Subspace methods

We start with the Carathacute{e}odory-Fejacute{e}r-Pisarenko result for Toeplitz matrices. Given a ntimes n positive definite Toeplitz matrix T, let c>0 be the smallest number such that T-cI is singular and has rank m<n, then T-cI has the formE

 T-cI=sum_{k=1}^m rho_k F(e^{itheta_k})F(e^{itheta_k})^*


 F(e^{itheta_k})= left[                     begin{array}{c}                       1                        e^{itheta_k}                        vdots                        e^{i(n-1)theta_k}                      end{array}                   right].

Accordingly, the power spectrum f(theta) deomposes as

 f(theta)= sum_{k=1}^m 2pirho_k delta(theta-theta_k)+c

where delta(cdot) is the Dirac function. The decomposition corresponds to a set of white noise. See MA decomposition for a decomposition corresponds moving-average (MA) noise.

The above result can be generalized to the case of state covariances [1]. More specifically, let Q be the unique solution to the Lyapunov equation


The matrix Q is the state covariance when the input is pure white noise. Let now R be an arbitray state covariance, let c be the smallest eigenvalues of the matrix pencil R-rho Q, and assme that the rank of R-c Q is m, then

 R-c Q=sum_{k=1}^m rho_k G(e^{itheta_k})  G(e^{itheta_k})^*,

where G(e^{itheta}) is generalization of F(e^{itheta}).

The subspace spectral analysis methods rely on the singular value decomposition

 R-cQ= U mbox{diag}{lambda_1, lambda_2, ldots, lambda_m, 0,ldots, 0 } U^*,

where U is a unitary matrix and lambda_k>0, k=1,ldots, m. Partition

 U=[U_{1:m},; U_{m+1:n}]

where U_{1:m} and U_{m+1:n} are the first m and the last n-m columns of U, respectively. Based on this decomposition, there are two ways we can proceed generalizing the MUSIC and ESPRIT methods, respectively [P. Stoica, R.L. Moses, 1997].

Noise subspace analysis

The columns of U_{m+1:n} span the null space of R-cQ, while the signal G(e^{itheta_k}), k=1,ldots,  m is in the span of the columns of U_{1:m} . So the nonnegative trigonometric polynomial

 d(e^{itheta},e^{-itheta})=G(e^{itheta})^* U_{m+1:n}U_{m+1:n}^*G(e^{itheta})

has m roots at {theta_1, ldots, theta_m }.

Given a sample state covariance matrix hat{R} and an estimate on the number of signals m<n, we let hat{U}_{m+1:n} be the matrix of singular vectors of the smallest singular values, and hat{d}(e^{itheta},e^{-itheta}) be the corresponding trigonometric polynomial for hat{U}_{m+1:n}. Two possible generalization of the MUSIC method are as follows.

  1. Spectral MUSIC:  identify theta_k, for k=1,ldots, m as the values on [-pi,; pi] where 1/hat{d}(e^{itheta},e^{-itheta}) achieves the m-highest local maxima.

  2. Root MUSIC:  identify theta_k, for k=1,ldots, m as the angle of the m-roots of hat{d}(z,  z^{-1}) which have amplitude <1 and are closest to unit circle.

Signal subspace analysis

A signal subspace method relies on the fact that for the pair A, B and U_{1:m} given above, there is a unique solution to the following matrix equation

 U_{1:m} =[B ~ AU_{1:m}] left[begin{array}{c}muPhi end{array} right],

where mu is 1times m row vector and Phi a mtimes m matrix. The eigenvalues of Phi are precisely e^{itheta_k} for k=1,ldots, m. If R is a Toeplitz matrix and A, B given as in (1) with A a companion matrix, then we recover the ESPRIT result [P. Stoica, R.L. Moses, 1997].

The signal subspace estimation computed using sm.m, whereas music.m and esprit.m implement the MUSIC and ESPRIT methods, respectively. For the example discussed above, the estimated spectral lines are shown in the following figure. For an extensive comparison of the high resolution method sm.m with MUSIC and ESPRIT methods, see the example.

% subspace method
arrowb(thetas,residues); hold on;
[th_esprit,r_esprit]=sm(That,Ac,Bc,2); % ESPRIT spectral lines