Circulant Feedback Delay Networks

Consider the class of circulant feedback matrices having the form

$${\bf A} = ~\left[ \begin{array}{llll} a(0) & a(1) & \dots &... ...ots \\ a(1) & \dots & a(N-1) & a(0) \\ \end{array} \right]$$

This class of matrices gives rise to a class of FDNs we call Circulant Feedback Delay Networks (CFDN). The following two facts can be proved [3]:

Fact 1: If a matrix is circulant, it is normal, i.e., ${\bf A}^\ast {\bf A}={\bf A}{\bf A}^\ast$.

Fact 2: If a matrix is circulant and lossless, it is unitary.

It is well known that every circulant matrix is diagonalized by the Discrete Fourier Transform (DFT) matrix [3]. This implies that the eigenvalues of ${\bf A}$ can be computed by means of the DFT of the first row:

$$\{\lambda ({\bf A})\} = \{{\bf A} (k)\} = DFT([a(0) \dots a(N-1) ]^T) \nonumber$$

where $\{\lambda ({\bf A})\}$ denotes the set of all eigenvalues of ${\bf A}$, and $\{{\bf A}(k)\}$ denotes the set of complex DFT samples obtained from taking the DFT of $\{a(\cdot)\}$.