Suitability of Correlation Arrays and Superresolution for Minehunting ...

DSTO-TN-0443
(italics added). In this last clause he implies that it is not difficult to achieve
considerably greater resolution than that given by delay-and-add; that is, it is not
difficult to do this in the context of radio direction finding.
Haykin then proceeds to the Capon method. ‘Capon realised that the poor
resolution of the delay-and-add technique can be attributed to the fact that the power
of the delay-and-sum processor at a given direction does not depend only on the
power of the source at that direction, but also on undesirable contributions from other
sources.’ Capon’s approach consists of two parts. First, the array is still steered to a
direction θ via a linear combination of the received signals, using a weighting vector
θ
w θ now contains an
w () (vector of augmented weights). However, each element of ()
jβ
additional phase factor e (and also an additional real factor).
Prior to the optimisation step, a ‘steering condition’ is invoked that puts a linear
constraint on w () θ . Though stated by Haykin, this constraint is better expressed by
S&S as follows: the constraint is that the array gain is unity at the look direction θ .
At this point we introduce terminology from Haykin as follows. At the look angle
θ , first there is the contribution from the desired target or source, which is very close
to θ . The remaining contribution to the received signal, called interference, is due to the
other signals entering through the sidelobes, together with noise.
Haykin moves on to the second part of Capon’s approach, as follows. ‘Now in
order to minimise the contribution to the output power of other sources at directions
different from θ , Capon proposed to select the vector w () θ so as to minimise the output
power’ (italics added) subject to the steering constraint. Here the output power concerned
is the power received at the look direction θ . One can make intuitive sense of this
minimisation principle as follows. The constraint fixes the output power received at θ
due to the desired signal. So (provided powers are additive) the minimising must
minimise the power received at θ due to the interfering signals and noise. This is just
what one wants to do.
Note that the ‘intuitive sense’ argument does not cover the case where an
interfering signal is correlated with the desired signal (since powers are then not
additive). Haykin comments: ‘[This] minimisation of the power output ... may create
serious difficulties when the interfering signals are correlated with the desired signals.
In this case the resulting vector [ w () θ ] will operate on the interfering signals so as to
partially or fully cancel the desired signals, as dictated by the minimisation
requirement.’ The possible existence of these ‘serious difficulties’ associated with
correlation suggests that, if an active system is used in conjunction with the Capon
method, some method of decorrelation (Sections 4.3.1, 11) will be essential.
After the optimum w () θ has been calculated by the two parts of Capon’s approach
as above, the angular power spectrum is obtained by inserting that value into the usual
beamforming formula for power. The result should be equal to the true (source) power
spectrum, combined with less smoothing that with delay-and-add; in other words the
result should be better resolution.
Haykin proceeds to describe an alternative version of the minimisation principle,
which is equivalent to the above one provided that the interfering signals are uncorrelated
with the desired signal. The alternative is that ‘the minimum variance technique can be
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