# Study of Generalized Phase Spectrum Time Delay Estimation Method for Source Positioning in Small Room Acoustic Environment

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Ideal Propagation Model

_{0}(t) be the signal emitted by the source. Then the signals of the receivers will be

_{a}, τ

_{b}are lag values; α

_{a}, α

_{b}are signal attenuation coefficients; n

_{A}(t), n

_{B}(t) are random uncorrelated additive microphone noises. The values of τ

_{a}, τ

_{b}are determined by the geometric distances r

_{a}, r

_{b}from the signal source to the corresponding receiver

_{a}, α

_{b}can be caused by various factors, however, in the simplest ideal case, exclusively source beam pattern and the scattering of the sound wave are considered and, so

_{ab}= τ

_{b}− τ

_{a}which is used further to determine the position of the sound source. Using the notations above and having redefined t = t − τ

_{b}, we can rewrite (1)

#### 2.2. Reverberation Model

_{a}(t), h

_{b}(t) are room impulse response (RIR) functions. The complexity of application of (5) is in the practical difficulty of RIR determination. Acoustic measurements [18] or mathematical methods can be used to solve this problem. The image model method, first proposed in [19], is the most widespread among the latter. Alternatively, statistical methods [20] or methods based on geometric acoustics and ray tracing [21] can be used. To create realistic sound signals in this work, the image model method was used in the implementation of Lehman, Johansson and Nordholm [22,23].

#### 2.3. Basic Phase Shift TDE

_{ab}of two signals. The algorithm for constructing the cross-phase spectrum is known from spectral analysis [14]. At the initial stage, the Fourier transforms S

_{a}(f

_{k}) and S

_{b}(f

_{k}) of the signals of each of the channels are determined

_{a}(t

_{i}) and s

_{b}(t

_{i}) are series of N real samples of s

_{a}(t) and s

_{b}(t) signals sampled with an interval Δ; F

_{D}is the operator of short-time discrete Fourier transform (DFT); S

_{a}(f

_{k}) and S

_{b}(f

_{k}) are spectrums of the signals.

_{k}) are calculated

_{q}= Δ∙N∙q of the beginning of the q-th time window; * is the element-wise complex conjugation; × is the element-wise product. The final measurement of the cross-spectrum S

_{ab}(f

_{k}) is obtained by averaging the Q instantaneous spectrums

_{ab}(f

_{k}) would not be correct. However, this assumption is normally relevant for the cross-spectrum. If we consider that neither source nor sensors are moving, the phase shift for each particular harmonic component will remain the same for all Q instantaneous spectrums. Therefore, coherent accumulation is applied this way to reduce the impact of the additive random noise.

_{ab}$($f

_{k}) is finally calculated

_{0}(t) will also be present in s

_{a}(t) and s

_{b}(t). In this case, the phase difference between the k-th harmonic components of s

_{a}(t) and s

_{b}(t) is determined by τ

_{ab}∙f

_{k}. Therefore, the estimation τ

_{ab}can be obtained as the coefficient of proportionality in the line equation of the approximating Φ

_{ab}$($f

_{k}).

_{0}(t).

#### 2.4. Generalized Phase Spectrum TDE

_{k}) which is used to determine ${\widehat{\tau}}_{ab}$. Similarly to (10), the weighted error in this case are introduced

_{k}) should be chosen in the way that its value is high if the useful signal prevails over noises at the f

_{k}frequency and differs little from zero in other cases. A set of five frequency weighting functions was investigated in [14]. Table 1 below shows the calculation formulas for these functions.

^{2}

_{ab}(f

_{k}) widely used for this purpose is calculated as

_{ab}in linear regression. This feature is practically important and will be addressed later. As far as W(f

_{k}) is based on spectral estimations, the generalized method should be applied carefully for signals that are non-stationary.

## 3. Results and Discussion

_{d}= 44,100 Hz).

#### 3.1. Experimental Setting

_{k}ϵ [100 Hz, 850 Hz]. The second set contained four non-overlapping frequency bands shown below. The choice of such frequency intervals was carried out in accordance with the form of power density spectrum of the raw signal shown in Figure 2. The presented characteristic was obtained by averaging all instantaneous power density spectrums with a window of N = 4096 samples. The position of the cut-off level was chosen empirically to optimize the TDE operation in the absence of reverberations. It should be noted that the power density spectrum for different speakers or even for different speech fragments by this speaker would not remain the same. However, the proposed procedure will remain applicable regardless.

