Filter-length word-length tradeoffs in FIR digital filter design

We give a theorem which shows that there is a lower bound on the Chebyshev approximation error for linear-phase direct-form FIR digital filters, when the coefficients are constrained to be b-bit numbers. We then investigate the tradeoff between filter-length N and coefficient word-length b, using the product Nb as a complexity measure, for both the usual direct form and the sharpening structures of Kaiser and Hamming. The sharpening structures usually provide no overall gain in Nb product, but achieve a given performance with a smaller value of b.