## 2 Methodology
The eigenvectors forming \(U_{b}\) corresponding to the smallest eigenvalues are orthogonal to the column vectors of the transfer matrix \(A\):  
\[\big{\{}\big{[}u_{p+1}u_{p+2}\ldots u_{N}\big{]}\big{\}}\,\perp\,\big{\{}a( \tau_{1}),a(\tau_{2}),\ldots,a(\tau_{p})\big{\}} \tag{8}\]  
Therefore, we have:  
\[a^{H}(\tau)U_{b}=0 \tag{9}\]  
This orthogonality is due to the fact that the vectors of the signal subspace generate the same subspace as the column vectors of the source transfer matrix \(a(\tau)\) and as \(U_{s}\) is orthogonal to \(U_{b}\) we therefore have the columns of \(a(\tau)\) which are also orthogonal to those of \(U_{b}\). To estimate the directions of arrival of the sources, the model of the source vector \(a(\tau)\) must be known. The orthogonality of the directional vectors with the eigenvectors of the noise subspace is then characterized by a projection of the signal \(\tau\) subspace on the noise subspace (this is why the MUSIC method is also called projection subspace algorithm) and we seek the values for which the noise subspace is orthogonal to the signal subspace, which corresponds to the directions of arrival of the waves. The angular spectral function obtained by the MUSIC method makes it possible to determine the values of \(\tau\) for which this function is maximum, and it is defined in the following form:  
\[P_{MUSIC}(\tau)=\frac{1}{a(\tau)^{H}U_{b}U_{b}^{H}a(\tau)} \tag{10}\]  
where \(u_{i}(i=P+1,\ldots,N)\) is the \(i\)th vector of the matrix \(U_{b}\) resulting from the noise subspace. It should be noted that \(P_{MUSIC}(\tau)\) is not a real spectrum (it is a measure of the distance between two subspaces), it gives us peaks corresponding to the exact directions of arrival of the waves but does not tell us about the power of the sources. The classical MUSIC method requires that the source signals be completely uncorrelated so that the estimation of their directions of arrival is correct. However, the rank of the covariance matrix \(R\) is always equal to the number of completely uncorrelated signals, so as soon as there is a strong correlation between two signals, the rank of this correlation matrix is reduced by one unit, which implies an underestimation of the number of directions of arrival of the waves and the MUSIC algorithm is no longer directly applicable in this case. As described in the previous part, the classical MUSIC method is based on the spectral decomposition of the covariance matrix \(R\) for the estimation of the directions of arrival of the waves. The covariance matrix \(R\) is decomposed into \(P\) larger eigenvalues as well as \(N-P\) smaller values. The eigenvectors corresponding to the \(P\) largest eigenvalues construct a signal subspace that is identical to the \(A\) source transfer matrix and the eigenvectors corresponding to the \(N-P\) smallest eigenvalues construct a noise subspace. Through the property of orthogonality between the directional vectors of the signals \(a(\tau)\) with the vectors constituting the noise subspace, the null points, i.e., the arrival directions, are thus determined [16; 17; 18]. Considering \(P=5\) incident sources with DOAs randomly selected from [0\({}^{\circ}\), 180\({}^{\circ}\)] and adopting \(M=10\) uniform linear array with element spacing \(\frac{3}{2}\). The number of snapshots, \(L=100\) and the SNR is 0 dB, we present the spectra and DOA estimation results,