This paper critically examines the potential performance benefits offered by motion of sparse arrays for direction-of-arrival (DOA) estimation. The motivation behind utilizing array motion is to increase the number of consecutive difference lags. However, creating a synthetic array also requires more temporal measurements compared to the static (non-synthetic) array. For the first time, we rigorously analyze the trade-off between the required number of temporal samples and the length of the difference co-array to understand when synthetic arrays offer distinct advantages. As a concrete result, we show that if the ratio of the number of consecutive lags of the difference coarray of sparse arrays with and without motion is above a universal threshold, the synthetic array outperforms its non-synthetic counterpart and has a smaller estimation error. Our claims are demonstrated both theoretically and through numerical experiments.

Sparse Array; Difference Co-Array; Array Motion; Synthetic Array; Sample Complexity Trade-Off