Frequency to eigendomain transform

Denote $E(\omega)=[1 \quad e^{-j\omega} \quad \ldots \quad e^{-j\omega (P-1)}]'$ Given the covariance matrix $R=toeplitz(r_0,\ldots,r_{P-1})$ where

$$\begin{array}{lll} r_p & = & \frac{1}{N} \sum_{n=0}^{N-1-p... ...i}^{\pi} \Phi(\omega)e^{-j \omega p} d\omega\\ \end{array}$$

Form $R=\sum_{i=1}^{P} \lambda_i v_i v_i'$. The Frequency to Eigendomain Transformation is given by

$$\begin{array}{lll} \lambda_i & = & v_i' R v_i\\ & = & v_... ...t^2 d\omega \textrm{ (zero-phase filtering)}\\ \end{array}$$

To find the Inverse Frequency to Eigendomain Transformation consider

$$E(\omega)' \textrm{ toeplitz}(r_0,\ldots,r_{P-1}) E(\omega) = P \sum_{i=-(P-1)}^{P-1} r_i w_b(i) e^{-j\omega i}$$

where $w_b$ is the Bartlett (triangular) window.

$$\begin{array}{lll} \Phi_b(\omega) & = & \sum_{i=-(P-1)}^{P... ...i=1}^{P} \lambda_i V_i(\omega) V_i^*(\omega)\\ \end{array}$$