing frequency (BPF). Preliminary results also show that pressure fluctuations at the impeller BPF generated by the interaction of the impeller blades with the diffuser vanes are propagated through the axial diffuser and enter the combustion chamber through the dilution holes and the swirler. The objective of this paper is to provide a deeper analysis of the interactions between components through the use of the novel operator-based analysis called dynamic mode tracking method (DMT). Indeed, this method facilitates the analysis of three-dimensional results despite the billion-size mesh and the complexity of the simulation, since it extracts modes at specific frequencies on-the-fly within the code. The frequencies corresponding to the fan, impeller and half the impeller BPF are analyzed in the domain and compared against traditional and more computationally demanding methods like the well-known Dynamic Mode Decomposition or the Direct Fourier transform.