The weakly stationary processes $X_t$, $\;t=0$, $\,\pm$$1$, $\,\pm$$2$,$\,\ldots$ and $Y_t$,$\;$ $t=0$, $\,\pm$$1$, $\,\pm$$2$,$\ldots$ are input and output of a linear filter according to
$Y_t\;$$+\;0.5Y_{t-1}\;$$\;=\;$$X_t$, $\quad$ for $t=0$, $\,\pm$$1$, $\,\pm$$2$,$\ldots$
The process $X_t\,$ has the covariance function $r_X(0)=1$,$\,$ $r_X(\pm2)=0.2$,$\,$ and zero for all other values. Determine the cross-covariance funtion
$r_{X,Y}(\tau)=C[X_t,\,Y_{t+\tau}]$
for all values of $\tau$.
I began by assigning the impulse function h(x) the following values:
$h(0) = 1\qquad$ $h(1) = 0.5$
Does it makes sense?
The impulse response $h_{t}$ is actually given by $h_{t} = -0.5^{t}u_{t}$ (where $u_{t}$ is the step function). To obtain this, you can rearrange to get $Y_{t} = X_{t} - 0.5Y_{t - 1}$ and plug in values for $X_{t}$ being the discrete delta function, or you can take the $z$-transform to get the transfer function of $H\left(z\right) = \dfrac{1}{1 + 0.5z^{-1}}$ and use a table of $z$-transform pairs to find the corresponding sequence. Either way yields the sequence $h_{0} = 1, h_{1} = -0.5, h_{2} = 0.25,$ etc.
We can then use the relation $$r_{X, Y}\left(\tau\right) = \sum_{i = -\infty}^{\infty}h_{i}r_{X}\left(\tau - i\right)$$ which is found in several textbooks (eg. Probability and Stochastic Processes: A Friendly Introduction for Electrical and Computer Engineers by Yates and Goodman, Theorem 11.5) with the provided covariance function for $X_{t}$ to obtain $$r_{X, Y}\left(\tau\right) = h_{\tau} + 0.2h_{\tau + 2} + 0.2h_{\tau - 2}$$