# Big-Delay Estimation for Speech Separation in Assisted Living Environments
## 5 Results
The estimate of the target source, \(\hat{s}_{\parallel}[n]\), can be decomposed into the sum of four separate components,  
\[\hat{s}_{\parallel}[n]=\hat{s}_{\parallel}[n]+\epsilon_{\parallel}^{\text{ interference}}[n]+\epsilon_{\parallel}^{\text{noise}}[n]+\epsilon_{\parallel}^{\text{ artifact}}[n], \tag{43}\]  
where \(\hat{s}_{\parallel}[n]\) is the clean speech utterance [102]. The Source-to-Distortion (SDR) separation quality is defined as  
\[\text{SDR}=\frac{\left|\left|\hat{s}_{\parallel}[n]\right|\right|^{2}}{\left| \left|\epsilon_{j}^{\text{target}}[n]+\epsilon_{\parallel}^{\text{interference}}[n]+ \epsilon_{j}^{\text{noise}}[n]\right|\right|^{2}}. \tag{44}\]  
It gives a measure of how well the target utterance has been recovered. The SIR is defined as  
\[\text{SIR}=\frac{\left|\left|\hat{s}_{\parallel}[n]\right|\right|^{2}}{\left| \left|\epsilon_{\parallel}^{\text{interference}}[n]\right|\right|^{2}}. \tag{45}\]  
It computes a measure of how much the energy leaks from the other interfering sources, or how loud they can be heard in the recovered speech utterance. The Source-to-Artifact Ratio (SAR) is given as,  
\[\text{SAR}=\frac{\left|\left|\hat{s}_{\parallel}[n]+\epsilon_{\parallel}^{ \text{interference}}[n]+\epsilon_{\parallel}^{\text{noise}}[n]\right|\right|^{ 2}}{\left|\left|\epsilon_{\parallel}^{\text{artifact}}[n]\right|\right|^{2}}. \tag{46}\]  
Figure 5: **(a–c)** AdRess, DUET, Elevato-DUET and Elevator-AdRess are evaluated in terms of their Source-to-Distortion Ratio (SDR), on the LHS, Source-to-Interference Ratio (SIR), in the center, and their Source-to-Artifact Ratio (SAR), on the RHS, as a function of relative delay. The Ideal Binary Mask is also illustrated as a benchmark method. The algorithms are evaluated using a four-source mixture, \(J=4\), with a sampling rate of \(F_{s}=16\) kHz. All the separation quality metrics are measured in dB. The Elevato-AdRess algorithm typically gives better performance than Elevato-DUET for all delays. The performance range of AdRess and DUET is limited to small delays in the case of DUET and zero delay for AdRess. Of particular interest in this setting, is that in scenarios with large delays, Elevato-DUET generally outperforms Elevato-AdRess. This difference can be attributed to the Mean Absolute Error (MAE) associated with \(\hat{s}_{\parallel}[n]\) that is used to estimate \(\hat{\mathrm{a}}_{j}\), and subsequently the TF mask, especially when the delay is large. This measure attempts to quantify burbling sounds, which are also referred to as musical-noise, in the reconstructed separated utterances. Figure 5 illustrates that Elevato-DUET and Elevato-AdRess can function effectively with relative delays of up to 200 samples. This capability arises from combining the Elevatogram technique, which was contributed in [5], with the classical DUET [64] and AdRess [65] methods, enhancing their performance in scenarios where the acoustic sources experience significant relative delays. The figure further highlights that Elevato-AdRess outperforms Elevato-DUET in terms of the SDR, SIR, and SAR separation quality metrics. As shown in Figure 5a, Elevato-AdRess gives better results than Elevato-DUET by a mean reconstruction SDR gain of 2.2 dB for relative delays of up to 120 samples. Beyond this point, a performance decrease is observed due to an increase in the Mean Absolute Error (MAE) between the true delay, \(\delta_{\text{j}}\), and its estimated value, \(\hat{\delta}_{\text{j}}\), which is calculated by the Elevatogram [5], e.g., \(|\delta_{j}-\hat{\delta}_{\text{j}}|\). As can be seen in Figure 5b, the mean reconstruction SIR of Elevato-AdRess exceeds that of Elevato-DUET by 4.3 dB for a relative delay of \(\delta_{j}=100\) samples. Figure 5c shows that the difference in mean reconstruction SAR between the two algorithms is small. It lies within 1.5 dB for relative delays of up to 120 samples. In general, Elevato-AdRess improves the separation quality, in terms of the SDR, SIR, and SAR, compared with the utterances recovered by Elevato-DUET.