A new exact method of measuring the Volterra kernels of finite-order discrete nonlinear systems is presented. The kernels are rearranged in terms of multivariate cross-products in vector form. The one-, two-,..., and l-dimensional kernel vectors are determined using a deterministic multilevel sequence with l distinct levels at the input of the system. It is shown that the defined multilevel sequence with l distinct levels is persistently exciting for a truncated Volterra filter with nonlinearities of polynomial degree l.. Examples demonstrating the rearrangement of the Volterra kernels and a novel method for estimation of the kernels are presented. Simulation results are given to illustrate the effectiveness of the proposed method.