Design of FIR filters as a tapped cascaded interconnection of identical subfilters

A general theory is presented for the design of linear-phase FIR digital filters as a tapped cascaded interconnection of identical FIR subfilters. The approach is an extension of the Kaiser-Hamming procedure [1] proposed for sharpening the response of an FIR filter. The new approach allows the subfilter and the tap coefficients to be simultaneously optimized to minimize either the number of subfilters for the given order of the subfilter or the subfilter order for the given number of subfilters. The optimization is based on the use of standard FIR filter design algorithms. Several examples demonstrate how the new approach leads to implementations requiring significantly fewer distinct multipliers than equivalent direct-form minimax FIR designs at the expense of a slight increase in the overall filter order. The number of distinct multipliers can be reduced to approximately
\sqrt{2.6L}
, where
L
is the order of the direct-form minimax design. Alternatively, the design of the subfilter and tap coefficients can be separated. This makes it possible to construct the subfilter so that it roughly meets the overall specifications with a highly reduced number of arithmetic operations. In this case, the tap coefficients are optimized to minimize the required number of subfilters to meet the given criteria. Even multiplier-free designs can be obtained by carefully constructing the subfilter and determining the tap coefficients. Several structures are discussed for implementing the overall filter. These structures are compared with each other and with equivalent directform minimax designs in terms of the number of distinct multipliers, overall filter order, overall multiplication rate, number of delay elements, coefficient sensitivity, and output noise variance.