Source localization using recursively applied and projected (RAP) MUSIC Page: 4 of 8
This article is part of the collection entitled: Office of Scientific & Technical Information Technical Reports and was provided to UNT Digital Library by the UNT Libraries Government Documents Department.
Extracted Text
The following text was automatically extracted from the image on this page using optical character recognition software:
To appear in: Proceedings Thirty-first Annual Asilomar Conference on Signals, Systems, and Computers, Pacific Grove, CA, Nov. 2-5, 1997. Page 2 of 5
the covariance matrix into signal subspace (span(s) ) and
noise-only subspace (span(4n)) terms.
Let f denote the sample covariance matrix estimate
of R obtained by averaging the outer products of the data
vectors. Accordingly, we designate the first r eigenvectors
of f as (Is, i.e., a set of vectors which span our estimate
of the signal subspacex similarly we designate the estimated
noise-only subspace On using the remaining eigenvectors.
Finally, we generalize the array manifold vector for the
case of vector sources representing, for instance, diverse
polarization [1, 2] in conventional array processing or cur-
rent dipoles in EEG and MEG source localization [9, 10,
11]. In this case, the array manifold vector is the product of
an array matrix and a polarization or orientation vector,
a(0) = G(p)$ (3)
and we may view the parameter set for each source as
0 = {p, $}, comprising quasi-linear orientation parame-
ters $ and nonlinear location parameters p .
3. MUSIC and principal angles
The MUSIC algorithm [1, 2] finds the source locations
as those for which the corresponding array vector is nearly
orthogonal to the noise-only subspace, or equivalently,
projects almost entirely into the estimated signal subspace.
For the diversely polarized case, the problem becomes
more complex since the signal or noise-only subspaces
must be compared with the entire span of the gain matrix
G(p). A natural way to compare these two subspaces is
through use of principal angles [3] or canonical correla-
tions (i.e. the cosines of the principal angles) (cf. [6]).
Let q denote the minimum of the ranks of two matri-
ces A and B. The "subspace correlation" is a vector con-
taining the cosines of the q principal angles that reflect the
similarity between the subspaces spanned by the columns
of the two matrices. The elements of the subspace correla-
tion vector are ranked in decreasing order, and we denote
the cosine of the smallest principal angle (i.e., the largest
canonical correlation) asand the second is already orthogonalized, the square of this
signal subspace correlation is easily shown to be
^2 (a(0)H s~4 a() 5
subcorr1(a(9), s) = (a()H a(0)) (5)
(a(0) a(0))
where the right hand side is the standard metric used in
MUSIC [1, 2]. Practical considerations in low-rank E/MEG
source localization lead us to prefer the use of the signal
rather than the noise-only subspace [9, 11, 12]. The devel-
opment below in terms of the signal subspace is readily
modified to computations in terms of the noise-only sub-
space.
Principal angles can also be used to represent the
MUSIC metric for diversely polarized sources [1, 2] and
E/MEG dipole localization [10]. In this case, the algorithm
must compare the entire space spanned by the gain matrix
G(p) with the signal subspace. It is again straightforward
to equate the subspace correlation with Schmidt's diversely
polarized MUSIC solution,
subcorr (G(p), (ds) = Xma(UG(Ds$Fs UG), (6)
where UG is the orthogonalization of G(p) and ax( )
is the maximum eigenvalue of the enclosed expression.
The source locations p can be found as those for
which (6) is approximately unity. The quasi-linear parame-
ters $ can then be found as the eigenvector corresponding
to the maximum eigenvalue in (6). Equivalently, the singu-
lar vectors from the SVDs performed to compute
subcorr ( ) can be used to form 0 (see the appendix in
[9] for further details).
4. RAP-MUSIC
If the r -dimensional signal subspace is estimated per-
fectly, then the sources are simply found as the r global
maximizers of (6). Errors in our estimate (S reduce (6) to
a function with a single global maximum and at least
(r - 1) local maxima. Finding the first source is simple:
over a sufficiently densely sampled grid of the nonlinear
parameter space p, find the global maximum of (6),subcorr (A, B)
(4)
If subcorr1(A, B) = 1, then the two subspaces have at
least a one dimensional subspace in common. Conversely,
if subcorr1(A, B) = 0, then the two subspaces are
orthogonal. These subspace correlations are readily com-
puted using SVDs as described in [3] and reviewed in [9].
The MUSIC algorithm finds the source locations as
those for which the principal angle between the array vec-
tor and the noise-only subspace is maximum. Equivalently
the sources are chosen as those that minimize the
noise-only subspace correlation subcorr (a(0), On) or
equivalently maximize the signal subspace correlation
subcorr (a(0), Ds). Since the first argument is a vectorp = arg max(subcorr1(G(p), (Ds))
P(7)
We then extract the corresponding eigenvector to form the
quasi-linear parameter estimate . The estimate of the
parameters of the first source is denoted =i f 1 $1}
and the first estimated array manifold vector is formed asa(01) = G(pl) 1
(8)
Identifying the remaining local maxima becomes more
difficult since nonlinear search techniques may miss shal-
low or adjacent peaks and return to a previous peak. We
also need to locate the r best peaks, rather than any r local
maxima. Numerous techniques have been proposed in the
past to enhance the "peak-like" nature of the spectrum (cf.
Upcoming Pages
Here’s what’s next.
Search Inside
This article can be searched. Note: Results may vary based on the legibility of text within the document.
Tools / Downloads
Get a copy of this page or view the extracted text.
Citing and Sharing
Basic information for referencing this web page. We also provide extended guidance on usage rights, references, copying or embedding.
Reference the current page of this Article.
Mosher, J.C. & Leahy, R.M. Source localization using recursively applied and projected (RAP) MUSIC, article, March 1, 1998; New Mexico. (https://digital.library.unt.edu/ark:/67531/metadc690712/m1/4/: accessed March 28, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.