Towards Gridless Sound Field Reconstruction

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Sound pressure varies over space and time. Knowledge about this exact behavior has many applications, e.g., room compensation, dereverberation and sound field reconstruction. Inside enclosures, the sound field is influenced by the surroundings, such as the geometry of the enclosure and the materials used.
Reconstructing a satisfying sound field in the whole enclosure by extrapolating from few measurements is thus not an obvious task. The sound field in a room can be represented by a weighted sum of room modes. Thus, we can estimate the room modes and compute the sound field from it. To estimate the room modes, compressive sensing literature uses on-the-grid, sparse reconstruction methods. However, these on-the-grid methods are known to suffer from basis mismatch. In this work, we investigate the use of a gridless framework for estimating room modes using atomic norm minimization, a gridless method. The advantage of this approach is that it does not suffer from this basis mismatch problem.
We derive a bound for the sound field reconstruction problem in a one-dimensional room with rigid walls and relate this to the frequency separation that is required by the atomic norm. We conclude that for perfect reconstruction of the room modes based on the investigated gridless approach, additional
prior knowledge about the signal model is required. For example, knowledge about the shape of the room modes can be used. We show how recovery is possible in a one-dimensional setting by exploiting both the structure of the sound field and the acquisition method.