This work develops two fast echo signal retrieval algorithms for sonar, which are able to efficiently work in the presence of impulse noise. First, using spatial sparsity of sonar echo signal, an L21-norm optimization model with lasso form is formulated in impulse noise environment. Then, in order to recover original signal quickly, a direct gradient L21-norm algorithm is employed to solve the optimization problem, where the solution has closed form but two variables need to be initialized. Further, the original problem is transformed into a constrained minimization problem and then Lagrange multiplier method is used to find the solution. Experimental results show that the proposed two methods have similar mean square error (MSE) performance compared with traditional method using CVX optimization tool, however, computational complexities are greatly reduced.

Lagrange multiplier; Impulse noise; L21-norm; Sparse signal; Gradient