Suitability of Correlation Arrays and Superresolution for Minehunting ...

DSTO-TN-0443
4.2 to 4.4 present ‘slightly new’ work that is about active systems and is ‘wideband’ in a
useful sense.
2.2 Possible Ways of Using the Cross-Shaped Array
Given a set of elements that forms a cross shape, there are at least three ways in which
those elements could be used. The first way is the correlation telescope mentioned
above. 1 A second way is to use the elements as a normal sonar receiving array.
However, the two-dimensional (2-D) array thus defined is quite unsatisfactory, for the
following reason. Since the grading, or weighting, function is the sum of the two
grading functions, the far-field beam pattern is the sum of the Fourier transforms of the
two grading functions. For simplicity, consider a system in which each arm is an
aperture (continuous array), so that the overall grading function [Steinberg 1976] is
g( x, y) = rect( x L) δ ( y) + δ ( x) rect( y L)
(2.1)
where L is the length of each of the two arms. Then the (amplitude) beam pattern is
−1
−1
G( l,
m) = Lsinc( Lλ
l) + Lsinc( Lλ
m)
(2.2)
where the direction cosines l and m are given in terms of the coordinates ( x , y,
z)
of
2 2 2
the image point by l = x r , m = y r , where r = x + y + z is the range. (An
image point is a point at which an image intensity is to be calculated.) (The expressions
(2.1) and (2.2) have not been normalised.) Then along the x axis, when we move from
the peak of the pattern to a point many sidelobes out along the x axis, the image
intensity drops off by only 6 dB. Such a pattern is clearly unsatisfactory.
A third way of applying the cross-shape arrangement is to use one of the arms on
transmit and the other arm on receive. This method works, because the combined
beam pattern is then the product of the two component beam patterns. The latter
feature results in the combined beam pattern intensity being small unless the
transverse displacement of the image point (from the point target) is small in both the x
and the y directions. Here ‘small’ means ‘not significantly greater than the
beamwidth.’
Thus, of the three methods described, the first and the third are sound while the
second is unsound. Yet the belief is widely held that the second method has been used,
commonly and successfully, in sonar. This cannot be. As far as the author is aware,
the cross-shaped arrays that have been used in sonar all utilise the third method.
1 As described in Section 3, the correlation telescope itself can be considered to come in a
‘traditional’ and an ‘updated’ version.
3