AIC and MDL tests

Direction finding algorithms such as MUSIC and
ESPRIT require knowledge of the number of sources of signals impinging
on the array or equivalently, the dimension, *d*,
of the signal subspace. The Akaike Information Criterion (AIC) and
the Minimum Description Length (MDL) formulas are two frequently-used
estimators for obtaining that dimension. Both estimators assume that,
besides the signals, the data contains spatially and temporally white
Gaussian random noise. Finding the number of sources is equivalent
to finding the multiplicity of the smallest eigenvalues of the sampled
spatial covariance matrix. The sample spatial covariance matrix constructed
from a data snapshot is used in place of the actual covariance matrix.

A requirement for both estimators is that the dimension of the
signal subspace be less than the number of sensors, *N*,
and that the number of time samples in the snapshot, *K*,
be much greater than *N*.

A variant of each estimator exists when forward-backward averaging
is employed to construct the spatial covariance matrix. Forward-backward
averaging is useful for the case when some of the sources are highly
correlated with each other. In that case, the spatial covariance matrix
may be ill conditioned. Forward-backward averaging can only be used
for certain types of symmetric arrays, called *centro-symmetric* arrays.
Then the forward-backward covariance matrix can be constructed from
the sample spatial covariance matrix, *S*, using *S*_{FB} =
S + JS*J where *J* is the exchange matrix.
The exchange matrix maps array elements into their symmetric counterparts.
For a line array, it would be the identity matrix flipped from left
to right.

All the estimators are based on a cost function

plus an added penalty
term. The value λ_{i} represent
the smallest *(N–d)* eigenvalues of the spatial
covariance matrix. For each specific estimator, the solution for *d* is
given by