A “3-channel” FDN feedback matrix can be represented as:
The inner loop calculations of the FDN shown in Fig. 12 can then expressed as:
and the loop output given by
These expressions can also be written in frequency-domain vector notation as
where
The matrix
is called the state transition matrix. is typically a diagonal matrix of lowpass filters, each having gain no greater than 1.
Stability of the FDN is assured when the norm of the state vector decreases over time when the input signal is zero:
for all , where
Stable feedback matrices can thus be parameterized in terms of
, where is any orthogonal matrix and is a diagonal matrix having entries less than 1 in magnitude.
A feedback matrix is lossless if and only if its eigenvalues have modulus 1 and its eigenvectors are linearly independent.