# An Efficient Framework for Estimating the Direction of Multiple Sound Sources Using Higher-Order Generalized Singular Value Decomposition

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

#### 2.1. Data Model

#### 2.2. Basic Assumptions

#### 2.3. Transformation Problem

## 3. Proposed Method

#### 3.1. Problem for Estimating the Transformation Matrix and Its Solution

**Lemma**

**1.**

**Proof.**

**Theorem**

**1.**

**Proof.**

#### 3.2. Estimation of the Transformation Matrices by HOGSVD

#### 3.3. DOA Estimation Scheme

#### 3.3.1. DOA Estimation Scheme via MUSIC

#### 3.3.2. DOA Estimation Scheme via ESPRIT

## 4. Numerical Simulations

^{®}R2017a, using PC with Debian GNU/Linux 9.4 × 86_64, Intel

^{®}Core

^{™}i5-4590 CPU 3.30 GHz, 16G RAM, Intel

^{®}Math Kernel Library 11.3.1 on BLAS and LAPACK 3.5.0. Each scenario is repeated 100 times, and simulation parameters are chosen as follows: sampling frequency is 48 kHz, an output of each microphone is captured at 1 s, speed of sound c is 343 m/s, the spacing of microphone elements d is 5 cm, STFT focusing frequency range is from 0.1 to 16 kHz, the reference frequency ${f}_{\mathrm{o}}$ is 3.43 kHz. Note that we used perturbations of the true angles by adding Gaussian random noise.

#### 4.1. Scenario 1: Performance with Respect to Source Types

#### 4.2. Scenario 2: Performance with Respect to the Number of Microphone Elements

#### 4.3. Scenario 3: Performance with Considering Automatic Pairing

#### 4.4. Scenario 4: Performance under Reverberation Environment

#### 4.5. Computational Complexity

## 5. Experimental Results

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Proof of Theorem 1

## References

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**Figure 2.**RMSE estimation performance versus SNR on Scenario 1; (

**a**) three different human speeches, and (

**b**) three uncorrelated musical sounds where six microphones are employed each subarray.

**Figure 3.**SD estimation performance versus SNR on Scenario 1; (

**a**) three different human speeches, and (

**b**) three uncorrelated musical sounds where six microphones is employed each subarray.

**Figure 4.**RMSE estimation performance versus SNR on Scenario 2; three human speeches are employed and the number of microphone elements each subarray on (

**a**) $N=4$, (

**b**) $N=8$, and (

**c**) $N=12$.

**Figure 5.**SD estimation performance versus SNR on Scenario 2; three human speeches are employed and the number of microphone elements each subarray on (

**a**) $N=4$, (

**b**) $N=8$, and (

**c**) $N=12$.

**Figure 8.**Performance evaluations of Scenario 4; (

**a**) RMSE estimation performance versus SNR, and (

**b**) SD estimation performance versus SNR, where three uncorrelated human speeches are employed along with a reverberant environment. The reverberations were simulated by the following procedure [53], where dimensions of enclosure room is $15\times 15\times 5$ m, a measurement protocol of reverberation time is RT60, and wall absorption coefficients are followed on Table 3.

**Figure 9.**Computational complexities; (

**a**) changing the number of microphone elements each subarray N, and (

**b**) the number of microphone elements including all subarray M where the number of incident sources $K=3$.

Command Name | Command Counts | |
---|---|---|

HOGSVD in Equation (20) | Optimized HOGSVD in Equation (32) | |

Matrix Addition/Subtraction | $P(P-1)-1$ | $P-1$ |

Element-wise Multiplication | 1 | 0 |

Matrix Multiplication | $3P(P-1)$ | $P+{\left\{1\right\}}^{\ast}$ |

Matrix Inversion | $P(P-1)$ | P |

QR Decomposition | 0 | 1 |

Eigenvalue Decomposition (EVD) | 1 | 1 |

Command Name | Complex Floating Point Operations per Command |
---|---|

Matrix Addition/Subtraction | ${M}^{2}$ |

Element-wise Multiplication | ${M}^{2}$ |

Matrix Multiplication | $2{M}^{3}-{M}^{2}$ |

Matrix Inversion (Gauss-Jordan elimination) | $\frac{2}{3}{M}^{3}+\frac{3}{2}{M}^{2}-\frac{7}{6}M$ |

QR Decomposition (Householder transformation) | $(2P-\frac{2}{3}){M}^{3}-2P{M}^{2}+\frac{2}{3}M$ |

