  Volterra series representation is a powerful mathematical model for nonlinear
circuits. However, the difficulties in determining higher-order Volterra
kernels limited its broader applications. In this work, a systematic approach
that enables a convenient extraction of Volterra kernels from X-parameters is
presented. A concise and general representation of the output response due to
arbitrary number of input tones is given. The relationship between Volterra
kernels and X-parameters is explicitly formulated. An efficient frequency sweep
scheme and an output frequency indexing scheme are provide. The least square
linear regression method is employed to separate different orders of Volterra
kernels at the same frequency, which leads to the obtained Volterra kernels
complete. The proposed Volterra series representation based on X-parameters is
further validated for time domain verification. The proposed method is
systematic and general-purpose. It paves the way for time domain simulation
with X-parameters and constitutes a powerful supplement to existing blackbox
macro-modeling methods for nonlinear circuits.
