Estimation of signal parameters via rotational invariance techniques

Let us define a signal vector as,

${\displaystyle a(w_{k})=[1\quad e^{-jw_{k}}\quad e^{-j2w_{k}}\quad ...\quad e^{-j(m-1)w_{k}}]^{T}}$

where ${\displaystyle w_{k}}$ represents the radial frequency of kth sinusoid. Let us construct a Vandermonde matrix for K number of sinusoids, such as

${\displaystyle A=[a(w_{1})\quad a(w_{2})\quad ...\quad a(w_{K})]}$

Let us divide the matrix A into two sets, such as

${\displaystyle A_{2}=[I_{m-1}\quad 0]A}$

and

${\displaystyle A_{1}=[0\quad I_{m-1}]A}$ where ${\displaystyle I_{m-1}}$ is an identity matrix of size (m-1)-by-(m-1). It is clear that, ${\displaystyle A_{1}}$ contains the last (m-1) rows of A, while ${\displaystyle A_{2}}$ contains the first (m-1) rows of A.

Now, let us write the following relation,

${\displaystyle A_{2}=A_{1}H}$ Here, H is a diagonal matrix, where its diagonal elements can be written in a vector, such as,

${\displaystyle diag(H)=[e^{-jw_{1}}\quad ...\quad e^{-jw_{K}}]}$

In other words, the diagonal elements of H, are the complex exponentials with the radial frequencies of the set ${\displaystyle \{w\}_{k=1}^{K}}$. Here, it is clear that H applies a rotation to the matrix ${\displaystyle A_{1}}$. ESPRIT exploits similar rotations from the covariance matrix of the measured data.

To understand the algorithm itself, let us denote R as the covariance matrix of the measured data. By computing the eigenvalue decomposition of R (via algorithms like singular value decomposition), the following can be written,

${\displaystyle R=UEV^{*}}$ where E is a diagonal matrix that contains the eigenvalues of R, in a decreasing order. Here, by finding the eigenvalues that are higher than the variance of the noise, we can separate the orthonormal eigenvectors from U, that correspond to these eigenvalues. This can be noted as ${\displaystyle S=U(:,1:K)}$ where we kept only the first K columns.

As similar before, we can make the following separation on S,

${\displaystyle S_{1}=[I_{m-1}\;0]\;S}$ and ${\displaystyle S_{2}=[0\;I_{m-1}]\;S}$.

Moreover, there exists a relation between S and A such as ${\displaystyle S\;=\;A\;F}$, where the content of the matrix F is known, but irrelevant for the current subject. We can derive the following relations,

${\displaystyle S_{2}=A_{2}\;F\;=\;A_{1}\;H\;F\;=\;S_{1}\;F^{-1}\;H\;F\;=\;S_{1}\;P}$ (where we made use of ${\displaystyle S_{1}\;=\;A_{1}\;F}$ and ${\displaystyle A_{1}\;=\;S_{1}\;F^{-1}}$).

It is clear that the matrix P contains rotational information with respect to the frequency contents, such that the rotation on the first set of orthonormal eigenvectors yield to the second set. Moreover, the eigenvalues of P are equal to the diagonal elements of H. Therefore, by solving the following equation for P,

${\displaystyle S_{2}=S_{1}P}$

we can estimate the frequency content. To achieve this, the above equation can be solved with pseudo inverse (via Least squares) method.

To do so, ${\displaystyle P=(S_{1}^{*}S_{1})^{-1}S_{1}^{*}S_{2}}$ can be written.

Finally, by finding the angles of the eigenvalues of P, one can estimate the set ${\displaystyle \{w\}_{k=1}^{K}}$.

A pseudo code is given below for the implementation of ESPRIT algorithm.

function esprit(y, model_order, number_of_sources):
    m = model_order
    n = number_of_sources
    create covariance matrix R, from the noisy measurements y. Size of R will be (m-by-m).
    compute the svd of R
    [U, E, V] = svd(R)
    
    obtain the orthonormal eigenvectors corresponding to the sources
    S = U(:, 1:n)                 
      
    split the orthonormal eigenvectors in two
    S1 = S(1:m-1, :) and S2 = S(2:m, :)
                                               
    compute P via LS (MATLAB's backslash operator)
    P = S1\S2 
       
    find the angles of the eigenvalues of P
    w = angle(eig(P))               
    return w

• Independent component analysis

• Paulraj, A.; Roy, R.; Kailath, T. (1985), "Estimation Of Signal Parameters Via Rotational Invariance Techniques - Esprit", Nineteenth Asilomar Conference on Circuits, Systems and Computers, pp. 83–89, doi:10.1109/ACSSC.1985.671426, ISBN 978-0-8186-0729-5.
• Roy, R.; Kailath, T. (1989). "Esprit - Estimation Of Signal Parameters Via Rotational Invariance Techniques" (PDF). IEEE Transactions on Acoustics, Speech, and Signal Processing. 37 (7): 984–995. doi:10.1109/29.32276..
• Ibrahim, A. M.; Marei, M. I.; Mekhamer, S. F.; Mansour, M. M. (2011). "An Artificial Neural Network Based Protection Approach Using Total Least Square Estimation of Signal Parameters via the Rotational Invariance Technique for Flexible AC Transmission System Compensated Transmission Lines". Electric Power Components and Systems. 39 (1): 64–79. doi:10.1080/15325008.2010.513363.
• Haardt, M., Zoltowski, M. D., Mathews, C. P., & Nossek, J. (1995, May). 2D unitary ESPRIT for efficient 2D parameter estimation. In icassp (pp. 2096-2099). IEEE.