MFIR Filters: Properties and Applications

The class of multiplicative FIR (MFIR) filters is considered and some properties and applications of MFIR filters are investigated. MFIR filter approximation of a given IIR filter is shown to reduce the number of multiplications and additions logarithmically, in comparison to the corresponding FIR filter in direct form. The pure multiplicity property is introduced and is shown to apply to a class of MFIR filters. This property results in a criterion for optimal ordering and expressions for roundoff noise when no scaling is used and also results in the in-variance of roundoff noise output under l2-scaling. Linear phase MFIR filter realization of a desired low-pass frequency response magnitude |Hd| with centered transition band is shown to require 0(n log2 N) each multipliers and adders. N is the order of the min-max FIR filter design of |Hd| and n is the order of the elliptic IIR filter design of |Hd|1/2. Several design examples of linear phase low-pass filters are used to compare MFIR filter designs versus those of min-max FIR filters in direct form. Comb filters of order N are shown to have an exact MFIR realization that requires fewer than 2 log2 N additions. Suggestions for further research and applications conclude the paper.