Signal Processing Tools for Radio Astronomy - Signal Processing Systems - page 421

( ) − ( )

The vector z i is the baseline : the (normalized) vector pointing from

telescope i to telescope j . In radio astronomy, it is usually expressed in coordinates
denoted by

m z j m

( , , )

. The objective in telescope design is often to have as many
different baselines as possible. In that case the entries of R m are different and
non-redundant. As the Earth turns, the baselines also turn, thus giving rise to new
baseline directions. We will see later that the set of baselines during an observation
determines the spatial sampling function by which the incoming wave field is
sampled, with important implications on the quality of the resulting image.
If we generalize now ( 11 ) to Q sources and add zero mean noise, uncorrelated
from antenna to antenna, as in the signal model ( 8 ) , we obtain what is known as the
measurement equation , or covariance data model,

u v w

s A m + ˙

= ˙ ,

R m A m (12) n

where A m =[ a 1 ( m ) ,···, a Q ( m )]

1 Q

˙ s = diag { [ σ ,···, σ ] }

n H

2
n

2
n

˙ n = E { n ( n ) ( n ) } = diag { [ σ ,···, σ ] }.

, 1 , J

q 2

Here, σ = E {| s q ( n , k ) | } is the variance of the q th source, ˙ s is the corresponding

signal covariance matrix, and n is the noise covariance matrix. (With abuse of

notation, subscript n is now used to signify “noise”.) Noise is assumed to be
independent but not evenly distributed across the array. The noise variances

˙

2
n , j

σ

are considered unknown.
Under ideal circumstances, the array response matrix A m is just a phase matrix:
its columns are given by the vectors a q

in ( 9 ) , and its entries express the phase
shifts due to the geometrical delays associated with the array and source geometry.
We will later generalize this and introduce directional disturbances due to non-
isotropic antennas, unequal antenna gains, and disturbances due to atmospheric
effects.

( m )

3.4 Image Formation for the Ideal Data Model

Ignoring the additive noise and using the ideal array response matrix A m ,the
measurement equation ( 12 ) , in its simplest form, can be written as

Q
q = 1 I ( p q ) e − j ( z i ( m ) − z j ( m )) T p q

( ) =

R m (13) i , j

where ( R m ) i , j is the correlation between antennas i and j at STI interval m ,

2

I ( p q )= σ

q is the brightness (power) of the source in direction p q , z i (

m ) is the