## 3 Experiment I: Minimal SBR for Full Sentence-Level Intelligibility
### Model Predictions of \(\text{SBR}_{min}\)  
A function that allows conversion from a desired BiDWGP to the mixing level SBR is required. The upper panel of Figure 4 illustrates the BiDWGP scores as a function of SBR in various background sounds for talker F6. Each data point on the plotted lines (in steps of 3-dB SBR) was calculated as the mean of the BiDWGP scores for all samples used in the \(\text{SBR}_{min}\) psychoacoustic experiment. In order to formulate a function relating BiDWGP and SBR, a two-parameter sigmoid function in the form of \(BiDWGP=f(SBR)=1/(1+exp(-a-b\cdot SBR))\) with variables \(a\) and \(b\) was constructed using the MATLAB _glmfit_ function with a normal distribution and _logic_ link function. The fitted functions are shown in the lower panel of Figure 4. Subsequently, given a desired BiDWGP score, the required SBR in the presence of each background sound can be calculated using the inverse of the function \(f\). Due to the masking effect varying between background sounds, different SBR values for a given BiDWGP score are found. In the lower panel of Figure 4 the SBRs required in order to achieve a BiDWGP score of 0.5 in different background sounds is shown (The reason for choosing a BiDWGP score of 0.5 is explained below). This procedure was also applied to the other talkers. Figure 4: (**a**): Mean BiDWGP as a function of SBR. (**b**): A two-parameter logistic function fitted to the data in the upper panel. Speech produced by Talker F6 in different background sounds. The dotted lines in the lower panel indicate the SBR required for a BiDWGP score of 0.5 in each background. Drawing on the perceptual experiment that determined listeners' \(\text{SBR}_{min}\), it is possible to derive the BiDWGP score which corresponds to the predicted \(\text{SBR}_{min}\) in all tested conditions. This was performed by minimising the root-mean-square error (RMSE) between the measured and the predicted \(\text{SBR}_{min}\) over 12 conditions (3 background sounds \(\times\) 4 talkers) from SSN, BAB and CS. In order to investigate the robustness of this process for unseen conditions, 8 conditions from SNG and MSC were excluded from the RMSE minimisation and were reserved for testing only. Figure 5 shows RMSE as a function of BiDWGP scores. It suggests that SBRs leading to a BiDWGP of 0.5 provides the closest match to the measured \(\text{SBR}_{min}\), with a RMSE of 2.6 dB across SSN, BAB and CS. When predicting the \(\text{SBR}_{min}\) for the previously unseen SNG and MSC cases using 0.5 BiDWGP, the RMSEs are 1.0 and 2.8 dB across all the talkers, respectively. The BiDWGP scores corresponding to \(\text{SBR}_{min}\) were also individually learnt for SNG and MSC. They are 0.48 (RMSE: 0.8 dB) and 0.52 (RMSE: 2.4 dB) for SNG and MSC, respectively--the RMSEs of the predictions using 0.5 BiDWGP are merely 0.2 and 0.4 dB higher for the two maskers. Consequently, the value of 0.5 was used for all background sounds to derive the \(\text{SBR}_{min}\). The predicted \(\text{SBR}_{min}\) levels in all conditions are displayed against the measured values for all speakers and markers in Figure 3. With a strong linear relationship [\(r^{2}=0.70,p<0.001\)] and the least RMSE (2.4 dB) between the measured and the predicted values, the BiDWGP metric was considered capable of making reasonable \(\text{SBR}_{min}\) predictions in tested background sounds, especially as this was performed across both factors of background sound and talker. Any OIM could be used in the place of BiDWGP here as the intelligibility predictor. However, the predictive power of the chosen OIM may affect the robustness of the system. As an illustration, the performance of another four OIMs was also investigated, including the SII and the STI--two standard intelligibility measures--and two more recent measures: NCM and STOI. The SII makes intelligibility predictions from the normalised frequency-dependent speech-to-noise ratios, which reflect the masked-audibility of each frequency in noise.