Phased arrays are diversely applied with some specific areas including biomedical imaging and therapy, non-destructive testing, radar and sonar. In this thesis, the matrix pencil method is employed to reduce the number of elements in a linear ultrasound phased array. The non-iterative, linear method begins with a specified pressure beam pattern, reduces the dimensionality of the problem, then calculates the element locations and apodization of a reduced array. Computer simulations demonstrate a close comparison between the initial array beam pattern and the reduced array beam pattern for four different linear arrays. The number of elements in a broadside-steered linear array is shown to decrease by approximately 50% with the reduced array beam pattern closely approximating the initial array beam pattern in the far-field. While the method returns a slightly tapered spacing between elements, for the arrays considered, replacing the tapered spacing with a suitably-selected uniform spacing provides very little change in the main beam and low-angle side lobes.
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