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8. Conclusion
The work reported in this chapter started with our work on compressive sensing for
direction of arrival (DOA) detection with a phased array (Shaw and Valley, 2010). In that
work, we realized that most work in compressive sensing concerned recovering signals on a
sparse grid. In the DOA domain, that meant that targets had to be on a set of grid angles,
which of course never happens in real problems. We found a recovery solution for a single
target in that work by scanning the sparsifying DFT over an offset index that was a measure
of the sine of the target angle but the solution was time consuming because the penalized
ell-1 norm recovery algorithm had to be run multiple times until the best offset and best
sparse solution was found and the procedure was not obviously extendable to multiple
targets. This work led us to the concepts of orthogonal matching pursuit and removing one
target (or sinusoid) at a time. But we still needed a reliable method to find arbitrary
frequencies or angles not on a grid. The next realization was that nonlinear least squares
could be substituted for the linear least squares used in most versions of OMP. This has
proved to be an extremely reliable method and we have now performed 10's of thousands of
calculations with this method. Since OMP is not restricted to finding sinusoids, it is natural
to ask if OMP with NLS embedded in it works for other functions as well. We have not tried
to prove this generally, but we have performed successful calculations using OMP-NLS with
signals composed of multi-dimensional sinusoids such as would be obtained with 2D
phased arrays (see also Li et al., 2001), signals composed of multiple sinusoids multiplied by
chirps (i.e. sums of terms of the form a k exp(i k t+b k t 2 ) and multiple Gaussian pulses within
the same time window.
9. Acknowledgment
This work was supported under The Aerospace Corporation's Independent Research and
Development Program.
10. References
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