Conventional source localization techniques when the source is located on a dispersive medium require both the time-of-arrival differences (TOADs) between the transducer signals and the group velocities. Furthermore, they are only practically applicable if we have a high signal-to-noise ratio (SNR). In practice, the material properties or the geometry of a medium are not fully known; therefore the group velocity is not available. The transducers' signals are usually very small or embedded in noise. In this paper, we propose a novel impact source localization method, in the case where we have the source on an elastic plate. The method is applicable even if we do not know the group velocity and we have a relatively small SNR. The group velocities are obtained by estimating a source location based on the measured TOADs. The estimated group velocities have a minimum variance at the impact source location. However, this estimation degrades as the SNR decreases. To reduce the noise effect, an exponential function is asymmetrically weighted in smoothed Wigner-Ville distributions (WVDs). Experiments and simulations are carried out to verify the validity of this technique. As a result, the proposed technique is found to be effective even in a relatively noisy environment.