Doppler frequency analysis for sound-field sampling with moving microphones

Moving microphones allow for the fast acquisition of sound-field data that encode acoustic impulse responses in time-invariant environments. Corresponding decoding algorithms use the knowledge of instantaneous microphone positions for relating the dynamic samples to a positional context and solving the involved spatio-temporal channel estimation problem subject to the particular parameterization model. Usually, the resulting parameter estimates are supposed to remain widely unaffected by the Doppler effect despite the continuously moving sensor. However, this assumption raises issues from the physical point of view. So far, mathematical investigations into the actual meaning of the Doppler effect for such dynamic sampling procedures have been barely provided. Therefore, in this paper, we propose a new generic concept for the dynamic sampling model, introducing a channel representation that is explicitly based on the instantaneous Doppler shift according to the microphone trajectory. Within this model, it can be clearly seen that exact trajectory tracking implies exact Doppler-shift rendering and, thus, enables unbiased parameter recovery. Further, we investigate the impact of non-perfect trajectory data and the resulting Doppler-shift mismatches. Also, we derive a general analysis scheme that decomposes the microphone signal along with the encoded parameters into particular subbands of Doppler-shifted frequency components. Finally, for periodic excitation, we exactly characterize the Doppler-shift influences in the sampled signal by convolution operations in the frequency domain with trajectory-dependent filters.