# Omnidirectional Haptic Guidance for the Hearing Impaired to Track Sound Sources
## 2 Methods
### Relationship between Mean Standard Deviation and Distance to Sound Source  
The distance to a sound source was estimated by using the variation of the frequency response of the short-term interaural phase difference of sound pressure. This section describes a method for evaluating the variation of the frequency response of the short-time interaural phase difference of sound pressure [40]. Let us explain its method briefly. In the frequency characteristics of the short-time interaural phase difference shown in Figure 4a, the amount of data included in frequency band \(\Delta f\) near center frequency \(f_{i}\) is \(2m+1\), and the phase difference at center frequency \(f_{i}\) is \(\varphi_{i}\). If the average value of the phase difference within \(\Delta f\) is \(\overline{\varphi}_{i}\), the standard deviation \(\rho_{i}\) is expressed by Equation (4):  
\[\rho_{i}=\sqrt{\frac{1}{2m+1}\sum_{j=i-m}^{i+m}(\varphi_{j}-\overline{\varphi} _{i})^{2}} \tag{4}\]  
If we calculate the standard deviation of the data in frequency band \(\Delta f\) while changing center frequency \(f_{i}\), we can obtain the frequency characteristics of the standard deviation, as shown in Figure 4b. The average value of the standard deviation within frequency band \(\Delta F\), where the standard deviation value fluctuates relatively with the distance to the sound source, was used to evaluate the variation. The relationship between the mean value of the standard deviation and the distance to the sound source was obtained experimentally, and then the distance to the sound source was estimated from the standard deviation value using Figure 5. For example, when the mean value of the standard deviation is 65\({}^{\circ}\), the estimated distance is about 2.4 m.  
Figure 4: Frequency characteristics of (**a**) short-time interaural phase difference of sound pressure and (**b**) standard deviation, when calculating standard deviation of the data in frequency band \(\Delta f\) while changing center frequency \(f_{i}\). ©(2021) IEEE, Reprinted, with permission.  
Figure 5: Relationship between mean value of standard deviation and distance to sound source was obtained experimentally; then, the distance to the sound source was estimated from the standard deviation value using this graph.  
[PERSON] et al. (2019) reported that the average value of this standard deviation depends on the direction of the sound source [41]. Let us briefly touch on this here. The directional characteristics of the mean standard deviation are shown in Figure 6. The radial axis is the mean value of the standard deviation (in degrees). It can be seen that this value is relatively low when the sound source is located either directly in front (180\({}^{\circ}\)) or directly behind (0\({}^{\circ}\)), and it becomes higher as the sound source locates to the side. The farther to the side the sound source is located, the more difficult it is to detect the distance, because the mean value of the standard deviation changes little even when the distance to the sound source changes and the dynamic range becomes narrower. To estimate the change in distance to the sound source, the direction of the sound source should be fixed, for example, directly in front or directly behind, where the dynamic range is large. This feature is one of the reasons for tracking the sound source in Section 4 below.