HOGSVD in Equation (20) without counting EVD | $\frac{20P(P-1)}{3}{M}^{3}-\frac{P(P-1)}{2}{M}^{2}-\frac{7P(P-1)}{6}M$ |

Optimized HOGSVD in Equation (32) without counting EVD | $\frac{(14P+4)}{3}{M}^{3}-\frac{(P+4)}{2}{M}^{2}-\frac{(7P-4)}{6}M$ |

**Table 3.**Wall absorption coefficients at various reverberation time in Scenario 4 [53].

Reverberation Time based on RT60 (Millisecond) | Axial Wall Plane | |||||
---|---|---|---|---|---|---|

Positive Direction | Negative Direction | |||||

$\mathit{x}-\mathit{z}$ | $\mathit{x}-\mathit{z}$ | $\mathit{x}-\mathit{y}$ | $\mathit{x}-\mathit{z}$ | $\mathit{x}-\mathit{z}$ | $\mathit{x}-\mathit{y}$ | |

200 | 0.7236 | 0.2021 | 0.6844 | 0.0792 | 0.2436 | 0.5586 |

300 | 0.7142 | 0.1687 | 0.7666 | 0.2650 | 0.2387 | 0.7043 |

400 | 0.7306 | 0.0555 | 0.7731 | 0.4091 | 0.8493 | 0.8587 |

500 | 0.5064 | 0.4974 | 0.8248 | 0.4189 | 0.8069 | 0.7572 |

600 | 0.6074 | 0.6299 | 0.8028 | 0.7599 | 0.6373 | 0.8209 |

700 | 0.7442 | 0.7624 | 0.8734 | 0.6922 | 0.6480 | 0.7893 |

800 | 0.6779 | 0.6827 | 0.7865 | 0.8045 | 0.8386 | 0.8430 |

900 | 0.6992 | 0.7111 | 0.7741 | 0.8752 | 0.8233 | 0.9081 |

1000 | 0.7622 | 0.7707 | 0.9394 | 0.8248 | 0.8192 | 0.8398 |

Hardware Type/Parameter | Specification/Value |
---|---|

Audio Interface | Roland^{®} Octa-capture (UA-1010) |

Sampling Frequency | 48,000 Hz |

Microphone Name | Behringer^{®} C-2 studio condenser microphone |

Number of Microphones | 8 |

Pickup Patterns | Cardioid (8.9 mV/Pa; 20–20,000 Hz) |

Diaphragm Diameter | 16 mm |

Equivalent Noise Level | 19.0 dBA (IEC 651) |

SNR Ratio | 75 dB |

Microphone Structure | L-shaped Array |

Spacing of Microphone | 9 cm |

**Table 5.**Performance evaluation on Experiment 1. The boldfaced results highlight the optimal minimum RMSE.

Incident Sources | RMSE of DOAs (Degree) | ||||||||
---|---|---|---|---|---|---|---|---|---|

Number | Position | Angle (Degree) | IMUSIC | TOFS | TOPS | Squared TOPS | WS-TOPS | Proposed Method with MUSIC | Proposed Method with ESPRIT |

1 | ${\varphi}_{1}$ | 96 | 0.3050 | 0.2050 | 1.0950 | 1.3350 | 0.5600 | 0.7750 | 0.7074 |

${\theta}_{1}$ | 86 | 0.5400 | 1.2600 | 1.2750 | 2.0150 | 0.6850 | 0.5700 | 0.6915 | |

Average | 0.4225 | 0.7325 | 1.1850 | 1.6750 | 0.6225 | 0.6725 | 0.6995 | ||

2 | ${\varphi}_{1}$ | 65 | 1.1857 | 1.7286 | 20.0143 | 28.5857 | 37.8714 | 1.5000 | 2.0284 |

${\theta}_{1}$ | 150 | 9.6000 | 6.6857 | 26.3571 | 39.7857 | 88.2000 | 8.8143 | 8.6800 | |

${\varphi}_{2}$ | 55 | 1.0714 | 1.6857 | 22.2571 | 19.4000 | 32.2429 | 2.9714 | 3.8695 | |

${\theta}_{2}$ | 100 | 8.3714 | 8.3857 | 5.0143 | 6.7857 | 60.2286 | 6.6714 | 3.1630 | |

Average | 5.0571 | 4.6214 | 18.4107 | 23.6393 | 54.6357 | 4.9893 | 4.4353 | ||

3 | ${\varphi}_{1}$ | 58 | 2.1400 | 2.3900 | 46.5500 | 52.8100 | 40.9500 | 3.6600 | 4.0334 |

