As shown in Figure 3.2, the eigenbeamformer can be logically divided into to separate cascaded sections. The first step of this two-stage beamforming process can be viewed as a preprocessor to the actual forming of the output
beam which is done in the modal-beamformer second stage. As will be shown in Section 4, introducing this preprocessor step results in many advantages for the beamforming itself.
The task of this preprocessor is to transform the microphone signals into an orthogonal beam-space. Since these beams are characteristics of the sound-field in a similar way as eigenvectors are for a matrix, they are referred to as eigenbeams. Hence, the preprocessor is referred to as an eigenbeamformer.
The decomposition of the sound-field is based on the orthonormal property of the spherical harmonics. To begin, assume a rigid sphere covered with position-dependent and described by the spherical harmonic
output of such a microphone is:
a continuous pressure sensitive surface and the sensitivity of this surface be The
where represents an integration over the surface of the sphere. This result dependance as its surface sensitivity function, namely
states that the far-field directivity of the microphone has the same directional The factor introduces a phase shift and scaling that can be easily compensated and is neglected in the further discussions. Another factor is the modal coefficient It introduces a frequency dependance that must taken into account and is further investigated in Section 3.3. Yet another problem to be solved is the change from the continuous microphone aperture as it is used in (3.8) to a sampled aperture for the spherical array. This step is necessary since a) a position dependent continuous sensitivity would be extremely difficult to manufacture and b) a separate sphere for every eigenbeam would be required. The step from a continuous to a discrete aperture is described in the next section.
Note that the subscript was previously used to identify a point on the spherical surface while in (3.9) it enumerates the sensors that are located on the surface.
It is a difficult task to find a set of sensor locations that fulfill the orthonormality.
To relax the constraint from orthonormality to orthogonality, a factor introduced. This factor can be further reduced by defining:
is
The factor becomes necessary since there is no integration over a sphere for the discrete case where it becomes a sum over S sensors which are normalized by the factor 1/S. Instead of including this re-normalization in an additional factor this new normalization can be included in the spherical harmonics immediately. To avoid unnecessary confusion this approach is not pursued here.
One sensor arrangement that fulfills the constraint of discrete orthonormality up to modes of 4th-order is the center of the 32 faces of a truncated icosahe-dron. Another arrangement using only 24 sensors was found, which achieves orthogonality up to 3rd-order modes (see Section 9).
The resulting structure of the eigenbeamformer can be derived from (3.9):
for a specific eigenbeam the microphone signals are first weighted by the sampled values of the corresponding surface sensitivity, then Corrected by and finally summed. The outputs of this beamformer are the eigenbeams.
Besides the orthonormal constraint, spatial aliasing has to be considered when sampling a discrete aperture. Just as sampling a time waveform requires a minimum number of samples per time interval in order to be able to recover the original signal, sampling the spatial aperture requires a minimum number of sample locations to recover the original spatial signal distribution. Since there are spherical harmonics for a spatial resolution of order N (see Section 3.2), a minimum of sample locations are required to distinguish the spherical harmonics.
3.2 THE EIGENBEAMS
The outputs of the eigenbeamformer are a set of orthonormal beam-patterns, the eigenbeams. These eigenbeams represent a spatially orthonormal
decom-position of the sound-field, as shown in (3.4). A complete set of eigenbeams contains all spatial information about the original sound-field.
Figure 3.3 shows some example eigenbeams. From (3.6) it can be seen that the elevation dependance follows the Legendre function while the azimuth dependence has a sine-cosine dependance. The order determines the number of zeros in while two times the degree gives the number of zeros in the
application. The total number of eigenbeams up to N-th order is
For example, in a directional microphone application the maximum achievable directional gain is
The number of eigenbeams depends on the desired spatial resolution for the
To obtain a maximum directional gain of 12 dB for an arbitrary direction, one would need all eigenbeams up to third-order, which is 16 eigenbeams. As mentioned before, the number of microphones needs to be equal or larger than the number of eigenbeams. The example assumes full 3D coverage. If one is interested only in the horizontal spatial resolution the number of eigenbeams can be reduced significantly to 2N + 1.
The beampattern of the eigenbeams is frequency independent. However the magnitude shows a dependance according to the modal coefficient. This dependence will be analyzed in the next section
3.3 THE MODAL COEFFICIENTS
From (3.8), it is seen that the eigenbeams exhibit a frequency dependence according to the modal coefficient The magnitude of the frequency response for these coefficient is plotted in Fig. 3.4 for various orders
In Fig. 3.4, it can be seen that at very low frequencies the zero-th order mode is dominant. For (for a sphere of radius 5cm, this would result in a frequency of 220 Hz), the first-order mode is down by 20 dB. At higher frequencies more modes emerge. The rising slope of the modal coefficients is 6N dB per octave. Once a mode has reached an adequate level, it can be used for further processing. The level depends on the desired SNR of the overall system.
To allow subsequent combination of the eigenbeams they should have a flat frequency response. This means that the output of the eigenbeamformer should be filtered with the inverse frequency response of the modal coefficients. For low frequencies this is basically an amplification. Depending on the performance of the microphones and associated hardware and the desired SNR, the maximum gain needs to be limited to a certain threshold. The result is that each eigenbeam has a low frequency cutoff below which the eigenbeam should not be used for further processing.
The sound field around the sphere contains modes of higher orders than the array can sample. For example, at 5 kHz the sound-field around a sphere of 5 cm radius contains spherical harmonics of significant strength up to fifth-order. A spherical array with 32 microphones is able to handle spherical harmonics up to fourth-order. To enable a wideband response while avoiding aliasing, one has to provide a spatial low-pass filter. Such a low-pass filter can be implemented using microphones with large membranes or patch-microphones.
The term patch-microphone refers to microphones that cover a continuous sec-tion of the spherical surface as opposed to point microphones. By integrating the sound over a large area, higher order modes will be attenuated. Such a patch microphone might be built by using pressure sensitive materials that can be placed conformingly onto the surface of a sphere or made by combining many closely-spaced pressure microphones.