Performing root-music, how do i retrieve frequency and phase data of a signal?
Subspace-based spectrum estimation methods give you a high resolution pseudospectrum that is used for frequency detection only. The complex amplitude (that includes the phase) can be calculated with linear least-squares once your frequency estimates are available.
In MUSIC and root-MUSIC,

You calculate the autocorrelation matrix of the sensor signals.
Calculate its eigenvalue decomposition. By looking at its largest magnitude eigenvalues, you can estimate how many frequencies are there in the signal.
Separate the noise subspace (i.e, the noise eigenvectors associated with the low magnitude noise eigenvalues) from the signal subspace.
Project a linear search-vector of candidate frequencies onto the noise subspace. The frequencies that are actually present in your signal will have 0 inner product with the noise eigenvectors since they are uncorrelated. So, the inverse of this projection (known as the pseudospectrum) will have peaks at those frequencies.
root-MUSIC is a more computationally robust because it models the pseudospectrum as a polynomial evaluated on the unit circle. The roots of this polynomial are our desired frequencies.

Okay, after this high-level description of the two methods, I want to discuss how you can calculate the complex amplitudes once you have the frequencies. So you have $N$ samples of a complex signal, $x$, composed of $L$ frequencies plus added noise $w$,
$$x(n) = \sum_{k=1}^{L} A_ke^{j\theta_k}e^{j\omega_k n} + w(n) \\
 = \sum_{k=1}^{L} \bar{A_k} e^{j\omega_k n} + w(n)$$
In vector form, this can be written as
$$\begin{bmatrix} x(1) \\ x(2) \\ \vdots \\ x(N)\end{bmatrix} =\begin{bmatrix} e^{j\omega_1 1} &amp; e^{j\omega_2 1} &amp; \cdots &amp; e^{j\omega_L 1} \\
\vdots &amp; \vdots &amp; \ddots &amp; \vdots \\
 e^{j\omega_1 N} &amp; e^{j\omega_2 N} &amp; \cdots &amp; e^{j\omega_L N}
\end{bmatrix} \begin{bmatrix} \bar{A_1} \\ \bar{A_2} \\ \vdots \\ \bar{A_L} \end{bmatrix} + \begin{bmatrix} w(1) \\ w(2) \\ \vdots \\ w(N)\end{bmatrix} \\
\mathbf{x} = \mathbf{\Omega \bar{A}} + \mathbf{w}
$$
Now, we know $\mathbf{x}$ and we have estimated the frequencies $\omega_k$ from root-MUSIC, so we know $\mathbf{\Omega}$. Now, the least squares estimate of $\mathbf{\bar{A}}$ is
$$\hat{\mathbf{\bar{A}}} = (\mathbf{\Omega^\top \Omega})^{-1} \mathbf{\Omega^\top x}
$$
Of course for this to work, $N &gt; L$, which is also a requirement for root-MUSIC. Now you can estimate the phase angles by calculating the angle of $\hat{\mathbf{\bar{A}}}$.