${\theta}_{1}$ | 55 | 55.0000 | 55.0000 | 55.0000 | 55.0000 | 55.0000 | 9.4300 | 4.1057 | |

${\varphi}_{2}$ | 100 | 1.8400 | 2.0000 | 41.5700 | 62.4000 | 70.9100 | 1.8700 | 2.4554 | |

${\theta}_{2}$ | 95 | 95.0000 | 83.4200 | 52.4500 | 71.4800 | 95.0000 | 9.7700 | 5.8638 | |

${\varphi}_{3}$ | 130 | 10.9300 | 11.8900 | 28.8300 | 32.2800 | 95.2400 | 8.2500 | 6.9071 | |

${\theta}_{3}$ | 120 | 26.9800 | 25.8400 | 16.1200 | 18.0100 | 91.2800 | 5.9400 | 7.3165 | |

Average | 31.9817 | 30.0900 | 40.0867 | 48.6633 | 74.7300 | 6.4867 | 5.1137 |

**Table 6.**Performance evaluation on Experiment 2. The boldfaced results highlight the optimal minimum RMSE.

Incident Sources | RMSE of DOAs (Degree) | ||||
---|---|---|---|---|---|

Number | Position | Angle (Degree) | 2D-IMUSIC | 2D-TOFS | Proposed Method with 2D-MUSIC |

1 | ${\varphi}_{1}$ | 96 | 0.9000 | 0.9000 | 0.9000 |

${\theta}_{1}$ | 86 | 0.4000 | 1.0500 | 0.7500 | |

Average | 0.6500 | 0.9750 | 0.8250 | ||

2 | ${\varphi}_{1}$ | 57 | 0.9500 | 1.1500 | 1.1000 |

${\theta}_{1}$ | 91 | 1.0500 | 1.8000 | 1.7000 | |

${\varphi}_{2}$ | 139 | 4.9500 | 5.2000 | 5.4500 | |

${\theta}_{2}$ | 96 | 3.1500 | 3.3000 | 2.0500 | |

Average | 2.5250 | 2.8625 | 2.5750 | ||

3 | ${\varphi}_{1}$ | 48 | 0.9500 | 1.5500 | 1.9500 |

${\theta}_{1}$ | 86 | 1.4500 | 0.8000 | 2.4500 | |

${\varphi}_{2}$ | 98 | 0.9000 | 1.8000 | 1.1500 | |

${\theta}_{2}$ | 95 | 1.4500 | 2.1500 | 2.6000 | |

${\varphi}_{3}$ | 152 | 2.7000 | 2.4000 | 5.9000 | |

${\theta}_{3}$ | 95 | 4.5000 | 3.9000 | 1.4500 | |

Average | 1.9917 | 2.1000 | 2.5833 | ||

4 | ${\varphi}_{1}$ | 100 | 5.8095 | 6.5238 | 3.2857 |

${\theta}_{1}$ | 94 | 2.4286 | 2.6190 | 1.6667 | |

${\varphi}_{2}$ | 51 | 1.2381 | 1.0952 | 2.5714 | |

${\theta}_{2}$ | 95 | 0.5714 | 0.6667 | 1.3333 | |

${\varphi}_{3}$ | 134 | 1.9524 | 1.8571 | 3.9524 | |

${\theta}_{3}$ | 103 | 10.0952 | 10.2857 | 9.2857 | |

${\varphi}_{4}$ | 153 | 7.4762 | 7.8095 | 7.8571 | |

${\theta}_{4}$ | 89 | 4.7143 | 4.7143 | 5.3810 | |

Average | 4.2857 | 4.4464 | 4.4167 |

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**MDPI and ACS Style**

Suksiri, B.; Fukumoto, M.
An Efficient Framework for Estimating the Direction of Multiple Sound Sources Using Higher-Order Generalized Singular Value Decomposition. *Sensors* **2019**, *19*, 2977.
https://doi.org/10.3390/s19132977

**AMA Style**

Suksiri B, Fukumoto M.
An Efficient Framework for Estimating the Direction of Multiple Sound Sources Using Higher-Order Generalized Singular Value Decomposition. *Sensors*. 2019; 19(13):2977.
https://doi.org/10.3390/s19132977

**Chicago/Turabian Style**

Suksiri, Bandhit, and Masahiro Fukumoto.
2019. "An Efficient Framework for Estimating the Direction of Multiple Sound Sources Using Higher-Order Generalized Singular Value Decomposition" *Sensors* 19, no. 13: 2977.
https://doi.org/10.3390/s19132977