Recently,　iteratively　reweighted　methods　have　attracted　much　interest　in　compressed　sensing,　outperforming　their　unweighted　counterparts　in　most　cases.　In　these　methods,　decision　variables　and　weights　are　optimized　alternatingly,　or　decision　variables　are　optimized　under　heuristically　chosen　weights.　In　this　paper,　we　present　a　novel　weighted　l1-norm　minimization　problem　for　the　sparsest　solution　of　underdetermined　linear　equations.　We　propose　an　iteratively　weighted　thresholding　method　for　this　problem,　wherein　decision　variables　and　weights　are　optimized　simultaneously.　Furthermore,　we　prove　that　the　iteration　process　will　converge　eventually.　Using　the　homotopy　technique,　we　enhance　the　performance　of　the　iteratively　weighted　thresholding　method.　Finally,　extensive　computational　experiments　show　that　our　method　performs　better　in　terms　of　both　running　time　and　recovery　accuracy　compared　with　some　state-of-the-art　methods.　©　2020,　Science　China　Press　and　Springer-Verlag　GmbH　Germany,　part　of　Springer　Nature.