Considerable research and development efforts are required to meet the targets of future gas turbine technologies in terms of performance, emissions, and operational flexibility. One of the recurring problems is the constructive coupling between flames and combustor's acoustics. These thermoacoustic interactions can cause high-amplitude dynamic pressure limit cycles, which reduce the lifetime of the hot gas path parts or in the worst-case scenario destroy these mechanical components as a result of a sudden catastrophic event. It is shown in this paper that the dynamics and the statistics of the acoustic signal envelope can be used to identify the linear growth rates hidden behind the observed pulsations, and the results are validated against numerical simulations. This is a major step forward and it will contribute to the development of future gas turbine combustors, because the knowledge of these linear growth rates is essential to develop robust active and passive systems to control these combustion instabilities.