The power spectrum of a randomly modulated harmonic oscillator

4 The power spectrum of a randomly modulated harmonic oscillator 

Having found a relationship between the absorption lineshape associated with a periodically modulated system variable and the power spectrum of this variable, we now consider a specific example. Consider a harmonic oscillator which is randomly perturbed so that its frequency changes in time as  

Problem 7.11. Show that a(Z)r = IE/ mar, where E is the oscillator energy. 

Once we evaluate / (t) we can obtain the power spectrum of the randomly modulated harmonic oscillator using the Wiener-Khintchine theorem (7.76) 

The last equality is obtained by changing the order of integration as in the transition from (7.127) to (7.128) in Appendix 7D. Equation (7.100) then becomes 

We now assume that the stochastic frequency modulation 8co t) is a stationary Gaussian process with Sco t) = 0 and Sco to)Sco to + /)) = 8co )s t), where s t) = s —t) is defined by this relationship. The parameter Sco ) and the fimction s(t) characterize the physics of the random frequency modulations and are assumed known. From Eq. (7.64) we get 

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