## Abstract

Transducer arrays combined with signal processing algorithms are called beamformers when they transmit or receive energy in a focused direction. Beamformers are widely used in radar, radio astronomy, communication, seismology, tomography, and sonar. They can, for example, enhance signal-to-noise ratio, track a moving source, scan the surrounding environment, or detect and locate an event.

Beamforming algorithms can be divided into three main categories: data independent, statistically optimum, and adaptive. This thesis deals with data-independent algorithms, which can provide computationally efficient methods to control the spatial sensitivity of a broadband microphone-array beamformer. Power consumption and hardware costs play an important role in consumer electronics and mobile telephony applications so designers want to keep these as low as possible. In this work, the ideal configuration consists of no more than four microphones fitted on the form factor of a hand-held device. However, the algorithm itself can accommodate any number of microphones and the array can be of any shape and size.

The author has derived a new steering method based on a combination of the conventional filter-and-sum FIR beamforming techniques and the well-known Farrow structure. The algorithm provides an efficient implementation of fractional delay filters using polynomial approximations of the FIR filter coefficients. The performance of this steerable beamformer is evaluated by exposing the given example designs to a simulated sound field. For comparison, a reference system is built with a set of conventional filter-and-sum beamformers, each of which is optimized for a fixed target direction.

The example design discussed here contains a Y-shaped array of four omnidirectional microphones with leg lengths of 2.54cm, a sampling rate fs = 8000Hz, FIR filters of length Nh = 21, and the Farrow degree Nq − 1 = 4. The desired output magnitude response is a frequency-invariant cardioid shape pattern over a frequency range of 300Hz – 3400Hz and steering angles of 0° ⩽ θ ⩽ 360°. It is also argued that full 3D control can be achieved by replacing the planar Y-shaped array with a 3-dimensional tetrahedron, for example, and adding a second variable in the steering function to enable elevation control.

The operation of the beamformer is validated according to the following criteria:

• computational complexity

• accuracy of the polynomial approximation

• white noise gain

Computational complexity is evaluated with the same number of microphones and equal FIR lengths in both designs. It is shown that the polynomial beamformer is computationally less complex than traditional configurations as long as the number of simultaneously processed target directions is greater than the length of the approximating polynomials, i.e. Nq = 5 for horizontal steering.

Accuracy of the polynomial approximation is evaluated by comparing the steered output response with the fixed single-direction optimized beampattern. Both the directivity index and target direction magnitude response are calculated and, regarding these measures, the difference between steered and fixed filter outputs is, on average, below ±1dB over the entire frequency range of 300Hz – 3400Hz. This difference is so small that it is barely recognizable, even when the two outputs are directly compared with one another by a trained listener.

White noise gain measures the algorithm’s sensitivity to thermal noise, which is inherently present in all analog electronics. For the Y-shaped array it mostly remains below 0dB, thus keeping the output noise at the same level as, or below the input noise. Only the low frequency end below 1kHz shows increased values up to 9dB at 300Hz. This is related to the super-directive performance inherent to small array apertures, and can be lowered, if desired, by reducing the requirements for spatial response or increasing the gain error tolerance in the optimization routine.

The most remarkable thing about this system is that it has no steering delay at all. The output is formed on a sample-by-sample basis, since the modal beamformers consist of fixed FIR filters. Therefore, any change in the control value immediately defines a new direction for the next output sample.

Beamforming algorithms can be divided into three main categories: data independent, statistically optimum, and adaptive. This thesis deals with data-independent algorithms, which can provide computationally efficient methods to control the spatial sensitivity of a broadband microphone-array beamformer. Power consumption and hardware costs play an important role in consumer electronics and mobile telephony applications so designers want to keep these as low as possible. In this work, the ideal configuration consists of no more than four microphones fitted on the form factor of a hand-held device. However, the algorithm itself can accommodate any number of microphones and the array can be of any shape and size.

The author has derived a new steering method based on a combination of the conventional filter-and-sum FIR beamforming techniques and the well-known Farrow structure. The algorithm provides an efficient implementation of fractional delay filters using polynomial approximations of the FIR filter coefficients. The performance of this steerable beamformer is evaluated by exposing the given example designs to a simulated sound field. For comparison, a reference system is built with a set of conventional filter-and-sum beamformers, each of which is optimized for a fixed target direction.

The example design discussed here contains a Y-shaped array of four omnidirectional microphones with leg lengths of 2.54cm, a sampling rate fs = 8000Hz, FIR filters of length Nh = 21, and the Farrow degree Nq − 1 = 4. The desired output magnitude response is a frequency-invariant cardioid shape pattern over a frequency range of 300Hz – 3400Hz and steering angles of 0° ⩽ θ ⩽ 360°. It is also argued that full 3D control can be achieved by replacing the planar Y-shaped array with a 3-dimensional tetrahedron, for example, and adding a second variable in the steering function to enable elevation control.

The operation of the beamformer is validated according to the following criteria:

• computational complexity

• accuracy of the polynomial approximation

• white noise gain

Computational complexity is evaluated with the same number of microphones and equal FIR lengths in both designs. It is shown that the polynomial beamformer is computationally less complex than traditional configurations as long as the number of simultaneously processed target directions is greater than the length of the approximating polynomials, i.e. Nq = 5 for horizontal steering.

Accuracy of the polynomial approximation is evaluated by comparing the steered output response with the fixed single-direction optimized beampattern. Both the directivity index and target direction magnitude response are calculated and, regarding these measures, the difference between steered and fixed filter outputs is, on average, below ±1dB over the entire frequency range of 300Hz – 3400Hz. This difference is so small that it is barely recognizable, even when the two outputs are directly compared with one another by a trained listener.

White noise gain measures the algorithm’s sensitivity to thermal noise, which is inherently present in all analog electronics. For the Y-shaped array it mostly remains below 0dB, thus keeping the output noise at the same level as, or below the input noise. Only the low frequency end below 1kHz shows increased values up to 9dB at 300Hz. This is related to the super-directive performance inherent to small array apertures, and can be lowered, if desired, by reducing the requirements for spatial response or increasing the gain error tolerance in the optimization routine.

The most remarkable thing about this system is that it has no steering delay at all. The output is formed on a sample-by-sample basis, since the modal beamformers consist of fixed FIR filters. Therefore, any change in the control value immediately defines a new direction for the next output sample.

Original language | English |
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Place of Publication | Tampere |

Publisher | Tampere University |

ISBN (Electronic) | 978-952-03-2060-7 |

ISBN (Print) | 978-952-03-2059-1 |

Publication status | Published - 2021 |

Publication type | G4 Doctoral dissertation (monograph) |

### Publication series

Name | Tampere University Dissertations - Tampereen yliopiston väitöskirjat |
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Volume | 453 |

ISSN (Print) | 2489-9860 |

ISSN (Electronic) | 2490-0028 |