OPTIMAL AND SUB-OPTIMAL SOURCE LOCALIZATION IN A WAVEGUIDE Joseph Tabrikian and Hagit Messer Department of Electrical Engineering - Systems, Tel-Aviv University Tel-Aviv 69978, Israel ABSTRACT In this paper, an optimal and a sub-optimal proce- dures for source localization in a waveguide are pre- sented. We assume that the array consists of vertical and non-vertical sub-arrays. The performance of the sub-optimal procedure is evaluated analytically and nu- merically and is compared to the performance of the optimal one. The sub-optimal method involves a two dimensional search procedure instead of a three dimen- sional search in the optimal one. Numerical examples with channel parameters which are typical to shallow water source localization, show that the sub-optimal performance loss is 0-3dB, depending on the scenario conditions. 1. INTRODUCTION Source localization in a waveguide, based on matched- field-processing (MFP), has received extensive atten- tion in recent years. The earliest MFP techniques were proposed by Hinich [3] and Bucker [2]. Following these works, many source range and depth estimation tech- niques using a vertical array were developed, based on the normal mode representation of the received data This paper deals with three dimensional passive lo- calization of a narrow-band point source in a waveguide using an array of sensors. The waveguide consists of t- wo parallel infinite plates. This model is widely used as an approximation to the shallow water environment for the source localization problem. In [6] performances of two spatial sampling meth- ods by Linear Equi-Spaced (LES) arrays in vertical and horizontal configurations have been studied, and fun- damental limits of these arrays were derived. Based on this spatial sampling study, two localization procedures using a hybrid array, consisting of the horizontal and the vertical arrays, are examined and compared. Many existing algorithms, mostly used in source localization in shallow water, are based on range and depth estima- tions using a vertical array and DOA estimation using a separate, horizontal array. The estimations of range and depth from the vertical array are used for DOA estimation. This sequential algorithm is sub-optimal, [I, 3,41. Figure 1: Problem geometry since it ignores the capability of a horizontal array to estimate source range and depth in addition to DOA. In this paper, we study the performance loss when us- ing the sub-optimal, sequential algorithm, relative to the performance of an optimal algorithm, in which the three location parameters are estimated simultaneously using the data received by both horizontal and vertical arrays. 2. PROBLEM FORMULATION We consider a point source, located in a waveguide at depth z, below the upper surface of the waveguide, radiating a non-monochromatic signal at angular fre- quency W. The waveguide is assumed to be determinis- tic and time invariant. The field, excited by the source, is sampled by a general three dimensional array of N sensors, where the range between its origin and the source is T, (see figure 1). The sensors are located at (a, zi), i = 1, . . . , N, where a is the position of sensor i in the horizontal plane and zi denotes its depth. The sensor locations are assumed to be known. In the far field case, where the range T, is large compared with the array aperture, the range ri from the source to sensor t where B is the source direction. This relation is depicted in the horizontal plane in figure 2. Under far field assumption, the field measured by sensor i at time t can be expressed by (see [7]): At i is: Ti z To - $)sine - z)2)cose = T, +L (e),,