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Part of the book series: Springer Topics in Signal Processing ((STSP,volume 9))

Abstract

The process of combining signals acquired by a microphone array in order to ‘focus’ on a signal in a specific direction is known as beamforming or spatial filtering. This chapter considers signal-independent (fixed) beamformers, controlled by weights only dependent on the direction of arrival of the source to be extracted, and which do not otherwise depend on the desired signal. Because the weights of these beamformers are given by simple expressions, they present the advantages of being straightforward to implement and of having low computational complexity.

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Notes

  1. 1.

    The dependency on time is omitted for brevity. In practice, the signals acquired using a spherical microphone array are usually processed in the short-time Fourier transform domain, as explained in Sect. 3.1, where the discrete frequency index is denoted by \(\nu \).

  2. 2.

    We use the complex conjugate weights \(W^*_{lm}\) rather than the weights \(W_{lm}\); this notational convention originates in the spatial domain [30].

  3. 3.

    If the real SHT is applied instead of the complex SHT, the complex spherical harmonics \(Y_{lm}\) used throughout this chapter should be replaced with the real spherical harmonics \(R_{lm}\), as defined in Sect. 3.3.

  4. 4.

    It should be noted that this simplified expression is only valid for beamformers that satisfy the distortionless constraint given in (6.5). It therefore does not apply to the plane-wave decomposition beamformer presented in Sect. 6.3.1.1, which satisfies a scaled version of this constraint.

  5. 5.

    Note that in some publications, such as [28], \(\mathcal {B}(k,\varOmega )\) is referred to as the beam pattern, and its square magnitude is referred to as the power pattern .

  6. 6.

    The main lobe width is sometimes also defined as the width of the region where the beam pattern is no less than half of its maximum value, or equivalently, no more than 3 dB below its maximum value.

  7. 7.

    This expression is identical to (12) in [22] if we substitute \(d_n = 1\).

  8. 8.

    This expression is identical to (11) in [22] if we substitute \(d_n = 1\), with the exception of the \((4 \pi )^2\) scaling factor, which is required due to the fact that in [22] a \(4 \pi \) scaling factor is included in the definition of the mode strength.

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Jarrett, D.P., Habets, E.A.P., Naylor, P.A. (2017). Signal-Independent Array Processing. In: Theory and Applications of Spherical Microphone Array Processing. Springer Topics in Signal Processing, vol 9. Springer, Cham. https://doi.org/10.1007/978-3-319-42211-4_6

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