# Omnidirectional Haptic Guidance for the Hearing Impaired to Track Sound Sources
## 2 Methods  
### Estimation Method of Short-Time Interaural Time Difference in Sound Pressure  
The direction of the sound source was estimated by using the arrival time difference between the two ears. The interaural time difference (ITD) between the ears was calculated from the short-time interaural phase difference of sound pressure [39]. In this section, we briefly explain the estimation method.  
First, let us explain the ambiguity of the phase difference of sound pressure. A schematic diagram of the pressure waveforms detected by the left and right microphones is shown in Figure 1. Comparing the phases of the two sound pressure waveforms, the left waveform is delayed by \(\Delta\varphi\) compared to the right waveform, suggesting that the phase difference is \(\Delta\varphi\). However, the left waveform is also considered delayed by \(\Delta\varphi+2\pi\) due to the periodicity of the waveform. This means that there are multiple phase differences to be defined and the phase difference can generally be expressed as \(\Delta\varphi+2n\pi\), where \(n\) is an arbitrary integer. This is known as the ambiguity of the phase difference of sound pressure. Since there are multiple short-time interaural phase differences of sound pressure, there are also multiple ITDs, and therefore, the direction of a sound source cannot be estimated accurately.  
Next, we describe a method for estimating the true short-time interaural time difference from the ambiguous difference using acoustic signals detected by two microphone systems. The short-time interaural phase difference of sound pressure at frequency \(f_{i}\), \(\Delta\varphi_{ni}\) is as follows:  
\[\Delta\varphi_{ni}=\angle\frac{X_{L}(f_{i})}{X_{R}(f_{i})}+2\pi n_{i}(f_{i}), \tag{1}\]  
where \(X_{L}(f_{i})\) and \(X_{R}(f_{i})\) are the spectra obtained by the discrete Fourier transform (DFT) of the two time functions \(x_{L}(t)\) and \(x_{R}(t)\), respectively, and \(n_{i}(f_{i})\). is an integer that is a function of frequency. The interaural time difference \(\Delta t_{ni}\) is as follows:  
\[\Delta t_{ni}=\frac{\Delta\varphi_{ni}}{2\pi f_{i}}. \tag{2}\]  
Due to the ambiguity of phase differences, multiple interaural time differences can be calculated for any given value of integer \(n\) at any given frequency. However, the direction of sound arriving from a single fixed source should be constant, independent of frequency. Therefore, in the histogram of the interaural time difference, the \(\Delta t\) where the evaluation function shown in Equation (3) is the maximum obtained, and this is regarded as the true interaural time difference.  
\[L(\Delta t)=\sum_{n}\sum_{i}\Delta t_{ni} \tag{3}\]  
Figure 2a shows an example of the frequency response of the interaural time difference when the sound source is located at \(140^{\circ}\) behind and to the left. Many streak patterns are seen, and multiple interaural time differences are displayed. Figure 2b shows the corresponding histogram. In this example, the interaural time difference is \(\Delta t\), where the evaluation function is the maximum. In this example, the time difference between the two ears is \(360\) us.  
Figure 1: Schematic diagram of pressure waveforms detected by left and right microphones. Comparing phases of left and right sound pressure waveforms, left waveform is delayed by \(\Delta\varphi\) or \(\Delta\varphi+2\pi\) due to waveform periodicity.