Signal Processing Tools for Radio Astronomy - Signal Processing Systems - page 446

It is important to note that G can be identified only up to a unitary factor V at the
right: G = GV would also be a solution. This factor makes the gains unidentifiable

unless we introduce more structure to the problem.
To make matters worse, note that this problem is used to fine-tune earlier coarser
models ( 33 ) . At this level of accuracy, the number of dominant sources Q is often
not small anymore, making G not identifiable.
As discussed in [ 30 ] and studied in more detail in [ 39 ] , more structure needs to be
introduced to be able to solve the problem. Typically, what helps is to consider the
problem for a complete observation (rather than for a single snapshot R )wherewe
have many different frequencies f k and time intervals m . The directional response
matrix A m , k varies with m and k in a known way, and the instrumental gains g and b
are relatively constant. The remaining part of G A is due to the ionospheric

perturbations, and models can be introduced to describe its fluctuation over time,
frequency, and space using some low order polynomials. We can also introduce
stochastic knowledge that describe a correlation of parameters over time and space.
New instruments such as LOFAR and SKA will only reach their full potential
if this general calibration problem is solved. For LOFAR, a complete calibration
method that incorporates many of the above techniques was recently proposed in
[ 16 ] . In general, calibration and imaging need to be considered in unison, leading
to many potential directions, approaches, and solutions. This promises to be a rich
research area in years to come.

gb H =

7 Factor Analysis

7.1 Introduction

Many array signal processing algorithms are at some point based on the eigenvalue
decomposition, which is used e.g., to make a distinction between the “signal
subspace” and the “noise subspace”. By using orthogonal projections, part of the
noise is projected out and only the signal subspace remains. This can then be used
for applications such as high-resolution direction-of-arrival estimation, blind source
separation, etc. In these applications, it is commonly assumed that the noise is
spatially white. However, this is valid only after suitable calibration.
Factor analysis considers covariance data models where the noise is uncorrelated
but has unknown powers at each sensor, i.e., the noise covariance matrix is an
arbitrary diagonal with positive real entries. In these cases the familiar eigenvalue
decomposition (EVD) has to be replaced by a more general “Factor Analysis”
decomposition (FAD), which then reveals all relevant information. It is a very
relevant model for the early stages of data processing in radio astronomy, because
at that point the instrument is not yet calibrated and the noise powers on the various
antennas may be quite different. We saw two examples in Sect. 6 .