## 3 Theoretical Background
### Proposed DCGOSM-Based Speech Enhancement Technique
#### 3.9.1 Process of Obtaining the Required Sensing Matrix
The selected noise frames are further used for the noise variance estimation (\(v_{n}^{final}\)), which is used during the final enhancement process. Estimating the noise variance from the multiple selected frames reduces the error possibilities, especially in the cases where the noise is not uniformly distributed throughout the entire signal duration. 8. PSO is initialized according to the number of particles and maximum generations. Each particle \(p_{i}^{j}\) (here, the superscript \(j\) represents the generation, and the subscript \(i\) represents the particle) is defined as an \(M\times N\) dimension row vector, where each element is bounded within the range of \([-1,\ 1]\). \[p_{i}^{j}=\left[e_{1}^{i},\ e_{2}^{i},\ \ldots,\ e_{M\times N}^{i}\right],\ where\ \forall i,\ j:\left\{-1\leq e_{j}^{i}\leq 1\right\}\] (18)
9. Each of these particles (\(p_{i}^{j}\)) represents a sensing matrix (\(\phi_{i}^{j}\)) when converted into an \(M\) rows and \(N\) columns matrix: \[\phi^{i}= \begin{bmatrix}e_{1}^{i}&e_{2}^{i}&\cdots&e_{N}^{i}\\ e_{N+1}^{i}&e_{N+2}^{i}&\cdots&e_{2\times N}^{i}\\ \vdots&\vdots&\ddots&\ldots\\ e_{(M-1)\cdot N+1}&e_{(M-1)\cdot N+2}&\cdots&e_{M\times N}^{i}\end{bmatrix} ,\] (19) where \[\forall i,j:\left\{-1\leq e_{j}^{i}\leq 1\right\}\]
10. However, before using this matrix as a sensing matrix, it is required to make \(\phi_{i}^{j}\) a sample from the uniform spherical ensemble (USE). This is done by dividing each element of each column of the matrix by the column's \(l_{2}\) norm: \[\phi^{i}= \begin{bmatrix}\frac{e_{1}^{i}}{r_{1}}&\frac{e_{2}^{i}}{r_{2}}&\cdots &\frac{e_{M}^{i}}{r_{N}}\\ \frac{e_{M+1}^{i}}{r_{1}}&\frac{e_{M+2}^{i}}{r_{2}}&\cdots&\frac{e_{M\times N}^ {i}}{r_{N}}\\ \vdots&\vdots&\ddots&\cdots\\ \frac{e_{(M-1)\cdot N+1}}{r_{1}}&\frac{e_{(M-1)\cdot N+2}}{r_{2}}&\cdots&\frac {e_{M\times N}^{i}}{r_{N}}\end{bmatrix} ,\] (20)where  
\[r_{i}=\left\|\left[e_{0\times N+j^{i}}^{i},\ e_{1\times N+j^{i}}^{i},\ldots,e_{(M-1 )\times N+j}^{i}\right]\right\|_{2}=\sqrt{\sum_{k=0}^{M-1}\left|e_{k\times N+j}^ {i}\right|^{2}}\]
11. Using the sensing matrix (\(\phi_{i,USE}^{j}\)) formed by \(i^{th}\) particle \(p_{i}^{j}\), the observation vector \(y_{s}^{i}=\phi_{i,USE}^{j}x_{s}\) for speech, and \(y_{n}^{i}=\phi_{i,USE}^{j}x_{n}\) for noise, frames are generated. 12. The OMP is applied to \(y_{s}^{i}\) and \(y_{n}^{i}\) independently to reconstruct \(x_{s}\) and \(x_{n}\), respectively. However, our aim is not to find \(x_{s}\) and \(x_{n}\), but to find the dominating basis vectors (columns) \(V_{s}^{i}\) and \(V_{n}^{i}\) in \(\phi_{USE}^{i}\) for \(y_{s}^{i}\) and \(y_{n}^{i}\). The OMP uses \(v_{n}\) as the termination threshold. 13. Our aim is to make \(\phi_{i,USE}^{j}\) a dominant columns group orthogonal sensing matrix where the columns used to represent the speech signals and noise signals are different. Therefore, speech signals can be extracted from the noisy samples using only the speech dominant columns group. 14. To match the above-mentioned condition, the \(\phi_{i,USE}^{j}\) matrix must satisfy the condition \(f_{obj}=0\) (Equation (17)). 15. However, the condition \(f_{obj}=0\) is only possible for the ideal cases. Therefore, the optimization is performed to minimize the \(f_{obj}\) value to its maximum extent or up to a certain termination threshold (\(\gamma^{th}\)) (Equation (18)). 16. \(f_{i,obj}^{j}\) is calculated through \(\phi_{i,USE}\) for each particle \(p_{i}^{j}\). 17. Once the \(f_{i,obj}^{j}\) is calculated for each particle in the current generation, based on the \(f_{i,obj}^{j}\) values obtained from all the generations (1 to \(j\)), the \(p_{i,best}\) (particle's best), and \(g_{best}\) (global best) values are estimated. 18. Based on these estimations, the new positions of particles \(p_{i}^{j+1}\) are calculated as described in Equations (7) and (8). 19. PSO repeats the process until the proper \(\phi_{i,USE}^{j}\) matrix is generated. 20.