Frequency-domain investigations of cochlear stability in the presence of active elements

For a linear, time-invariant cochlear response function, the requirements of causality and stability imply the absence of poles of the spectral response function g(omega) for Im omega greater than 0, with e-i omega t as the time-dependent factor. The existence of such poles invalidate conventional Hilbert transform (dispersion) relations. The testing of these dispersion relations provides a frequency-domain check of system stability for cochlear models which contain active elements such as negative resistance. The stability of several one- and two-dimensional cochlear models with active elements is checked using these dispersion relations. A simple circuit analog of the cochlea, which contains active elements, is used to illustrate the relationship between the pole locations of g(omega) and the strength of active elements.