On The Behavior of Delay Network Reverberator Modes

The mixing matrix of a Feedback Delay Network (FDN) reverberator is used to control the mixing time and echo density profile. In this work, we investigate the effect of the mixing matrix on the modes (poles) of the FDN with the goal of using this information to better design the various FDN parameters. We find the modal decomposition of delay network reverberators using a state space formulation, showing how modes of the system can be extracted by eigenvalue decomposition of the state transition matrix. These modes, and subsequently the FDN parameters, can be designed to mimic the modes in an actual room. We introduce a parameterized orthonormal mixing matrix which can be continuously varied from identity to Hadamard. We also study how continuously varying diffusion in the mixing matrix affects the damping and frequency of these modes. We observe that modes approach each other in damping and then deflect in frequency as the mixing matrix changes from identity to Hadamard. We also quantify the perceptual effect of increasing mixing by calculating the normalized echo density (NED) of the FDN impulse responses over time.