In this talk, I present a novel framework to do black-hole spectroscopy. This approach is based on a new technique so-called “quasinormal-mode filter”, which can be classified into two subsets: rational filter and full filter. On the theoretical level, I explain how to use the rational filter to understand the ringdown of numerical-relativity waveforms. On the observational level, I introduce a way to incorporate the filter into Bayesian analysis. The new Bayesian framework not only allows us to analyze the ringdown of a real gravitational-wave event without Markov chain Monte Carlo, but also yields a natural estimate of the ringdown start time. By applying our method to GW150914, we find strong and self-consistent evidence for the first overtone from multiple perspectives. On the other hand, the relationship between the full filter and the metric reconstruction is discussed. Its connection to a numerical-relativity technique “Cauchy-characteristic Matching” is provided. In the final part of the talk, I also briefly present our recent progress on fully relativistic 3D Cauchy-characteristic Matching.