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In this paper, the asymptotic waveform evaluation (AWE) technique in conjugation with higher-order basis functions method based on the best uniform approximation theory is applied to solve scattering problems over a broad frequency band. To select the best uniform approximation higher-order basis functions, 3-order Chebyshev basis function is used as an example and applied to solving the electrical field integral equation (EFIE) by the method of moments(MOM). Finally, using AWE technique to solving the discrete linear system obtained by high order basis function to accomplish the scattering analysis of perfectly electrical conductor (PEC) over a broad frequency band. It is shown that the presented method has higher computational precision with lower discretization, and greatly improve the computational efficiency.