Let us assume that the time series $\{X_t\}$ is an irreversible MA(1): $$X_t=Z_t+bZ_{t-1},$$ where $\{Z_t\}$ is white noise with mean $0$ and variance $\sigma^2$. Let define a second time series $$Y_t=\sum_{j=0}^{+\infty}(-b)^{-j}X_{t-j}.$$ Calculate mean and the covariance function for $\{Y_t\}$. Verify if $\{X_t\}$ is a white noise with mean $0$ and variance $V$. Show that $X_t=Y_t+aY_{t-1}$ for a certain choice of $a$.
I started with mean and it is easy to show that $\mathbb{E}[Y_t]=0.$ For the covariance function I have $$\mathrm{Cov}(K_t,K_{t+h})=\mathbb{E}[\sum_{j=0}^{+\infty}(-b)^{-j}Z_{t-j}\sum_{j=0}^{+\infty}(-b)^{-j}Z_{t+h-j}+\sum_{j=0}^{+\infty}(-b)^{-j}Z_{t-j}\sum_{j=0}^{+\infty}(-b)^{-j}bZ_{t+h-j-1}+\sum_{j=0}^{+\infty}(-b)^{-j}bZ_{t-j-1}\sum_{j=0}^{+\infty}(-b)^{-j}Z_{t+h-j}+\sum_{j=0}^{+\infty}(-b)^{-j}bZ_{t-j-1}\sum_{j=0}^{+\infty}(-b)^{-j}bZ_{t+h-j-1}]=...$$ And I do not know how to continue... I would be grateful for any hints.