Post date: Jun 04, 2015 9:46:16 PM
MUSIC is a popular algorithm used to estimate the angle of arrivals (AOA) (a vector with dimension = the # of sources) in array signal processing. It is based on the following data model
where is the signal subspace for a particular AOA , is the received signal coefficient (source signal), and is assumed to be WGN. is the dimension of the vector-valued signal coming fromÂ
sensors, is the # of sources. A straightforward analysis shows that the data autocorrelation matrix is
where is the signal autocorrelation matrix.
Let
and denote the eigenvalues and eigenvectors of , respectively. Then
Notice that,
=> spans the same space as the signal subspace , hence the name the signal subspace.
because only has rank
=> the grouping which means linear combination of
is orthogonal to ( is Hermitian), defined as the noise subspace.
Thus to find , search for a that is in the signal subspace/orthogonal to the noise subspace
where . MUSIC can be extended to exploit the symmetry of many sensor arrays with ESPRIT/rotation-invariance technique [2].
Finally, since eigenvalues of autocorrelation matrix of a signal form its power spectrum [1] (Szego's theorem), it is interesting to know that MUSIC is simply the frequency domain version of the
cross-correlation (CC) method. The following is a summarization of key points in [1].
- Complex exponentials uniformly spaced on a unit circle in the complex plane is the building block:
- Sinusoidal in, sinusoidal out, with changes in magnitude and phase: , where is the frequency index, is the time index (reversed from [1]), and are LTI impulse response in time and frequency domains, respectively.
- Filtering in circulant matrix form: If
, circulant, then . In other words, any circulant has complex eigenvalues given by and the (complex exp) eigenvectors.
- Apply the same analogy for the circulant correlation matrix
. The power spectrum density , i.e.
[1] http://www.eecs.berkeley.edu/Pubs/TechRpts/2006/EECS-2006-90.pdf
[2] http://www.girdsystems.com/pdf/GIRD_Systems_Intro_to_MUSIC_ESPRIT.pdf