# Big-Delay Estimation for Speech Separation in Assisted Living Environments
## 4 TF Masking
### Elevato-Dulet  
Assuming that speech sources are sparse in the TF mixture suggests that at any bin, one source is active. The RTF in Equation (6) for a stereo-anechoic setting, where the number of sensors is \(I=2\), in Equations (8) and (9), gives us the mixing model,  
\[ \begin{bmatrix}\textbf{X}_{1}[k,\tau]\\ \textbf{X}_{2}[k,\tau]\end{bmatrix} = \begin{bmatrix}1\\ \alpha_{j}e^{-j\frac{\pi}{K}k_{j}}\end{bmatrix} \textbf{S}_{j}[k,\tau], \tag{33}\]  
where \(\textbf{S}_{j}[k,\tau]\) is the speaker of interest that must be isolated from the mixture and directed to the hearing-aid. In many papers where the DUET-based sparsity assumption is made, for example [95; 96; 97; 98; 99], the TF-bins where one source is active are referred to as Single Source Points (SSPs). This assumption is restricted to noiseless anechoic scenarios. Given a sampling rate of \(F_{s}=16\) kHz, DUET can accurately estimate delays of up to 7 samples. A recent contribution in [91] extended DUET to echo scenarios and used a Head Related Transfer Function (HRTF) data lookup table, where the spatial cues of different speakers in the room relative to the listener are predefined. This approach is an extension of the TF dictionary approach in [95] which constructs spatial signatures for different room locations. The Google-held patent [100] underlines the efficacy, quality and robustness of DUET  
Figure 3: **(a)** Both halves of the frequency-attenuation matrix are illustrated [84]. (**b**) Considering the second-half of the frequency-attenuation matrix, when the frequency components of the sources are non-overlapping, nulls appear at appropriate attenuation coefficients. (**c**) Shows that when the sources overlap at 300 Hz a new peak is formed at \(\hat{g}\approx 0.5\). The interested reader is directed to [84] for a more in-depth discussion of this process. -based masking techniques. The classical DUET technique described in [64] constructs a 2-D weighted histogram from the ratio of the TF mixtures,  
\[\frac{\mathbf{X}_{2}[k,\tau]}{\mathbf{X}_{1}[k,\tau]}\approx\alpha_{j}e^{-j\frac {2\pi}{K}k\delta_{j}},\forall[k,\tau]\text{ belongs to }\mathbf{S}_{j}. \tag{34}\]  
The two axes represent a range of relative attenuations and delays, which are denoted by the discrete indices, \(\bar{\alpha}\) and \(\bar{\delta}\). Each axis is evenly divided with resolution widths of \(\Delta_{\alpha}\) and \(\Delta_{\delta}\). The set of points contributing to a specific location, \([\alpha,\delta]\), in the histogram are defined as  
\[I[\alpha,\delta]:=\big{\{}|\alpha[k,\tau]-\bar{\alpha}|<\Delta_{\alpha},|\delta [k,\tau]-\bar{\delta}|<\Delta_{\delta}\big{\}}. \tag{35}\]  
This 2-D histogram,  
\[H[\alpha,\delta]:=\int I[\alpha,\delta]\,d\alpha d\delta, \tag{36}\]  
aggregates membership of the set of \((\alpha,\delta)\) that belong to one source, \(\mathbf{S}_{j}[n]\). The most prominent peaks on \(H[\alpha,\delta]\) are the estimates of the mixing parameters, \((\hat{\alpha}_{j},\hat{\delta}_{j})\), associated with the \(J\) sources, \(\{(\hat{\alpha}_{j},\hat{\delta}_{j}),j=1,2\dots,J\}\). The underlying assumption is that the peaks located are sufficiently far apart and identifiable. More details about this technique can be found in [64]. In large delay scenarios, where \(d_{s}>\lambda_{\min}/2\), the DUET mixing parameter estimation step using the 2-D histogram plane is often not the best approach. The Elevatogram described in [5] provides good estimates of \(\hat{\delta}_{j}\) via Equation (22). This estimate can be used to subtract the delayed source in the second mixture using Equation (32) via the AdRess step in Equation (29), by estimating the attenuation, \(\hat{\alpha}_{j}\). This relative attenuation-delay pair can be used to generate a mask that partitions the TF-mixture  
\[\mathbf{J}[k,\tau]:=\underset{j}{\text{argmin}}\frac{\left|\hat{\alpha}_{j}e^ {-j\hat{\delta}_{j}\Omega}\mathbf{x}_{1}[k,\tau]-\mathbf{x}_{2}[k,\tau]\right|^ {2}}{1+\hat{\alpha}_{j}^{2}}. \tag{37}\]  
The variable \(\Omega=\frac{2\pi}{K}k\), is the angular frequency.