Suitability of Correlation Arrays and Superresolution for Minehunting ...

DSTO-TN-0443
Appendix B: Thoughts on Application of Capon’s
Method to the Minefinder System
Here an attempt is made to write down a procedure by which the Capon method
might be generalised to deal with the Minefinder problem. The appendix offers ideas
only, and is to be regarded as quite speculative.
In the standard problem treated in Section 8.1.1, the system is passive and the
sources are narrowband and far-field. We need to generalise the problem to an active
system, a wideband transmitter and targets in the near field, at the same time allowing
the directions to be specified by a 2-D angle (a pair of angles). Note as a preliminary
that, if a coded or chirped signal is used, decoding of the received signals is to be
carried out prior to other processing.
For the more general situation, we first discuss the specification of the problem—
i.e. ‘what parameters are to be found?’ We later look at the question of how to find
these parameters. At first sight, the specification of the problem appears to be: (A)
Given the received voltage streams, and given any image point in the 3-D field, to
determine the basic weight (real) and the delay 21 to be applied to each element. But
this formulation has a worrying feature. The delay is closely tied to estimated range. If
the delays can be varied by arbitrarily large amounts, a solution may be obtained in
which the estimated ranges of the targets are in gross error.
While (A) may still turn out to be satisfactory, we put forward a second and a third
alternative specification of the problem. The second is: (B) ‘Given the received voltage
streams, and given the image point in 3-D: (i) let us choose each delay to equal the time
for go-and-return via the image point; (ii) despite the explicit presence of delays, let us
choose the weights to be applied to each element to be complex; then (iii) the problem is
to determine these complex weights. 22 In (ii), it is taken that the image amplitude is the
sum of the analytic signals at the various elements, delayed as in choice (i) and
weighted by the complex weight. 23 ’ Note that the phase factor in the complex weight is
not accurately equivalent to an additional delay, since the system is wideband.
This last comment suggests the third specification (C), in which, as in (A), one
determines a real weight and a delay, but with the delay now confined to some narrow
interval about the delay-and-add value. The deviation ∆ t from the delay-and-add
value could be confined to, say, ∆ t ≤ 1 2 f
c
or ∆ t ≤ 3 4 f
c
, where f c
is the central
frequency. But the introduction of inequalities is messy and so this third formulation
(C) is not favoured.
Having specified (through (A), (B) or (C)) what parameters are to be found, we
need to specify a Capon-type criterion that yields a unique solution for those
parameters. Let us attempt to state what the criterion becomes when the formulation
21 The delay is the time interval from the time of projection of the ‘ping’ to the time at which the voltage
stream is evaluated for the purposes of beamforming.
22 Possibly the weights should be chosen to be independent of the range of the image point, but still
dependent on the 2-D angle.
23 Note that, though the weight is complex, it is combined with a delay.
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