Abstract
We all know from our own previous work and other works that pressure differences among microphones is another fundamental way to looking at beamforming, especially when sensors are close to each others. Conventional beamforming does not include this important information in its formulation and in an explicit way. Although, there are different manners to derive differential beamformers, none of the developed and available approaches do it in an elegant and systematic way. This is what we attempt to do in this chapter in the simplest case, which is based on first-order linear difference equations.
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Notes
- 1.
Notice that the last diagonal element of A, i.e., A M, is completely irrelevant since it vanishes in all equations but, for convenience, it is included in the definition of the matrix.
- 2.
Another way to look at this is to observe that the dimension of the nullspace of the matrix \({\mathbf {J}}_{-1} \left ( \mathbf {A} - \mathbf {S} \right )\) is exactly equal to 1. Therefore, the elements of A must be found in such a way that \({\mathbf {d}}_{\theta _0}\) is a vector of this nullspace.
- 3.
Notice that in the particular case of θ n = θ 0, we will have a null of multiplicity 2 in the direction θ 0.
References
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Benesty, J., Cohen, I., Chen, J. (2021). Beamforming with First-Order Linear Difference Equations. In: Array Beamforming with Linear Difference Equations. Springer Topics in Signal Processing, vol 20. Springer, Cham. https://doi.org/10.1007/978-3-030-68273-6_3
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DOI: https://doi.org/10.1007/978-3-030-68273-6_3
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