Inverse Z transform of symmetric function $R_{x}(n) = 3\cdot (0.8)^{|n|}$

Question

On Z-transform table, most of the pairs are only valid for $n≥0$.

My question is to find PSD (Z transform) of $$R_{x}(n) = (0.8)^{|n|}$$

Note that $n$ is an integer span from $-\infty$ to $\infty$.

Answer 1

For $|z|<\tfrac{5}{4}$ we have $$\sum_{n=0}^{\infty}\left(\frac{4z}{5}\right)^n=\frac{5}{5-4z}$$ and for $|z|>\tfrac{4}{5}$ we have $$\sum_{n=0}^{\infty}\left(\frac{4}{5z}\right)^n=\frac{5z}{5z-4}.$$ So for $\tfrac{4}{5}<|z|<\tfrac{5}{4}$ $$\sum_{n=-\infty}^{\infty}\left(\frac{4}{5}\right)^{|n|}z^n=\frac{5}{5-4z}+\frac{5z}{5z-4}-1=\frac{5}{5-4z}+\frac{4}{5z-4}$$