#### 3.2. Simulation of the Small Room Environment

_{a}(t), h

_{b}(t). The MATLAB program prepared by Eric Lehman [22] was used to obtain these characteristics. When calculating the RIR, the room parameters and the configuration of the sensors were specified as shown in Figure 3. The dimensions of the room were 5 × 3.5 × 2.25 m. The source has coordinates (1.5, 2.75, 1.8), and the microphones (4.5, 1.25, 1.8) and (4.5, 2.25, 1.8).

_{60}) was assumed to be 50 msec and 120 msec. The first value is compliant with the standards of a room intentionally designed for voice broadcasting. The second value is compliant with the requirements for verbal communication in an office space [26]. The synthesized RIRs are shown in Figure 4.

#### 3.3. Comparison of GPS TDE Methods in Anechoic Environment

_{PHAT}and W

_{ML}. When the noise intensity is not sufficient to go over the threshold, the estimators demonstrate the best possible performance in terms of accuracy regardless the noise level. When the SNR drops below the threshold level, the accuracy degrades gradually with the intensification of the noise. However, using a reduced set of frequency bins makes the contaminating effect of in-band noise less harsh. Notably, this is more obvious for W

_{PHAT}than for W

_{ML}. That can be explained by the fact that frequency weighting applied with ML estimator compensates for frequency bins where noise prevails over the signal. Despite the fact, that threshold SNR level appears in Figure 6 to be better for PHAT than for ML, the latter estimator surpasses the former in terms of accuracy in the single path scenario regardless of noise intensity. The frequency weighting function for the ML estimator is in Figure 7.

_{ab}(f

_{k}) and all W(f

_{k}) in the absence of noise (SNR = 32 dB) and their presence (SNR = 4 dB). A part of the curve that is close to linear shape is clearly distinguished at Φ

_{ab}, in both cases, however, in the presence of noise, the corresponding frequency range is significantly narrower. It should be noted that Φ

_{ab}in the absence of noise passes through the origin and behaves as described in [14]. However, when the signal is contaminated with the noise, Φ

_{ab}is offset relative to the abscissa axis. This can be explained by the fact that there is no voice signal on frequencies up to 100 Hz, so the prevalence of the noise in this band results in an unpredictable offset of the unwrapped phase spectrum. That makes the estimation technique proposed in [14] not relevant for this task.

_{SCOT}and W

_{COH}is close to a line parallel to the time axis in the absence of noise. In the presence of noise, a high level of W

_{SCOT}and W

_{COH}is observed in the intervals where the cross-power spectrum |S

_{ab}| has high values. W

_{BCC}form follows the shape of |S

_{ab}| and does not differ significantly in the presence of noise and their absence. Four areas of high values are visible at the W

_{ML}corresponding to the Φ

_{ab}regions that are best approximated by the line.

#### 3.4. Comparison of GPS TDE Methods in Reverberant Environment

_{ab}(f

_{k}) and all W(f

_{k}) for different values of reverberation time (T

_{60}). All graphics in Figure 7 and Figure 10 are obtained for one and the same fragment of the original signal. It can be seen from the form of Φ

_{ab}that an increase in the reverberation time leads to a distortion of the frequency response form and a decrease in the estimate accuracy. At the same time, the distortions observed for W

_{SCOT}and W

_{COH}are not as significant as they were in the absence of reverberations and the presence of noises. This can be explained by the fact that the reflected signals are mutually correlated, and their presence does not contribute to a significant decrease in the level of signal coherence. The correlation of the reflected signals also affects at the shape |S

_{ab}| and, therefore, at the W

_{BCC}form. The W

_{ML}form also changes significantly with an increase in the reverberation time, while the regions of high values also correspond to the linear sections Φ

_{ab}. At T

_{60}= 120 msec, the number of such sections becomes smaller which negatively affects the accuracy.

## 4. Conclusions

_{d}= 44,100 Hz (about 0.01 s). A decrease in accuracy is expected in the absence of echo but at an increase in the intensity of additive noise. However, narrowing of the frequency range over which TDE is performed helps to maintain accuracy under moderate noises (SNR > 4 dB). The best accuracy characteristics are provided by the ML GPS estimator.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Raw signal power density spectrum. Frequency bins that are included in highlighted areas comprise the second set. Highlighted frequency bands are: 127–237 Hz, 285–305 Hz, 476–496 Hz, 531–580 Hz.

**Figure 3.**Source and microphones configuration in the model room. Source located in position S. Microphones are in positions A, B. Distances are r

_{A}= 3.041 m, r

_{B}= 3.354 m.

