**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 *akexp(i*Z*kt+bkt2 )* and multiple Gaussian pulses within the same time window.
