Array design based on non-linear equation constraints

Pulse-echo airborne ultrasound is widely used as a ranging technique. The only direct information available from this method is the time of flight (TOF) measurement. A 2D target position can be represented as a linear equation system when the bearing is measured by an array of sensors located on a line. To calculate the 3D position of targets a nonlinear equation system has to be solved. The solution of this system can be done by several numerical methods, mainly iterative processes. In this paper the solution is given by representing a non-linear system as a linear system using the Taylor expansion series. This approximate solution restricts the array geometry where no singularities appear in the solution. The proposed 2D array geometry, given by the approximate solution and the characteristics of the capacitive transducer built at Nottingham University, in conjunction with a composite algorithm of envelope beam-forming technique and Taylor expansion series algorithm, are used as to model a 3D ranging system. The results of 3D position of multiple targets are within the expected values given by range and angular resolutions of the ultrasonic system.