Limits of finite wordlength FIR digital filter design

It has been known for some time that it is not possible to meet arbitrarily severe FIR lter specications with xed b-bit wordlength by suciently increasing the lter length N. For any given non-trivial specication there is a nonzero lower bound on the approximation error, below which i t i s not possible to go, no matter how large the value of N. For practical purposes it is even more useful to know a lower bound for given N and b. This bound represents a nite wordlength FIR lter design limit which is of theoretical importance and has not been known so far. This paper presents a method for computing this limit. The method is based on a lower bound theorem and can be used to estimate the approximation error limit in practical nite wordlength FIR design cases. It is also useful in the algorithm for the optimal nite wordlength design.