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April 25th, 2015, 07:52 PM   #1
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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.
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