This paper is concerned with stability analysis of linear discrete time-delay systems. Firstly, by JSH transformation, we developed a new necessary and sufficient condition for the asymptotic delay-dependent stability. Secondly, in order to make the designed results of this work more applicable, an unified matrix polynomial approach and numerical radius are used to get a sufficient delay-independent asymptotic stability criterion. The effectiveness and the conservatism reduction of the proposed asymptotic stability results are shown by means of three numerical examples.