# Big-Delay Estimation for Speech Separation in Assisted Living Environments
## 2 Mixing Model
Recent work in [4] demonstrated that the accuracy of widely used delay estimation techniques decreased when tasked with estimating the relative delays of all constituent sources simultaneously. The advantage of a close-talk setup is the ability to estimate the most prominent peak associated with a relative delay [63], which corresponds to the reference microphone positioned near the target speaker's mouth. This provides a reliable estimate of the IPD for the speaker of interest, which can then be used to parameterize a TF mask to isolate the speech effectively. Assuming that an arbitrary source, \(s_{j}[n]\), is in physical proximity to any one RM, as shown in Figure 1b, then the corresponding discrete-time, anechoic stereo-mixing scenario, which consists of \(J\) sources is,  
\[x_{1}[n]=s_{j}[n], \tag{8}\] \[x_{2}[n]=\sum_{j=1}^{I}\alpha_{2j}s_{j}[n-\delta_{2j}], \tag{9}\]  
where the discrete time index is \(n\), where \(1\leq n\leq N\) and \(n\in\mathbb{Z}_{+}\). Given a binaural mixture, the relative attenuation and delay coefficients are redefined, \(\alpha_{2j}\) as \(\alpha_{j}\) and \(\delta_{2j}\) as \(\delta_{j}\)for simplicity. In the mixture \(x_{2}[n]\), the source, \(s_{j}[n]\), is attenuated and delayed by \(\alpha_{j}\) and \(\delta_{j}\), respectively, relative to \(x_{1}[n]\). The relative attenuation lies in the range \(0\leq\alpha_{j}\leq 1\), and the relative delay \(\delta_{j}\) is measured in samples. The TF transform of \(x_{1}[n]\) provides the mapping as follows, \(\mathbf{X}_{1}:x_{1}[n]\in\mathbb{R}\rightarrow\mathbf{X}_{1}[k,\tau]\in \mathbb{C}\), where the discrete frequency is \(k\), and where \(1\leq k\leq K\). The discrete-time frame is \(\tau\), where \(1\leq\tau\leq T\). Similarly, \(x_{2}[n]\) is mapped to \(\mathbf{X}_{2}[k,\tau]\), using the TF transform stated in [48]. It is assumed that the target speaker is always the most intense among all speakers present. As proposed in [4], speech enhancement with interference reduction using an RM can enable speech demixing. The demixing of a mixture of voice signals at the far-end microphone is illustrated in Figure 1b. In a simple anechoic model, an acoustic speech wave at one microphone differs from the reference RM signal in terms of IPD and IID cues. We hypothesize that DUCT-based masking techniques, which include the classical DUCT [64] and AdRess [65] algorithms--where the latter was originally designed for separating music mixed using a pan-pot model--can be adapted to a large-delay anechoic model, with delays of up to 200 samples. The AdRess technique is a brute force version of the DUCT mask, where the IID cues are localized on a frequency-azimuth plane relative to a gain scale. In the time domain, the silent interval between two interacting speakers is typically small. This causes the majority of the signal to have coefficients which are non-zero [66]. In the TF domain, a small percentage of the TF coefficients contain a majority of the energy of the acoustic signal [67]. Speech is typically sparse and disjoint in the TF domain [68]. The TF representation is commonly employed for speech processing because sources tend to exhibit less overlap in the TF domain compared to the temporal waveform [69; 70]. While speech in the time domain is not sparse during active speech, it is typically sparse in the TF domain. This implies that a small percentage of the TF Gabor coefficients capture a large part of the energy. Specifically, the magnitudes of most of these Gabor coefficients are generally small. According to the authors of [69; 71] it is unlikely that multiple speech signals will overlap in a particular TF bin of the mixture of speech utterances. This characteristic makes the signals approximately Windowed-Disjoint Orthogonal (WDO) [69; 71]. The authors in [67] observed that the TF bins that contain 90% of the energy of the \(j^{th}\) source contain less than 1% of the energy of all the interference combined together.