Implementations of adaptive IIR filters with lowest complexity

The problem of implementing adaptive IIR filters of minimum complexity is considered. The complexity is determined by the number of multiplications in the implementation of the structures generating both the adaptive filter output and the sensitivities to be used in any gradient-based algorithm. This complexity is independent of the specific adaptive algorithm used. It is established that the sensitivity generation requires a minimum of N additional states, where N is the order of the filter, and that the lowest complexity is ₃N+1 multiplications. This result is used to show minimum complexity implementations of direct-form and cascade-form adaptive IIR filters