# Big-Delay Estimation for Speech Separation in Assisted Living Environments
## 4 TF Masking
### Elevato-AdRess: Separation via Frequency-Azimuth Plane
A plane is constructed by finding the best \(\hat{g}\) for a particular source, where  
\[\hat{g}\leftarrow\text{find}\bigl{(}\mathbf{A}_{1}=\bigl{|} \mathbf{X}_{1}\bigl{[}k,\tau\bigr{]}-g\mathbf{X}_{2}\bigl{[}k,\tau\bigr{]} \bigr{|}\approx 0,\] \[\text{and }\mathbf{A}_{2}=\bigl{|}\mathbf{X}_{2}\bigl{[}k,\tau \bigr{]}-g\mathbf{X}_{1}\bigl{[}k,\tau\bigr{]}\bigr{|}\approx 0\bigr{)}, \tag{29}\]  
where the spectral energies of the sources are canceled, according to [84], producing  
\[\mathbf{A}_{1}= \begin{cases}\alpha_{1}=\frac{1}{\hat{g}}\text{ if }k=k_{1}\\ \alpha_{2}=\frac{1}{\hat{g}}\text{ if }k=k_{3}\\ 0,\text{ otherwise,}\end{cases}  \tag{30}\]  
\[\mathbf{A}_{2}= \begin{cases}\alpha_{1}=g,\text{ if }k=k_{2}\\ \alpha_{2}=g,\text{ if }k=k_{4}\\ 0,\text{ otherwise,}\end{cases}  \tag{31}\]  
provided \(g<0\). In Figure 3a, it is observed that a null forms at \(\hat{g}=\alpha_{1}=0.2\) for the frequencies of \(s_{1}[n]\), \(k=100\) Hz and \(k=300\) Hz. Similarly, the source \(s_{2}[n]\) contains the nulls at \(\hat{g}=\alpha_{2}=0.8\) for the frequencies, \(k=200\) Hz and \(k=400\) Hz, for non-overlapping frequencies. This frequency-gain plane is also called the frequency-azimuth plane because it provides information on the source's azimuth in the stereo mixture. The foundational step is to isolate the target source. A two-microphone scenario gives the azimuth location of the source and not the elevation. Peaks constructed at these nulls determine the energy content of the sources. If they overlap, the corresponding null forms at \(\hat{g}\approx\frac{\alpha_{1}+\alpha_{2}}{2}\) or \(\frac{2}{\alpha_{1}+\alpha_{2}}\), subject to \(|\hat{g}|\leq 1\). The initial, detailed discussion justifying this assertion is given in [94]. Figure 3c shows that the sources overlap at \(300\) Hz and have peaks at \(\hat{g}\approx 0.5\). AdRess is restricted to pan-pot mixing scenarios. Detection of peaks at \(\hat{g}\) becomes challenging in anechoic mixing scenarios. We formulate a technique for delay cancellation in the second mixture, \(\mathbf{X}_{2}\), using the delay estimated via the Elevatogram [5] in Equation (22) in [83], and scale \(\mathbf{X}_{2}\), based on the approach in [94] according to  
\[\mathbf{X}_{2}=e^{+j\omega\hat{g}_{j}}\odot\mathbf{X}_{2}. \tag{32}\]  
This operation localizes the spectral energies at accurate gain locations on the frequency-azimuth plane. Extracting the subportion of \(\mathbf{A}[:,:]\) where these peaks occur corresponding to a particular source and repeating this for all the available time frames, \(T\), results in an augmented TF representation. Inverting this TF representation gives the demixed source in the temporal domain. The spectral energy of sources at overlapping frequency locations cannot be accurately extracted using this approach. For real speech signals, the spectral energies at these overlapping frequencies are dispersed across **A** in Figure 3, making their precise estimation ambiguous. Spatial and spectral information must be highly correlated for AdRess to function effectively in an anechoic scenario. Any inaccuracy in estimating the relative delay can result in discrepancies when localizing the spectral energy of the corresponding source on the frequency-azimuth plane, ultimately hindering the technique's ability to isolate the target speaker. ```
1: Find the delay estimate, \(\hat{\delta}_{j}\), using Equation (22) as per the Elevatogram [5]. 2: Subtract the delayed sources in \(\textbf{X}_{2}\) using Equation (32). 3: Extract the peaks corresponding to the target source using Equation (29) and generate an augmented TF representation. 4: Compute the inverse TF (ITF) transform to recover the target utterance in time domain. ```  
**Algorithm 1** Elevato-AdRess algorithm