Abstract: To solve the underdetermined noise equations, high-dimensional discrete signals need to be reconstructed from noisy and sampled data. The zero-space characteristics of the perception matrix of compressed sensing theory ensure that the sparse representation can be restored through l₁ minimization, which can be achieved by convex optimization methods or estimation theories. On the basis of Kalman filtering mode algorithm (KML1 algorithm), the external threshold method with Aitken-based delta-squared is used to improve it. An improved Kalman filter mode acceleration algorithm (acceleration algorithm) was proposed. It can be used in speech signal reconstruction. Experimental results show that after 500 iterations, the solution reconstructed by the KML1 algorithm is basically consistent with the true value, and after 100 iterations, the solution reconstructed by acceleration algorithm is basically consistent with the true value. And compared with the traditional orthogonal matching tracking (OMP) algorithm, and the recovery time of the acceleration algorithm is nearly 20 times smaller than that of the OMP algorithm. The Kalman filter through l₁ minimization with external thresholds provides \vec xshorter time and higher accuracy for reconstruction.