Sparse arrays have been studied mainly to reduce the large numbers of elements in 2-D arrays. However, they can also provide an effective means of miniaturizing ultrasound 1-D array systems for point-of-care applications. Although a variety of sparse array design strategies have been proposed, designing an optimum sparse array to simultaneously satisfy the system specification requirements and performance criteria remains a challenge. This article presents an analytical approach for the design of an optimum pair of periodic sparse arrays (PSAs), one for transmission and the other for reception. The approach is based on three newly derived theorems that describe the most important properties of the two PSAs forming the sparse array pair and their relationship pertaining to the overall beam pattern. The proposed approach can be used to design 1-D sparse array pairs with arbitrary sparseness factors while meeting given performance criteria. The computer simulation verified that the spatial resolution of a 64-element phased array can be obtained with a PSA pair consisting of transmit and receive sparse arrays, of which the number of elements is reduced to 32 and 22, respectively.