Assuming you have the following $AR(1)$ process, $X_t=\phi X_{t-1}+\epsilon_t$, where $\{\epsilon_t\}$ is a white noise sequence and $|\phi|>1$, it is easy to find that the non-causal solution is $X_t=\sum_{i=1}^\infty -\phi^{-i}\epsilon_{t+i}$.
According to Brockwell's book this solution should also satisfy the causal $AR(1)$ equation $X_t=\phi^{-1}X_{t-1}+\bar\epsilon_t$ for another white noise sequence (appropiately chosen).
By replacing the non-causal solution in the above equation I got, $X_t-\phi^{-1}X_{t-1}=\sum_{i=1}^\infty -\phi^{-i}\epsilon_{t+i}+\phi^{-2}\epsilon_t-\phi^{-2}\sum_{i=1}^\infty -\phi^{-i}\epsilon_{t+i}$
However, I am not sure how to proceed from here on to define the $\{\bar\epsilon_t\}$ sequence. Any hint would be appreciated.