A note on covariance-invariant digital filter design and autoregressive moving average spectrum analysis

Consider an autoregressive-moving average (ARMA) discrete-time sequence
{x_{k}}
with covariance sequence
{R_{k}}
. Equations are given for the solution of the AR coefficients
{a_{k}}\min{1}\max{n}
in terms of the covariances
{R_{k}}\min{0}\max{2n-1}
, and subsequent solution for the MA coefficients
{b_{k}}\min{1}\max{n}
in terms of the AR coefficients and the covariances
{R_{k}}\min{0}\max{n-1}
. The results are derived and presented somewhat differently than usual to complement the results of [1] for the synthesis of covariance-invariant digital filters. In the context of spectrum analysis, the results provide a means of performing ARMA spectrum analysis on data that arise as sampled data from a rational continuous-time process [2]. An important result, originally derived in [2], shows that the ARMA spectrum can be obtained without actually solving the nonlinear factorization problem for the MA coefficients.