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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 crossspectrum 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 $$ r_t=u_t+\psi u_{t1}+\psi^2 u_{t2}+... $$ 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. 

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