On the mapping by a cross-correlation antenna system of partially coherent radio sources

In this paper it is shown that by means of a crosscorrelation antenna system it is possible to measure the mutual power distribution,
T(u, v)
, of partially coherent radio sources. The two linear antennas of the system are scanned independently to give a two-dimensional output which can be written as
\bar{R}(u, v) = T(u, v) \ast A(-u)B^{\ast}(-v)
where
A(u)
and
B(v)
are the patterns of the two antennas. A two-dimensional Fourier analysis of the output shows it to be a smoothed version of the true mutual power distribution. One can define a principal solution,
T_{0}(u, v)
, which is a generalization to two dimensions of the well-known principal solution,
T_{0}(u)
, which occurs when the sources are incoherent. The limiting cases of complete coherence and complete incoherence are considered. It is shown that for coherent sources the principal solution is factorable,
T_{0}(u, v)=\varepsilon_{0}(u)\varepsilon_{0}^{\ast}(v)
. For incoherent sources it is shown that the spatial frequency spectrum of
T_{0}(u, v)
is a function of only the difference,
x-y
, of the spatial co-ordinates,
t_{0}(x, y) = t_{0}(x- y)
.