Computational spectrometers based on coded measurement and computational reconstruction have great application prospects. This paper proposes a computational spectrometer that has a low cost, is easy to implement in hardware, and has high reconstruction accuracy. The proposed computational spectrometer uses low-cost but highly correlated polymethyl methacrylate (PMMA) material as broadband encoding filters, which could affect spectral reconstruction accuracy. To alleviate this issue, we decoupled the sensing matrix, which is the product of the measurement matrix and sparse base matrix, and subsequently optimized the sparse base matrix independently. Enlightened by the neural network method, an over-complete dictionary was trained based on the public spectral dataset, which was used as the required sparse base matrix for reconstruction. Through this method, we achieved good reconstruction results in simulation. In experiments, the spectrometer prototype can achieve a high-resolution spectral measurements, demonstrating the feasibility of a low-cost computational spectrometer based on the trained sparse base matrix.