What's the principle of Power Beamforming?

Power Beamforming (PBF) is another frequency-based mapping method in NoiseImage. It combines several advanced algorithmic beamforming approaches in one operating principle. Using eigenvalue decomposition of the cross-spectral matrix (CSM), the acoustic map is divided into single (pseudo) sources and their point-spread functions (PSF). An exponentiation of the PSF while applying opposite manipulation to the sound source strengths results in the attenuation of the side lobes and sharpens the main lobes. This in turn results in an increase of dynamic and a sharpening of the source representation.

The algorithmic method the calculation is based on is determined by the exponent \(ν\) which is selected with a slider. The further the slider is moved to the left, the more the sources are sharpened and the dynamic are increased. To obtain meaningful results, however, a sufficiently large time range must be selected.

  • Slider far left:
    \(ν=-1\) corresponds to adaptive beamforming (Minimum Variance Beamforming1)
  • Slider in the left half:
    \(-1<1/ν<0\) with defined -2,-4,-8,-16 and -32 is here called inverse functional beamforming
  • Slider in middle position:
    \(ν=∞\) is here called asymptotic beamforming
  • Slider in the right half:
    \(0<1/ν<1\) with defined 2,4,8,16 and 32 corresponds to functional beamforming
  • Slider far right:
    \(ν =1\) corresponds to standard beamforming based on the cross spectral matrix

Compared to the robust standard beamforming, the most aggressive stage of the adaptive beamforming method can deliver best results, however, is in terms of artefacts more susceptible to e.g. too high frequencies or time domains selected too short. Small artefacts require high accuracy of the steering vectors which in turn may require the more accurate determination of the focus. Therefore, in most practical cases, the ideal adaptive-beamforming stage often lies between these.

Application Example: Vacuum Cleaner Measurement with Fibonacci AC Pro with 120 Microphones

Using a vacuum cleaner as an example, some results obtained are shown in the frequency range of 5.6 kHz to 7.0 kHz (6300 Hz third octave) and at a dynamic range of 30 dB. If the exponent is increased, a sharpening of the most important main sound sources with simultaneous suppression of the side lobes can be seen. This leads to an improved localization of the sound sources and increases the dynamic of the image.

Figure 1: Adaptive Beamforming at &nu;=-1
Figure 1: Adaptive Beamforming at ν=-1
Figure 2: Inverse functional beamforming at &nu;=-4
Figure 2: Inverse functional beamforming at ν=-4
Figure 3: Inverse functional beamforming at &nu;=-32
Figure 3: Inverse functional beamforming at ν=-32
Figure 4: Asymptotic beamforming at &nu;→∞
Figure 4: Asymptotic beamforming at ν→∞
Figure 5: Functional beamforming at &nu;=32
Figure 5: Functional beamforming at ν=32
Figure 6: Functional beamforming at &nu;=4
Figure 6: Functional beamforming at ν=4
Figure 7: Standard beamforming at &nu;=1
Figure 7: Standard beamforming at ν=1

Further Reading

Johnson, D.H. und Dudgeon, D.E. (1993). Array Signal Processing: Concepts and Techniques. Prentice Hall.
Dougherty, R.P. (2018). A New Derivation of the Adaptive Beamforming Formula. Berlin Beamforming Conference. Berlin.
Dougherty, R.P. (2014). Functional Beamforming. Berlin Beamforming Conference. Berlin.
Puhle, C. (2021). Demonstration of a unified approach to beamforming. Internoise 2021. Washington D.C.

Visit the website Berlin Beamforming Conference held by GFaI e. V. https://www.bebec.eu.

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