# Big-Delay Estimation for Speech Separation in Assisted Living Environments
## 2 Mixing Model  
A reverberant stereo noise-free pair of mixtures is expressed as  
\[x_{1}[n] =\sum_{j=1}^{J}h_{1j}[n]*s_{j}[n], \tag{2}\] \[x_{2}[n] =\sum_{j=1}^{J}h_{2j}[n]*s_{j}[n], \tag{3}\]  
where \(*\) is the linear convolution operator and the filters, \(h_{ij}[n]\), are the channel Received Impulse Responses (RIR) for source \(s_{j}[n]\) at microphone, \(x_{i}[n]\). The source-to-microphone distance is \(d_{ij}\) and \(K\) is the number of DFT points. For an anechoic or low reverberant scenario, the mixing filter for the \(i^{th}\) channel, \(h_{ij}[n]\), simplifies to a combination of delays, \(\delta_{ij}\), and attenuations, \(\alpha_{ij}\), which are expressed as,  
\[\alpha_{ij}=\frac{1}{\sqrt{4\pi}d_{ij}}, \tag{4}\]  
Figure 1: (**a**) Motivational Scenario: Multi-microphone arrays and a smart hearing aid (HA) are interconnected via a wireless network. The microphones capture speech mixtures, which are processed by powerful demixing algorithms, isolating the speaker of interest and directing their speech to the HA. (**b**) Overhead schematic view of the experimental environment: the distance, \(d_{s}=B^{\prime}M\), is the effective inter-sensor gap in \(d_{s}\) meters. Relative to the RM, the speech signal, \(s_{1}[n]\), travels an extra distance, \(B^{\prime}M\), giving it a larger relative delay. As the distance, \(d_{s}\), increases, the target signal \(s_{1}[n]\) becomes more prone to phase wraparound. where \(d_{ij}\) is the distance between the \(j^{th}\) source and the \(i^{th}\) microphone [49]. The Acoustic Transfer Function (ATF) of the \(j^{th}\) source, which is the frequency domain transform of the RIR, which is called its steering vector in a scenario with \(I\) microphones is,  
\[\mathbf{d}_{j}[k]=\left[\frac{1}{\sqrt{4\pi}d_{1j}}e^{-j\frac{2\pi}{k}k\frac{d _{1j}}{c}F_{s}},\frac{1}{\sqrt{4\pi}d_{2j}}e^{-j\frac{2\pi}{k}k\frac{d_{2j}}{c }F_{s}},\dots,\frac{1}{\sqrt{4\pi}d_{lj}}e^{-j\frac{2\pi}{k}k\frac{d_{lj}}{c}F_ {s}}\right]. \tag{5}\]  
The gain normalized ATF of the \(j^{th}\) source, called the Relative Transfer Function (RTF) is,  
\[\mathbf{d}_{j}^{\text{Rel}}[k]=\left[1,\frac{d_{1j}}{d_{2j}}e^{-j\frac{2\pi}{ k}k\frac{(d_{2j}-d_{1j})}{c}F_{s}},\dots,\frac{d_{1j}}{d_{lj}}e^{-j\frac{2\pi}{k}k \frac{(d_{lj}-d_{1j})}{c}F_{s}}\right], \tag{6}\]  
and can be uniquely determined by the relative gain and delay estimates [52], in a low reverberant room [53]. Equation (6) helps us fix the scaling problem [54]. Substituting Equation (4) in Equation (6) gives the RTF,  
\[\mathbf{d}_{j}^{\text{Rel}}[k]= \left[1,\alpha_{2j}e^{-j\frac{2\pi}{k}kk_{2j}},\dots,\alpha_{lj}e ^{-j\frac{2\pi}{k}k\delta_{lj}}\right] \tag{7}\]  
where \(\delta_{ij}=\frac{\left(d_{ij}-d_{ij}\right)}{c}F_{s}\), is the relative delay measured in samples and the attenuation of the \(j^{th}\) source impinging on the \(i^{th}\) microphone is \(\alpha_{ij}=\frac{d_{1j}}{d_{1j}}\), which is bounded by \(0\leq\alpha_{ij}\leq 1\), and is used to define an anechoic mixing model. The anechoic mixing model is a special case of the reverberant model. In a room which has low reverberance, or which has no echoes, the RTF in Equation (7) can be uniquely determined by the relative gain and delay estimates [52; 53]. The terms relative gain, attenuation and IID are used interchangeably in the literature. Conventional relative delay estimation techniques, for example, the Generalized Cross-Correlation-PHAse Transform (GCC)-PHAT [55], Steered Power Response (SRP)-PHAT [56], and MUltiple Signal Classification (MUSIC) [57] are designed to estimate the loudest source in an indoor acoustic environment. The low power of the target speech source often impedes the accuracy of the delay estimate [58] in a multi-talker scenario because the phase information is lost due to other speech interference [59]. Joint multi-source delay estimation is challenging when two sources are spatially close to one another [46; 60; 61]. In addition, speech enhancement techniques are not useful when the interferer is in proximity to the target speaker of interest [62].