The MUSIC algorithm (multiple signal classification) is a method of spectral estimation that estimates signal parameters from the eigen decomposition of a covariance matrix. It exploits the orthogonality between a signal subspace and a noise subspace (Schmidt 1986, pp. 277-278).
Consider the model
 |
(1)
|
where 
,
, 
, and the columns of 
are known vectors 
, ..., 
evaluated at unknown parameters.
Assume 
and 
have zero mean, are uncorrelated,
and have covariance matrices 
and 
, respectively, where 
and 
is the identity matrix.
Then
 |
(2)
|
where 
denotes expectation value and the superscript
H denotes conjugate transpose. If 
has matrix rank 
and 
is positive definite,
the eigenvectors corresponding to the 
largest eigenvalues of 
span
the signal subspace. The remaining 
eigenvectors have eigenvalue 
and span
its orthogonal complement, the noise subspace.
Let 
have orthonormal columns spanning
the noise subspace. The defining relation
 |
(3)
|
for 
,
..., 
leads to the MUSIC pseudospectrum
 |
(4)
|
In practice, the covariance matrix is estimated from observations and the parameters are estimated
from the largest peaks of 
. Unique recovery additionally requires that no other
parameter value produce a vector 
in the signal subspace.
The pseudospectrum is a parameter-location criterion rather than an estimate of signal
power (Schmidt 1986, pp. 277-279; Stoica and Nehorai 1989, pp. 721-723).
For frequency estimation, one uses
 |
(5)
|
where 
is measured in radians per sample and the superscript
T denotes transpose. Xie and Mo (2026) apply MUSIC
to selecting poles for expansions using Takenaka-Malmquist
systems.
