In the numerical solution for nonlinear hyperbolic equations, numerical oscillation often appears and contaminates the real solution, and sometimes can even make the computation divergent. Using a signal processing approach, a dual wavelet shrinkage procedure is proposed, which allows one to extract the real solution hidden in the numerical solution with oscillation. The dual wavelet shrinkage procedure is introduced after applying the local differential quadrature (LDQ) method, which is a straightforward technique to calculate the spatial derivatives. Results free from numerical oscillation can be obtained, which can not only capture the position of shock and rarefaction waves, but also keep the sharp gradient structure within the shock wave. Three model problems, a one-dimensional dam break flow governed by shallow water equations, and the propagation of a one-dimensional and a two-dimensional shock wave controlled by the Euler equations, are used to confirm the validity of the proposed procedure.