Signal Processing Tools for Radio Astronomy - Signal Processing Systems - page 444

diagonal ˙ n . This is recognized as a “rank-1 factor analysis” model in multivariate

analysis theory [ 18 , 26 ] . Given R , we can solve for g and ˙ n in several ways

[ 4 , 5 , 48 ] . For example, any submatrix away from the diagonal is only dependent
on g and is rank 1. This allows direct estimation of g . This property is related to
the gain and phase closure relations often used in the radio astronomy literature for
calibration (in particular, these relations express that the determinant of any 2

2
submatrix away from the main diagonal will be zero, which is the same as saying
that this submatrix is rank 1).
In general, there are more calibrator sources ( Q ) in the field of view, and we have
to solve ( 35 ) . We resort to an Alternating Least Squares approach. If

×

would be
known, then we can correct R for it, so that we have precisely the same problem as
we considered before, ( 27 ) , and we can solve for ˙ and ˙ n using the techniques

discussed in Sect. 5.3 . Alternatively, with ˙ known, we can say we know a reference

A H , and the problem is to identify the element gains

model R 0 = A ˙ = diag ( g )

from a model of the form

H

R = R 0 + ˙ n

or, after applying the vec ( · ) -operation,

vec ( R )= diag ( vec ( R 0 ))( g ⊗ g )+ vec (˙ n ) .

This leads to the Least Squares problem

R

2

g = arg mi g vec ( − ˙ n ) − diag ( vec ( R 0 ))( g ⊗ g ) .

This problem cannot be solved in c losed form. Alternatively, we can first solve an
unstructured problem: define x = g ⊗ g and solve

R 0 )) − 1 vec

R

x = diag ( vec ( ( − ˙ n )

gg H ,

or equivalently, if we define X =

X R

=( − ˙ n ) R 0 .

where denotes an entrywise matrix division. After estimating the unstructured

X , we enforce the rank-1 structure X gg H , via a rank-1 approximation, and find

an estimate for g . The pointwise division can lead to noise enhancement; this is
remediated by only using the result as an initial estimate for a Gauss-Newton
iteration [ 13 ] or by formulating a weighted least squares problem instead [ 45 , 48 ] .
With g known, we can again estimate

=

n , and make an iteration. Overall
we then obtain an alternating least squares solution. A more optimal solution can
be found by solving the overall problem ( 35 ) as a covariance matching problem
with a suitable parametrization, and the more general algorithms in [ 31 ] lead to an
asymptotically unbiased and statistically efficient solution.

˙ and ˙