# Interaural Coherence Estimation for Speech Processing in Reverberant Environment
## 2 Interaural Coherence Estimation for Reverberant Speech
### Interaural Coherence
The estimated auto- and cross-PSD is expressed as  
\[\Phi_{11}(n,f)=\frac{1}{L_{\rm{T}}}\sum_{k=-L}^{L}\left|X_{1}(n+k,f)\right|^{ 2}, \tag{5}\]  
\[\Phi_{22}(n,f)=\frac{1}{L_{\rm{T}}}\sum_{k=-L}^{L}\left|X_{2}(n+k,f)\right|^{ 2}, \tag{6}\]  
\[\Phi_{12}(n,f)=\frac{1}{L_{\rm{T}}}\sum_{k=-L}^{L}\left|X_{1}(n+k,f)\right| \left|X_{2}(n+k,f)\right|e^{j2n/\tau_{0}}. \tag{7}\]  
If \(x_{i}(t)\) is steady within the ensemble average interval \(L_{\rm{T}}\), \(\left|X_{i}(n-k,f)\right|\) is constant for \(-L\leq k\leq L\), and the IC appears as one using Equations (5)-(7). Conversely, if \(x_{i}(t)\) is unsteady, we get \(k\) with different values of \(\left|X_{i}(n-k,f)\right|\) for \(-L\leq k\leq L\), and the following inequality is satisfied:  
\[\left(\sum_{k=-L}^{L}\left|X_{1}(n+k,f)\right|\left|X_{2}(n+k,f)\right|\right) ^{2}<\left(\sum_{k=-L}^{L}\left|X_{1}(n+k,f)\right|^{2}\right)\left(\sum_{k=- L}^{L}\left|X_{2}(n+k,f)\right|^{2}\right). \tag{8}\]  
The IC calculated based on Equation (8) appears to be less than 1. In the case of speech, it can be seen that the IC is very close to one when the ensemble averaging interval \(L_{\rm{T}}\) is determined to be the same length as the quasi-steady interval. Secondly, we suppose that two microphones are used to record speech with static locations of a speaker and two microphones in an echoic environment. When \(s(t)\) is a speech signal coming into the first microphone without reflection, the two microphone signals \(x_{1}(t)\) and \(x_{2}(t)\) are expressed as follows based on the image source method [15]:  
\[x_{1}(t)=s(t)+\sum_{k=1}^{\infty}a_{1k}s(t-\tau_{1k}), \tag{9}\]  
\[x_{2}(t)=a_{0}s(t-\tau_{0})+\sum_{k=1}^{\infty}a_{2k}s(t-\tau_{2k}). \tag{10}\]  
\(a_{0}\) and \(\tau_{0}\) are the attenuation and time delay due to the distance and azimuth between the speaker and microphones, and \(a_{k}\) and \(\tau_{k}\) (\(i=1,2\)) are the attenuation and time delay for the speech signal, respectively, reflected from the wall. Diffuse speech, the sum of the reflected speech except for the direct speech \(s(t)\) and \(a_{0}s(t-\tau_{0})\), represents noise for the direct speech. To see only the effect of diffuse speech, we assume that \(s(t)\) and \(a_{0}s(t-\tau_{0})\) for the interval \(L_{\text{T}}\) are quasi-steady. Diffuse speech is an infinite sum of \(a_{i\delta}s(t-\tau_{iR})\); thus, it also satisfies the quasi-steady state. However, diffuse speech is a random signal compared to direct speech because the speech continues to change over time. Consequently, the coherence between two microphones is far from one due to the effect of reverberation, and it deteriorates as the energy of diffuse speech is greater than that of direct speech. This shows that the IC in an echoic environment is lower than that in an anechoic environment. In summary, if direct speech is dominant, it is similar to an anechoic environment; thus, the IC is close to one. If there is only diffuse speech without direct speech, the IC appears near zero. Also, direct speech should be quasi-steady and diffuse speech is an almost random signal within the averaging interval \(L_{\text{T}}\); thus, the IC can quantify the effects of reverberation. However, contrary to the theory, the quasi-steady interval cannot be perfectly steady in real speech. To compensate for this, an exponentially decreasing weight away from the \(n\)-th time frame is applied, and it estimates the PSD rather than giving the same weight in the interval as in Equation (4). The final proposed PSD estimation equation is as follows:  
\[\Phi_{\mathrm{ij}}(n,f)=E\big{[}X_{i}^{\star}(n,f)X_{j}(n,f)\big{]}\ (i,j=1,2). \tag{2}\]0  
\[\Phi_{\mathrm{ij}}(n,f)=E\big{[}X_{i}^{\star}(n,f)X_{j}(n,f)\big{]}\ (i,j=1,2). \tag{2}\]1  
Here, \(\beta\) is the decaying ratio, and it has a range of \(0<\beta\leq 1\). Also, \(A\) is a value to make the sum of weights equal to one; therefore, we can derive \(A\) according to \(L\) and \(\beta\) from Equation (11b).