This paper investigates the design of the asynchronous $l_2-l_\infty$ filter for discrete-time switched systems under admissible edge-dependent integrated dwell time(AED-IDT). Different from the previous research results, it is the first time to propose the asynchronous transferring-dependent convex Lyapunov function to obtain the less conservative stability criteria. The novelty of the Lyapunov function is that it is constructed in a way that does not depend on the system mode, but depend on the currently activated filter mode and the freshly deactivated filter mode. Given the convex property of the designed Lyapunov function, this provides the possibility for the proposed method to improve the degree of freedom and flexibility. By using the transferring-dependent convex Lyapunov function and AED-IDT switching strategies, the sufficient conditions are derived to ensure the global uniform exponential stability and $l_2-l_\infty$ performance of the filtering error system. Furthermore, the method of constructing asynchronous $l_2-l_\infty$ filter is given. Finally, a numerical example and a switched RLC application circuit are provided to demonstrate the potential and validity of the obtained results. The simulation results verify that the proposed design approach yields tighter dwell time bounds and smaller filtering error.