Application of a state variable technique to the optimization of the SNR of an underwater array subject to a constraint on the super-gain ratio

Consider the design of a point detector array to extract a signal from background noise, using the criterion of maximizing the SNR. For certain array configurations, although we can theoretically find a particular set of excitations that result in a very high SNR, the SNR may be extremely sensitive to small random variations in the excitations about their nominal values. The super-gain ratio (or Q-factor) is a measure of this sensitivity. A recent formulation for the optimization of the SNR with a constraint on the Q-factor, reduces the problem to one of solving for the roots of a polynomial whose coefficients are, in general, complex. However, only real polynomials were considered, probably because of the excessive computation time involved. This paper (1) considers what happens when the coefficients are complex, and more importantly (2) demonstrates that by making a certain transformation it becomes possible to employ a state variable technique which changes the numerical problem into one of finding the eigenvalues of a real matrix which is considerably faster to do.