A multiplicative realization of FIR systems that is logarithmically efficient

A simple mathematical identity results in an expansion of the transfer function matrices of linear systems into an infinite product. We apply this expansion to realize an FIR approximation of a given IIR structure with a number of multipliers that is proportional to the logarithm to the base 2 of the length of the FIR required. This is a fundamental reduction in the computational complexity and has far-reaching implications for other applications such as on-line convolution. Also, it now becomes feasible to realize exponentially longer FIR structures using the new method which also allows design flexibility since many of the terms involved could be realized with slow multipliers. An additional advantage of the fewer number of multipliers used is the reduction of round-off noise as we demonstrate in this work.