## 6 Modified Unitary R-D Root MUSIC
For example, considering \(\lambda_{i}\in\mathbb{R},i=1,\ldots,M\) to be eigenvalues of a diagonalizable matrix, if \(|\lambda_{1}|>|\lambda_{2}|\geq\cdots\geq|\lambda_{M}|\) then the cited matrix satisfies the convergence requirements. The matrix \(\mathbf{C}_{x}\) is a real covariance matrix, thus it is symmetric [41]; therefore, it is diagonalizable, and its all eigenvalues are real numbers [42]. The Power Method outputs the eigenvector and its associated eigenvalue, which is the greatest in magnitude. Here, we prove that the greatest eigenvalue in magnitude is the one of the signal subspace. Note that \(\mathbf{C}_{x}\) is a real covariance matrix, hence, it is positive semi-definite [41], which means all eigenvalues are non-negative. The line-of-sight (LOS) component of the received signal that constitutes the eigenvalue of the signal subspace is greater than the eigenvalues of the noise subspace [24], and since they are all non-negative, it is not possible to have eigenvalues of the noise subspace equal to or greater than the one of signal subspace in magnitude. Therefore, the eigenvalue of the signal subspace is the greatest in magnitude, hence its corresponding eigenvector is the signal subspace. The same analysis goes to \(\mathbf{C}_{y}\). The implemented Power Method (Algorithm 2) does not compute the eigenvalue since the solution only needs the eigenvector. We considered \(K=30\) and \(tol=10^{-6}\). We carried out thousands of experiments and we verified that in most cases the Algorithm 2 takes 4 to 5 iterations to converge, and 30 iterations are much more than enough in all experimental instances, hence for \(k>30\) we assume the algorithm fails to compute the signal subspace. Finding all roots of a polynomial is a difficult computational task that requires time-consuming methods, and to aggravate the problem, the polynomial in question has complex coefficients with complex roots. Classical algorithms that operate directly on the polynomial function, such as the Newton-Raphson method, Secant method, and [PERSON]'s method, may be hard to work in practice. They only estimate one single root, some are guaranteed to converge if only certain conditions are satisfied and might not work on complex-valued polynomials, and the initial point must be chosen wisely since it could impact their convergence [43; 44]. Although they can be extended to estimate multiple roots, they are highly sensitive to computing error since they operate directly on the polynomial function. That is the reason practical computer eigenvalue solvers, such as in MATLAB [45], hardly resemble these root-finding algorithms [46]. Instead, they apply EVD on the polynomial's companion matrix to find the roots. Since such a matrix is non-symmetric, the implemented solution estimates the roots of polynomial \(p_{x}(z)\) (or \(p_{y}(z)\)) defined in Equation (22) (or (23)) by applying the Shifted QR Algorithm in which its implementation is an adaptation of the algorithm found in [33]. The Shifted QR Algorithm is a well-known method that performs  
Figure 6: Algorithm overview of the two covariance noise subspace computations. The left method calculates that covariance directly, but the right one calculates indirectly via the signal subspace. exceptionally well in practice and works even on non-symmetric matrices. It is an EVD method and an improved version of the standard QR Algorithm by including the shifting technique for rapid convergence. This algorithm calculates the eigenvalues of the companion matrix of \(p_{x}(z)\) (or \(p_{y}(z)\)), which are its roots. To simplify, let us consider a polynomial of the form \(p(z)\), where it could be either \(p_{x}(z)\) or \(p_{y}(z)\). ```
Input: covariance matrix \(\mathbf{C}=\mathbf{C}_{x}\) or \(\mathbf{C}=\mathbf{C}_{y}\). Output: signal subspace \(\mathbf{U}_{S,x}\) (or \(\mathbf{U}_{S,y}\)). 1 Define \(\mathbf{v}_{1}=[1,1,\ldots,1]^{T}\in\mathbb{R}^{M}\), \(\mathbf{v}_{0}=\mathbf{0}_{M}\), \(k=1\), \(tol\ll 1\in\mathbb{R}_{>0}\), and \(K\in\mathbb{Z}_{>0}\).