Design of FIR Filters Using Identical Subfilters of Even Length

Digital FIR Hilbert Transformers: Fundamentals and Efficient Design Methods

“…From (18), it can be seen that the length L increases inversely proportional with the transition band ωL/2π. Hence, for cases with narrow transition band and small ripple the implementation cost becomes very high.…”

“…For Hilbert transformers, the value ωL/2π is considered the transition band. (18), L is estimated as L  24.3266. For convenience, we use L = 4k+3 with k integer and the closest value for L, higher than the estimated value, is chosen.…”

“…Consider a function φ(δ, ωL), which can estimate with an acceptable exactitude the length of a HT in terms of its ripple δ and its lower passband edge ωL, such as the one presented in (18). We have, for the prototype filter and the subfilter,…”

“…The Frequency Transformation (FT) method, proposed first in [17] and extended in [18], was modified to design FIR Hilbert transformers in [12] based on a tapped cascaded interconnection of repeated simple basic building blocks constituted by two identical subfilters. Taking advantage of the repetitive use of identical subfilters, the recent proposal [13] gives a simple and efficient method to design multiplierless Hilbert transformers, where a combination of the FT method with the Pipelining-Interleaving (PI) technique of [19] allows getting a time-multiplexed architecture which only requires three subfilters.…”