**Figure 5.**Absolute error vs SNR for anechoic room environment for: complete (

**a**); and reduced (

**b**) sets of frequency bins.

**Figure 6.**Absolute error vs SNR for anechoic room environment: (

**a**) maximum likelihood weighting function (W

_{ML}); (

**b**) no weighting was applied (W

_{PHAT}).

**Figure 7.**Sample phase cross spectrum Φ

_{ab}(f

_{k}) and weighting functions W$($f

_{k}) for various SNR: (

**a**,

**b**) Φ

_{ab}(f

_{k}), (

**c**,

**d**) W

_{BCC}(f

_{k}), (

**e**,

**f**) W

_{SCOT}(f

_{k}), (

**g**,

**h**) W

_{ML}(f

_{k}), (

**i**,

**j**) W

_{COH}(f

_{k}). Figures (

**a**,

**c**,

**e**,

**g**,

**i**) are obtained for SNR = 32 dB. Figures (

**b**,

**d**,

**f**,

**h**,

**j**) are obtained for SNR = 4 dB. For W

_{ML}(f

_{k}) all values are normalized with the maximum value on the frequency band of interest.

**Figure 8.**Absolute error vs SNR for reverberant room environment. For subfigures (

**a**,

**b**) T

_{60}= 50 msec. For figures (

**c**,

**d**) T

_{60}= 120 msec. Reduced set was used for (

**b**,

**d**). Complete set was used for (

**a**,

**c**).

**Figure 9.**Absolute error vs SNR for various reverberation times and the complete set of frequency bins: (

**a**) W

_{ML}; and (

**b**) W

_{COH}frequency weighting functions were applied.

**Figure 10.**Sample phase cross spectrum Φ

_{ab}(f

_{k}) and weighting functions W$($f

_{k}) for various reverberation times: (

**a**,

**b**) Φ

_{ab}(f

_{k}), (

**c**,

**d**) W

_{BCC}(f

_{k}), (

**e**,

**f**) W

_{SCOT}(f

_{k}), (

**g**,

**h**) W

_{ML}(f

_{k}), (

**i**,

**j**) W

_{COH}(f

_{k}). Figures (

**a**,

**c**,

**e**,

**g**,

**i**) are obtained for T

_{60}= 50 msec. Figures (

**b**,

**d**,

**f**,

**h**,

**j**) are obtained for T

_{60}= 120 msec. For W

_{ML}(f

_{k}) all values are normalized with the maximum value on the frequency band of interest.

Method | Nomenclature | Formula |
---|---|---|

BCC | W_{BCC} (f_{k}) | |S_{ab} (f_{k})|/max(|S_{ab} (f_{k})|) |

PHAT | W_{PHAT} (f_{k}) | 1 |

SCOT | W_{SCOT} (f_{k}) | γ_{ab} (f_{k}) |

ML | W_{ML} (f_{k}) | γ^{2}_{ab} (f_{k})/[1 − γ^{2}_{ab} (f_{k})] |

COH | W_{COH} (f_{k}) | γ^{2}_{ab} (f_{k}) |

Set | SNR | Mean Absolute Error (msec) | ||||
---|---|---|---|---|---|---|

(dB) | W_{BCC} (f_{k}) | W_{PHAT} (f_{k}) | W_{SCOT} (f_{k}) | W_{ML} (f_{k}) | W_{COH} (f_{k}) | |

First | 32 | 0.008 | 0.013 | 0.012 | 0.007 | 0.011 |

24 | 0.007 | 0.014 | 0.016 | 0.007 | 0.013 | |

16 | 0.005 | 0.017 | 0.020 | 0.005 | 0.016 | |

12 | 0.008 | 0.049 | 0.031 | 0.008 | 0.023 | |

8 | 0.020 | 0.080 | 0.053 | 0.020 | 0.038 | |

4 | 0.425 | 0.782 | 0.626 | 0.399 | 0.600 | |

0 | 0.623 | 1.139 | 0.893 | 0.735 | 0.687 | |

−8 | 1.961 | 2.171 | 2.100 | 1.931 | 2.006 | |

Second | 32 | 0.005 | 0.008 | 0.009 | 0.005 | 0.009 |

24 | 0.005 | 0.008 | 0.009 | 0.005 | 0.009 | |

16 | 0.004 | 0.012 | 0.011 | 0.003 | 0.012 | |

12 | 0.008 | 0.019 | 0.017 | 0.005 | 0.015 | |

8 | 0.011 | 0.016 | 0.018 | 0.010 | 0.015 | |

4 | 0.020 | 0.053 | 0.030 | 0.022 | 0.024 | |

0 | 0.634 | 0.497 | 0.514 | 0.631 | 0.637 | |

−8 | 0.973 | 0.631 | 0.672 | 0.934 | 0.921 |

Set | SNR | Mean Absolute Error (msec) | ||||
---|---|---|---|---|---|---|

(dB) | W_{BCC} (f_{k}) | W_{PHAT} (f_{k}) | W_{SCOT} (f_{k}) | W_{ML} (f_{k}) | W_{COH} (f_{k}) | |

