Implementations of adaptive IIR filters with lowest complexity

The problem of implementing adaptive IIR filters of minimum complexity is considered, The complexity used here is 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. This result is used to show a minimum complexity of 3N + 1 multiplications for an order N filter. Principles to use in the construction of such lowest complexity implementations are provided, and examples of minimum complexity direct-form, cascade-form, and parallel-form adaptive IIR Biters are given.