# Interaural Coherence Estimation for Speech Processing in Reverberant Environment
## 2 Interaural Coherence Estimation for Reverberant Speech  
### Interaural Coherence  
We assume that the speech at a specific speaker location is recorded using two microphones apart from each other, and each recording signal is \(x_{1}(t)\) and \(x_{2}(t)\). To analyze the speech signal in both the time and the frequency domain, short-time Fourier transform (STFT) using overlap is applied, and the time-frequency bin values of the two microphones for the \(n\)-th time frame with frequency \(f\) can be expressed as \(X_{1}(n,f)\) and \(X_{2}(n,f)\), respectively. The auto- and cross-power spectral density (PSD) of the two microphone signals is represented by \(\Phi_{\mathrm{ij}}(n,f)(i,j=1,2)\). In this case, this value and the IC \(\Gamma^{2}(n,f)\) are defined as  
\[\Gamma^{2}(n,f)=\frac{\left|\Phi_{12}(n,\ f)\right|^{2}}{\Phi_{11}(n,f)\Phi_{2 2}(n,f)}, \tag{1}\]  
\[\Phi_{\mathrm{ij}}(n,f)=E\big{[}X_{i}^{\star}(n,f)X_{j}(n,f)\big{]}\ (i,j=1,2). \tag{2}\]  
It is hard to calculate accurate auto- and cross-PSD (i.e., \(\Phi_{\mathrm{ij}}(n,f)\)) using Equation (2) with finite-length \(X_{1}(n,f)\) and \(X_{2}(n,f)\). In previous studies [9, 10, 11], the PSD was estimated by multiplying exponentially decaying weight and summing continuous time-frequency bins over time as  
\[\Phi_{\mathrm{ij}}(n,f)=\alpha X_{i}^{\star}(n,f)X_{j}(n,f)+(1-\alpha)\Phi_{ \mathrm{ij}}(n-1,f). \tag{3}\]  
In this case, \(\alpha\) has a range of \(0\leq\alpha\leq 1\). As a recursion formula, all the time-frequency bin information for the previous time affects the estimation of the PSD for the time frame \(n\). However, since the speech changes continuously over time, the equation using all previous data can estimate the inaccurate PSD. An incorrect IC may be provided, which can cause performance problems. In addition,the proposed method is calculated recursively using all speech bins, which can take more time. To solve these problems, this paper proposes a new IC estimation method using finite time-frequency bin data, and it is compared with the conventional method in terms of the estimation. Interaural Coherence Estimation Using Finite Time-Frequency Bins  
In digital signal processing, the PSD of a stationary time-domain signal \(x(t)\) is derived by ensemble averaging of \(X(n,f)\) based on the assumption of the ergodic process, and this is called [PERSON]'s method [PERSON] (1954). Based on this method, the estimated PSD for a specific time \(n\) and frequency \(f\) is calculated as follows by averaging \(L_{1}+L_{2}+1\) data for time \((L_{1},L_{2}\geq 0)\):  
\[\Phi_{\rm{ij}}(n,f)=\frac{1}{L_{\rm{T}}}\sum_{k=-L_{1}}^{L_{2}}X_{t}^{*}(n+k,f )X_{j}(n+k,f). \tag{4}\]  
Here, \(L_{\rm{T}}\) is the total length of the interval to apply averaging, and it is equal to \(L_{1}+L_{2}+1\). In the case of speech, it does not satisfy the ergodic process because it is an unsteady state that continually changes with time. Nevertheless, a speech signal within short time intervals can be almost steady, and it is called a quasi-steady-state signal in this paper. The number of time frames \(L_{\rm{T}}\) is a quasi-steady-state interval of the speech signal, and it is applied to Equation (4) to estimate the correct PSD. Also, in the case of using finite data \(L_{\rm{T}}\), the uniform estimation method is applied to most time-frequency bins and provides accurate IC. Before estimating the PSD for a particular time frame \(n\), it is more quasi-steady for the interval close to \(n\) for time; thus, we apply \(L_{1}\) and \(L_{2}\) as \(L\) and estimate the accurate PSD using the bin data in the interval of \(2L+1\). By applying the IC based on the proposed PSD estimation from an anechoic to an echoic room, the theoretical validity is assessed. Firstly, we suppose that two microphones are used to record speech in an anechoic environment. Here, \(X_{1}\) and \(X_{2}\) have a phase difference of \(e^{j2n/\tau_{0}}\) for all \(n\) due to time delay \(\tau_{0}\) according to the azimuth of the speaker, since only the direct speech exists.