# Superdirective Robust Algorithms’ Comparison for Linear Arrays

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Proposed Methods

#### 2.1.1. Robust Least-Squares Frequency-Invariant Beamformer Design (RLSFIB)

#### 2.1.2. Frequency-Invariant Beam Pattern Design (FIBP)

## 3. Results

- Aperture of the array 12 cm (length for superdirective beamforming).
- 8 equi-spaced microphones (good compromise between number of microphones and final price of the array).
- Sampling frequency 16 kHz (twice the maximum frequency of the signals involved to respect the Nyquist’s theorem).
- L i.e., FIR’s filter length 128 taps.
- Frequency band of interest (100–8000) Hz (typical bandwidth for audio applications).
- Value of the speed of sound $c=340$ m/s.
- We used a grid of input data dividing the frequency bandwidth of interest $(100;8000)$ Hz in values with a constant step of 50 Hz, and the range of the angles of the direction of arrival $(-{90}^{\circ};\phantom{\rule{3.33333pt}{0ex}}{90}^{\circ})$ in values with a constant step of ${1}^{\circ}$.

#### 3.1. RLSFIB Algorithm

#### 3.2. FIBP Algorithm

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Veen, B.D.V.; Buckley, K.M. Beamforming: A versatile approach to spatial filtering. IEEE Assp Mag.
**1998**, 5, 4–24. [Google Scholar] [CrossRef] - Ward, D.B.; Kennedy, R.A.; Williamson, R.C. FIR filter design for frequency invariant beamformers. IEEE Signal Process. Lett.
**1996**, 3, 69–71. [Google Scholar] [CrossRef] [Green Version] - Ward, D.B.; Kennedy, R.A.; Williamson, R.C.; Brandstein, M. Microphone Arrays Signal Processing Techniques and Applications (Digital Signal Processing); Springer: Berlin/Heidelberg, Germany, 2001. [Google Scholar]
- Trees, H.L.V. Optimum Array Processing: Part IV of Detection, Estimation, and Modulation Theory; John Wiley & Sons, Inc.: Hoboken, NJ, USA, 2002. [Google Scholar]
- Ward, D.B.; Kennedy, R.; Williamson, R. Theory and design of broadband sensor arrays with frequency invariant far-field beam patterns. J. Acoust. Soc. Am.
**1999**, 97, 1023–1034. [Google Scholar] [CrossRef] - Mabande, E.; Schad, A.; Kellermann, W. Design of robust superdirective beamformers as a convex optimization problem. In Proceedings of the 2009 IEEE International Conference on Acoustics, Speech and Signal Processing, Taipei, Taiwan, 19–24 April 2009; pp. 77–80. [Google Scholar] [CrossRef]
- Crocco, M.; Trucco, A. Design of robust superdirective arrays with a tunable tradeoff between directivity and frequency-invariance. IEEE Trans. Signal Process.
**2011**, 59, 2169–2181. [Google Scholar] [CrossRef] - Doclo, S.; Moonen, M. Superdirective Beamforming Robust Against Microphone Mismatch. IEEE Trans. Audio Speech Lang. Process.
**2007**, 15, 617–631. [Google Scholar] [CrossRef] - Doclo, S. Multi-Microphone Noise Reduction and Dereverberation Techniques for Speech Applications. Ph.D. Thesis. Available online: ftp://ftp.esat.kuleuven.ac.be/stadius/doclo/phd/ (accessed on 23 May 2003).
- Crocco, M.; Trucco, A. The synthesis of robust broadband beamformers for equallyspaced array. J. Acoust. Soc. Am.
**2010**, 128, 691–701. [Google Scholar] [CrossRef] [PubMed] - Crocco, M.; Trucco, A. Stochastic and Analytic Optimization of Sparse Aperiodic Arrays and Broadband Beamformers With Robust Superdirective Patterns. IEEE Trans. Audio Speech Lang. Process.
**2012**, 20, 2433–2447. [Google Scholar] [CrossRef] - Trucco, A.; Crocco, M.; Traverso, F. Avoiding the imposition of a desired beam pattern in superdirective frequency-invariant beamformers. In Proceedings of the 26th Annual Review of Progress in Applied Computational Electromagnetics, Tampere, Finland, 26–29 April 2010; pp. 952–957. [Google Scholar]
- Doclo, S.; Moonen, M. Design of Broadband Beamformers Robust Against Gain and Phase Errors in the Microphone Array Characteristics. IEEE Trans. Signal Process.
**2003**, 51, 2511–2526. [Google Scholar] [CrossRef] - Mabande, E. Robust Time-Invariant Broadband Beamforming as a Convex Optimization Problem. Ph.D. Thesis. Available online: https://opus4.kobv.de/opus4-fau/frontdoor/index/index/year/2015/docId/6138/ (accessed on 10 April 2015).
- Trucco, A.; Traverso, F.; Crocco, M. Robust Superdirective End-Fire Arrays. In Proceedings of the OCEANS’13 IEEE Bergen Conference, Bergen, Norway, 10–14 June 2013; pp. 1–6. [Google Scholar] [CrossRef]
- Traverso, F.; Crocco, M.; Trucco, A. Design of Frequency-Invariant Robust Beam Patterns by the Oversteering of End-Fire Arrays. Signal Process.
**2014**, 99, 129–135. [Google Scholar] [CrossRef] - Trucco, A.; Crocco, M. Design of an Optimum Superdirective Beamformer Through Generalized Directivity Maximization. IEEE Trans. Signal Process.
**2014**, 62, 12. [Google Scholar] [CrossRef] - Trucco, A.; Traverso, F.; Crocco, M. Broadband performance of superdirective delay-and-sum beamformers steered to end-fire. J. Acoust. Soc. Am.
**2014**, 135, EL331–EL337. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Crocco, M.; Trucco, A. A Synthesis Method for Robust Frequency-Invariant Very Large Bandwidth Beamforming. In Proceedings of the 18th European Signal Processing Conference (EUSIPCO 2010), Aalborg, Denmark, 23–27 August 2010; pp. 2096–2100. [Google Scholar]

**Figure 2.**Broadside end end-of-fire steering directions (broadside orthogonal to the linear array, end-of-fire parallel to the direction of the array).

**Figure 3.**Directivity comparison: frequency-invariant beam pattern design (FIBP) blue, robust least-squares frequency-invariant beamformer design (RLSFIB) red.

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**MDPI and ACS Style**

Greco, D.; Trucco, A.
Superdirective Robust Algorithms’ Comparison for Linear Arrays. *Acoustics* **2020**, *2*, 707-718.
https://doi.org/10.3390/acoustics2030038

**AMA Style**

Greco D, Trucco A.
Superdirective Robust Algorithms’ Comparison for Linear Arrays. *Acoustics*. 2020; 2(3):707-718.
https://doi.org/10.3390/acoustics2030038

**Chicago/Turabian Style**

Greco, Danilo, and Andrea Trucco.
2020. "Superdirective Robust Algorithms’ Comparison for Linear Arrays" *Acoustics* 2, no. 3: 707-718.
https://doi.org/10.3390/acoustics2030038