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 filter specifications with fixed b-bit wordlength by sufficiently increasing the filter length N. For any given non-trivial specification there is a nonzero lower bound on the approximation error, below which it is 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 finite wordlength FIR filter 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 finite wordlength FIR design cases. It is also useful in the algorithm for the optimal finite wordlength design.