Length limit of optimal finite wordlength FIR filters

In practical FIR digital filter applications it is often necessary to represent the filter coefficients with a finite number of bits. The optimal finite wordlength coefficients have an interesting property. For all finite wordlength filters there exists a maximum filter length NmaxNmax and the corresponding cosine polynomial degree nmaxnmax. Increasing the filter length N   beyond NmaxNmax gives additional coefficients that are all zero. A theoretical explanation for the existence of NmaxNmax is given in the paper. The influence of the filter specifications on NmaxNmax is investigated. In addition, a simple method that gives a reasonably accurate estimate of NmaxNmax is also given. Knowing NmaxNmax and its relationship to the filter specifications is important in the finite wordlength FIR design because it can reduce the time needed to compute the optimal coefficients.