In section II, we briefly review the FDN and discuss some of its algebraic properties. In section III, we explore connections between FDNs and DWNs: It is shown how a single-junction DWN created by the intersection of waveguides can be interpreted as an order FDN; conversely, it is shown that any FDN can be interpreted as a DWN, although its scattering junction is not necessarily physical. We derive general conditions for lossless FDN feedback matrices in which the unitary matrix normally used in FDNs is extended to any matrix having unit-modulus eigenvalues and linearly independent eigenvectors. The extension corresponds to a generalization of signal energy by replacing the norm with an elliptic norm induced by any Hermitian, positive-definite matrix . In section IV, circulant matrices are proposed as good choices for FDNs due to their efficiency and versatility in practice. It is straightforward to control the eigenvalues of circulant feedback matrices, and therefore they can be optimized to yield best reverberation according to specified criteria. Finally, in section V we present applications in artificial reverberation and use of more general purpose resonators.
In section IV, we introduce the circulant FDN (CFDN) and show how CFDNs can be used to reduce computational complexity and give unique control over time-frequency behavior. In section V, we focus on applications of CFDNs in artificial reverberation and resonator design. For generality, we treat the complex case, although real numbers are typically used in practice.