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Signal sparse representation (SR) is an evolving research topic. The sparse coding performance is highly dependent on the properties of the system matrix that are usually hard to be guaranteed. Preconditioning is a technique that transforms a linear system into another one with more favorable properties for sparse solution. In this paper, we investigate the problem of constructing suitable preconditioner to improve the performance of the SR system. We formulate the model for designing the preconditioner by making the product of preconditioner and dictionary strictly approximate to a unit-norm tight frame (UNTF) which has been proved valid in facilitating sparse coding. A parametrization and gradient based approach is presented to solve for the UNTF-based preconditioner. Experiments on speech signals are carried out to demonstrate the performance of the proposed method.