Co-prime arrays and difference set analysis

Co-prime arrays have gained in popularity as an efficient way to estimate second order statistics at the Nyquist rate from sub-Nyquist samples without any sparsity constraint. We derive an expression for the degrees of freedom and the number of consecutive values in the difference set for the prototype co-prime array. This work shows that, under the wide sense stationarity (WSS) condition, larger consecutive difference values can be achieved by using the union of all the difference sets. We provide a closed-form expression in order to determine the number of sample pairs that are available for estimating the statistics for each value of the difference set, also known as the weight function. The estimation accuracy and latency depends on the number of sample pairs used for estimating the second order statistic. We also obtain the closed-form expression for the bias of the correlogram spectral estimate. Simulation results show that the co-prime based periodogram and biased correlogram estimate are equivalent, and the reconstruction using our proposed formulation provides lower latency.