Analysis of the transient spectrum of a random system

We de…ne the transient spectrum as the time-frequency spectrum of a random system undergoing a transient behavior. We show that the transient spectrum approaches the classical frequency spectrum when time goes to in…nity. We prove that it is always possible to decompose the transient spectrum into the sum of a stationary spectrum and a decaying spectrum. The stationary spectrum is, up to a constant, the classical power spectrum, while the decaying spectrum accounts for the nonstationary behavior of the transient. All the results are valid for random LTI systems de…ned by stochastic di¤erential equations of n-th order. The Langevin equation is studied as an example. 1 Introduction Consider the following stochastic di¤erential equation1 dx(t) dt + x(t) = f(t) (1) where > 0 and f(t) is a white Gaussian noise with autocorrelation given by Rf ( ) = E[f(t)f(t+ )] = N0 ( ) (2) Also, consider the solution for all times and take zero initial conditions, x( 1) = 0. A physicist would call it a Langevin equation [11], since it rules the velocity of a particle undergoing Brownian motion, while an Electrical Engineer would recognize the governing equation of an RC …lter [1], whose input is the white Gaussian noise f(t) and whose output is the solution x(t). In reality Eq. (1) is widely used to model random quantities belonging to several di¤erent …elds, such as economic time series, biology, atomic clocks, just to mention a few of them [2]. What is the spectrum Sx(!) of the output random process x(t)? Since we are considering the solution for all times, then x(t) is a Wide Sense Stationary process, and the power spectrum is given by2 [10] Sx(!) = N0 2 + !2 (3) which is the well known Cauchy or Lorentzian distribution, represented in Fig. 1. Suppose that now we turn on the driving term f(t) at t = t0 dx(t) dt + x(t) = u(t t0)f(t) (4) We use bold letters for stochastic quantities. We use the following Fourier transform pair: x(t) = 1 2 R +1 1 X(!)e d!, X(!) = R +1 1 x(t)e dt .