Poisson-kernel expansion of an AR(1) spectrum
= Poisson-kernel expansion of an AR(1) spectrum
{c}
{title2=$\sum_{k\in\mathbb Z}\phi^{|k|}z^k=\frac{1-\phi^2}{(1-\phi z)(1-\phi z^{-1})}$}
For $|\phi|<1$, the bilateral sum $\sum_{k\in\mathbb Z}\phi^{|k|}z^k$ equals $(1-\phi^2)/((1-\phi z)(1-\phi z^{-1}))$ on the unit circle. Multiplying by $\sigma^2/(\pi(1-\phi^2))$ gives the one-sided autoregressive spectrum and reads off its <covariance> coefficients.