Estimating the mixing time of concert halls using the eXtensible Fourier Transform

In room acoustics, we measure room impulse responses (RIRs) in order to fully describe the hall. RIRs are composed of a succession of arrivals (i.e., some sound rays which have undergone one or more reflections on their way from the source to the receiver). We propose the eXtensible Fourier Transform (XFT) in order to investigate the time evolution of spectral components of RIRs. The phase evolution versus time allows to estimate the mixing time, which is defined as the time it takes for initially adjacent sound rays to spread uniformly across the room. After presenting some properties of the XFT, we show why one must compensate the natural energy decay of the RIR in order to obtain stationary signals. We estimate mixing times for a set of experimental RIRs and several that are synthesized using a stochastic model. Then, we estimate the dependance of mixing time upon the source/receiver distance in all these RIRs. Results are consistent up to the lack of reproducibility of the sound sources, but are strongly dependent on some parameters used for computing the XFT. We finally discuss the relevance of the name mixing time with respect to the theory and in regard to the time we estimate, that we propose to call cross-over time.