A one-dimensional wave equation is considered with a boundary feedback controller, which is known to have exponentially stable solutions. However, the reduced models by the semi-discretized Finite Differences and Finite Elements lacks of exponential stability uniformly as the discretization parameter tends to zero. This is due to the loss of uniform gap among the high-frequency eigenvalues as the discretization parameter tends to zero. One remedy to overcome this discrepancy is the direct Fourier filtering technique where the high-frequency spurious eigenvalues are directly filtered. The exponential decay rate, mimicking the PDE counterpart, can be retained uniformly with the filtered solutions. However, the existing proof technique in the literature is solely based on an observability inequality. In this paper, exponential stability results for both filtered Finite Difference and Finite Element approximations are established by a Lyapunov-based direct approach. The maximal decay rate in terms of the filtering parameter and the optimal feedback gain is explicitly provided. Our results mimic the PDE counterpart. Several numerical tests are provided to support our results.