My Math Forum Problems to find the cross spectrum of a time series system
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 April 25th, 2015, 08:52 PM #1 Newbie   Joined: Apr 2015 From: Australia Posts: 1 Thanks: 0 Problems to find the cross spectrum of a time series system I'm looking for some advice to find the cross-spectrum for this system. $$y_t= r_t+\epsilon_t$$ $$r_t=\frac{\theta}{1-\psi z^{-1}}u_t$$ and the input is $$u_t=(1-\gamma z^{-1})w_t$$ $\epsilon_t$ and $w_t$ are both independent white noise signals with zero mean and independent variances and I'm looking for the cross_spectrum $G_{yu}(\omega)$ I've tried two ways, but the system configuration (the fact that there are two WN signals with different variances) confuses me and don't know how to proceed. First I tried to advance calculating the cross covariance, writing $r_t$ as an infinite summation $$r_t=u_t+\psi u_{t-1}+\psi^2 u_{t-2}+...$$ but don't how to include $\epsilon_{t}$ Also tried to use the identity $$G_{yu}=h(e^{-j\omega})G_x(\omega)$$ where $h(z^{-1})$ is the relationship between $y$ and $u$ and $G_u$ is the spectrum of $u$ but here don't know how to account for $w_t$ I would appreciate your advice on how to face this problem.

 Tags cross, find, problems, series, spectrum, system, time, time series

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