In this paper, a spectral finite element method (SFEM) based on the alternating frequency–time (AFT) framework is extended to study impact wave propagation in a rod structure with a general material nonlinearity. The novelty of combining AFT and SFEM successfully solves the computational issue of existing nonlinear versions of SFEM and creates a high-fidelity method to study impact response behavior. The validity and efficiency of the method are studied through comparison with the prediction of a qualitative analytical study and a time-domain finite element method (FEM). A new analytical approach is also proposed to derive an analytical formula for the wavenumber. By using the wavenumber equation and with the help of time–frequency analysis techniques, the physical meaning of the nonlinear behavior is studied. Through this combined effort with both analytical and numerical components, distortion of the wave shape and dispersive behavior have been identified in the nonlinear response. The advantages of AFT-FEM are (1) high-fidelity results can be obtained with fewer elements for high-frequency impact shock response conditions; (2) dispersion or dissipation is not erroneously introduced into the response as can occur with time-domain FEM; (3) the high-fidelity properties of SFEM enable it to provide a better interpretation of nonlinear behavior in the response; and (4) the AFT framework makes it more computationally efficient when compared to existing nonlinear versions of SFEM which often involve convolution operations.