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Exact Discrete-time Realizations of the Gammatone Filter

The paper derives an exact discrete-time state space realization of the popular gammatone filter. No such realization appears to be available in the literature. The proposed realization is computationally attractive: a gammatone filter with exponent N requires less than 6N multiplications and additions per sample. The integer coefficients of the realization can be computed by a simple recursion. The proposed realization also yields a closed-form expression for the frequency response. The proposed primary realization is not quite in a standard form, but it is easily transformed into another realization whose state transition matrix is in Jordan canonical form.