First | 32 | 0.081 | 0.012 | 0.009 | 0.054 | 0.009 |

24 | 0.080 | 0.014 | 0.010 | 0.065 | 0.012 | |

16 | 0.063 | 0.019 | 0.017 | 0.059 | 0.019 | |

12 | 0.066 | 0.021 | 0.015 | 0.061 | 0.021 | |

8 | 0.072 | 0.027 | 0.017 | 0.073 | 0.022 | |

4 | 0.062 | 0.177 | 0.097 | 0.067 | 0.069 | |

0 | 0.272 | 0.505 | 0.407 | 0.247 | 0.324 | |

−8 | 1.735 | 1.775 | 1.746 | 1.736 | 1.747 | |

Second | 32 | 0.166 | 0.195 | 0.195 | 0.182 | 0.194 |

24 | 0.165 | 0.190 | 0.192 | 0.178 | 0.192 | |

16 | 0.163 | 0.181 | 0.183 | 0.169 | 0.181 | |

12 | 0.161 | 0.171 | 0.171 | 0.163 | 0.168 | |

8 | 0.155 | 0.122 | 0.148 | 0.153 | 0.150 | |

4 | 0.165 | 0.151 | 0.156 | 0.157 | 0.150 | |

0 | 0.190 | 0.199 | 0.168 | 0.162 | 0.137 | |

−8 | 1.392 | 1.727 | 1.598 | 1.460 | 1.478 |

Set | SNR | Mean Absolute Error (msec) | ||||
---|---|---|---|---|---|---|

(dB) | W_{BCC} (f_{k}) | W_{PHAT} (f_{k}) | W_{SCOT} (f_{k}) | W_{ML} (f_{k}) | W_{COH} (f_{k}) | |

First | 32 | 0.346 | 0.129 | 0.146 | 0.243 | 0.164 |

24 | 0.364 | 0.198 | 0.208 | 0.347 | 0.251 | |

16 | 0.315 | 0.327 | 0.320 | 0.299 | 0.282 | |

12 | 0.365 | 0.194 | 0.212 | 0.366 | 0.251 | |

8 | 0.348 | 0.658 | 0.578 | 0.361 | 0.433 | |

4 | 0.531 | 0.825 | 0.768 | 0.606 | 0.592 | |

0 | 0.572 | 0.897 | 0.842 | 0.723 | 0.611 | |

−8 | 1.169 | 1.297 | 1.291 | 1.181 | 1.263 | |

Second | 32 | 0.519 | 0.558 | 0.567 | 0.584 | 0.571 |

24 | 0.520 | 0.559 | 0.574 | 0.592 | 0.580 | |

16 | 0.522 | 0.577 | 0.586 | 0.583 | 0.586 | |

12 | 0.525 | 0.560 | 0.586 | 0.574 | 0.587 | |

8 | 0.570 | 0.594 | 0.604 | 0.629 | 0.619 | |

4 | 0.631 | 0.550 | 0.651 | 0.682 | 0.683 | |

0 | 0.884 | 0.829 | 0.942 | 0.921 | 0.935 | |

−8 | 1.057 | 0.832 | 0.914 | 0.985 | 0.971 |

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**MDPI and ACS Style**

Faerman, V.; Avramchuk, V.; Voevodin, K.; Sidorov, I.; Kostyuchenko, E. Study of Generalized Phase Spectrum Time Delay Estimation Method for Source Positioning in Small Room Acoustic Environment. *Sensors* **2022**, *22*, 965.
https://doi.org/10.3390/s22030965

**AMA Style**

Faerman V, Avramchuk V, Voevodin K, Sidorov I, Kostyuchenko E. Study of Generalized Phase Spectrum Time Delay Estimation Method for Source Positioning in Small Room Acoustic Environment. *Sensors*. 2022; 22(3):965.
https://doi.org/10.3390/s22030965

**Chicago/Turabian Style**

Faerman, Vladimir, Valeriy Avramchuk, Kirill Voevodin, Ivan Sidorov, and Evgeny Kostyuchenko. 2022. "Study of Generalized Phase Spectrum Time Delay Estimation Method for Source Positioning in Small Room Acoustic Environment" *Sensors* 22, no. 3: 965.
https://doi.org/10.3390/s22030965