The Transient Spectrum of a Random System

The frequency spectrum is a fundamental quantity for the analysis and design of a system. When a system is turned on or off, the frequency spectrum of its output changes with time. We define the transient spectrum as the time-frequency spectrum of the system output during the transient phase. We obtain the exact transient spectrum for a wide class of random systems, and we formulate it with respect to the classical power spectrum, which is reached for large times. We apply the developed method to the case of a harmonic oscillator and of a system with two resonances.