In this paper H-infinity, a posteriori filter (HIPF) is converted to a continuous time H-infinity (HI) filter. Then, an observer is presented which uses the gain and the state error Gramian from the continuous time HI filter (CTHF). Asymptotic stability result of the observer's error dynamics is derived using the Lyapunov energy functional. The performance of the CTHF is evaluated using numerical simulations carried out in MATLAB. These results establish the local stability of the underlying CTHF. This type of result is novel in the literature on HI theory of filters and the observers.