### ASA 124th Meeting New Orleans 1992 October

## 4aUW11. LS multi-line curved array signal localization.

**Homer Bucker
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*Code 541, NRaD, NCCOSC, San Diego, CA 92152
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Let F[sub j] be the analytic signal received at sensor j of a hydrophone
array of unknown shape and let g[inf jk] =< F[inf j]F[inf k][sup *]> be an
element of the covariance matrix. Above, the * means complex conjugate and < >
indicates a time average. If there are N plane wave signals incident upon the
array and g[sub jk] is the expected covariance element after ``sufficient''
time averaging then g[sub jk] is equal to (summation)[sub n]A[sub n][sup 2]
exp{i(2(pi)/(lambda))[(x[sub j]-x[sub k])cos (phi)[sub n]+(y[sub j]-y[sub
k])sin (phi)[sub n]]}. In the above equation, A[sub n] and (phi)[sub n] are the
amplitude and bearing of signal n, x[sub j], and y[sub j] are the horizontal
coordinates of sensor j, and (lambda) is the wavelength. An LS (least-squares)
iteration takes place to reduce the error function E=(summation)[sub
j](summation)[sub k](cursive beta)sub jk]-g[sub jk]|[sup 2] to a minimum.
Values of {A[sub n]}, {(phi)[sub n]}, and several harmonic coefficients that
define the sensor coordinates relative to straight lines are adjusted to
minimize the error function. Several examples will be presented to illustrate
the method.