1 Spatio-Temporal Channels From the Wave Equation Have Butterfly Spectra Thibaut Ajdler, Student member, IEEE, Luciano Sbaiz, Member, IEEE, Andrea Ridolfi, Martin Vetterli, Fellow, IEEE. Abstract Multipath channels between moving senders and receivers are of interest in many signal pro- cessing and communications scenarios, from acoustic echo cancellation to wireless mobile commu- nications. Two physical constraints characterize these channels: (i) propagation is governed by the wave equation, (ii) movements of sender and / or receiver are smooth and slow as compared to propagation. In this paper, we show that the resulting time-varying impulse response has a butterfly shaped spectrum, that is, its spatial bandwidth increases linearly with temporal frequency. The method to show this uses smooth trajectories given by continuous-time autoregressive processes, in order to generate a suitable stochastic process corresponding to time-varying impulse responses. The power spectrum of such a process can be evaluated explicitly in certain simple cases and approximated in more general cases. The distinguishing features of the power spectrum is its essential butterfly shape and the widening of the shape as a function of the speed of movement. We specifically derive a Carson’s like approximation rule that predicts the butterfly’s bandwidth as a function of the parameters of the movement and of the channel. Experimental results in both the acoustic and the electro-magnetic case verify the established shape of the power spectra. Both the theoretical and experimental results are relevant for modeling of wideband channels (e.g. acoustic or ultra-wide band), since they predict the “variation bandwidth” as a function of temporal frequency. This in turn is important for channel adaptation, equalization and power allocation. Index Terms The work presented in this paper was supported by the National Competence Center in Research on Mobile Information and Communication Systems (NCCR-MICS), a center supported by the Swiss National Science Foundation under grant number 5005-67322. T. Ajdler, L. Sbaiz, A. Ridolfi and M. Vetterli are with the Audiovisual Communications Laboratory, EPFL, Lausanne, Switzerland (e-mail: {thibaut.ajdler, luciano.sbaiz, andrea.ridolfi, martin.vetterli}@epfl.ch). M. Vetterli is also with the Dept. of EECS, UC Berkeley, CA 94720.