The expression

is often encountered in literature or text books where the limit-cycle bounds for digital filters or l
1norm for impulse responce sequence

is used. Several authors have tried and derived some useful upper bounds for direct form digital filters mathematically ( 1,2,3 ). In this paper, novel approximations to the sum for direct form and normal form digital filters will be given, with the only constant factor

appeared. Plots for an example are given for comparison.