Fundamental results are established for truncation order and its effect in the Volterra class of nonlinear systems. These results provide a significant insight into the analytical relationships among truncation order, truncation error, and estimation error for the output spectra of nonlinear systems, and also lead to the development of novel methods for choosing the truncation order and excitation magnitudes for accurate determination of nonlinear frequency response components. These results should be important in the kernel estimation and frequency domain analysis of nonlinear systems on the basis of the Volterra series, structure and order selection of block-oriented nonlinear systems with polynomial nonlinearity. The results also provide a much improved version of the multi-level excitation method for system analysis and identification. (c) 2012 Elsevier Ltd. All rights reserved.