## 3 Theoretical Background
### Proposed DCGOSM-Based Speech Enhancement Technique
#### 3.9.1 Process of Obtaining the Required Sensing Matrix
\begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline  & \multicolumn{10}{c}{**Techniques**} \\ \hline  & \multicolumn{2}{c}{**OMP**} & \multicolumn{2}{c}{**CoSaMP**} & \multicolumn{2}{c}{**StOMP**} & \multicolumn{2}{c}{**K-SVDCS**} & \multicolumn{2}{c}{**Proposed**} \\ \hline
**SNR** & **SNR** & **SSNR** & **SNR** & **SNR** & **SNR** & **SNR** & **SNR** & **SNR** & **SNR** \\
**(dB)** & **(dB)** & **(dB)** & **(dB)** & **(dB)** & **(dB)** & **(dB)** & **(dB)** & **(dB)** & **(dB)** & **(dB)** & **(dB)** \\ \hline
0 & 2.560 & 2.385 & 0.758 & 1.638 & 1.388 & 1.853 & **–** & 3.80 & **3.112** & 3.072 \\
5 & 6.201 & 5.835 & 5.193 & 4.602 & 5.681 & 4.965 & **–** & 3.50 & **6.673** & **6.592** \\
10 & 10.119 & 9.345 & 9.349 & 8.471 & 9.532 & 8.708 & **–** & 3.06 & **10.176** & **9.885** \\
15 & 13.118 & 12.438 & 13.119 & 12.195 & 13.074 & 12.168 & **–** & 1.60 & **13.361** & **13.056** \\
20 & 15.383 & 14.873 & 15.453 & 14.890 & 15.156 & 14.685 & **–** & -0.94 & **15.516** & **15.325** \\ \hline SNR & \multirow{2}{*}{PESQ} & \multirow{2}{*}{STOI} & \multirow{2}{*}{PESQ} & \multirow{2}{*}{STOI} & \multirow{2}{*}{PESQ} & \multirow{2}{*}{STOI} & \multirow{2}{*}{PESQ} & \multirow{2}{*}{STOI} & \multirow{2}{*}{PESQ} & \multirow{2}{*}{STOI} & \multirow{2}{*}{PESQ} & \multirow{2}{*}{STOI} \\ (dB) & & & & & & & & & & \\ \hline
0 & 1.780 & 0.652 & 1.917 & **0.683** & 1.969 & 0.644 & 1.96 & 0.66 & **2.304** & 0.615 \\
5 & 2.216 & **0.736** & 2.251 & 0.726 & 2.308 & 0.715 & 2.28 & 0.72 & **2.384** & 0.689 \\
10 & 2.434 & **0.811** & 2.513 & 0.759 & 2.513 & 0.771 & 2.52 & 0.79 & **2.674** & 0.748 \\
15 & 2.606 & **0.827** & 2.646 & 0.774 & 2.709 & 0.805 & 2.69 & 0.81 & **2.903** & 0.785 \\
20 & 2.667 & **0.859** & 2.772 & 0.816 & 2.853 & 0.847 & 2.85 & 0.83 & **3.194** & 0.824 \\ \hline \hline \end{tabular} The bold font indicates the best obtained result for the selected measure in that row.  
\end{table} 
Table 2: Performance comparison for babble noise using the NOIZEUS dataset sample sp05.wav. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline  & \multicolumn{6}{c}{**Techniques**} \\ \hline  & \multicolumn{2}{c}{**OMP**} & \multicolumn{2}{c}{**CoSaMP**} & \multicolumn{2}{c}{**StOMP**} & \multicolumn{2}{c}{**Proposed**} \\ \hline
**SNR (dB)** & **SNR** & **SSNR** & **SNR** & **SSNR** & **SNR** & **SNR** & **SNR** & **SSNR** \\
**(dB)** & **(dB)** & **(dB)** & **(dB)** & **(dB)** & **(dB)** & **(dB)** & **(dB)** & **(dB)** & **(dB)** \\ \hline
0 & 3.288 & 2.490 & 1.292 & 1.744 & 2.056 & 2.036 & **4.687** & **4.058** \\
5 & 7.220 & 6.101 & 5.639 & 4.656 & 6.266 & 5.129 & **7.926** & **improving efficiency. The framing of the signal causes spectral leakage at the edges of the frame. The overlapping Hamming windows are used to mitigate spectral leakage by smoothly transitioning between adjacent frames, reducing the abrupt changes at the frame boundaries. 4. Signal sparsity is the fundamental condition for CS. Therefore, to make the frame sparse, each frame is converted into the frequency domain using the discrete cosine transform (DCT), then the insignificant DCT components within the transformed frame are set to zero. The term \"significance\" refers to the components that encompass the most energy, specifically those exceeding 90%. This objective can be accomplished by strategically selecting the highest 25% of the coefficients. 5. The non-transformed speech frames are sent to the VAD black for identification of the speech and non-speech frames. In addition, the preliminary noise variance (\(v_{n}\)) estimation is performed using the initial frame, which is considered to carry only noise. 6. The noisy speech and non-speech frames are then grouped into separate databases for \(DB_{spech}\) and \(DB_{non-speech}\). This helps in obtaining dominant columns separately for noisy speech and non-speech frames. 7. To minimize the optimization time, only a small number of frames are chosen from these databases.