DOA estimation exploiting a uniform linear array with multiple co-prime frequencies

The co-prime array, which utilizes a co-prime pair of uniform linear sub-arrays, provides a systematical means for sparse array construction. By choosing two co-prime integers M and N , O(MN) co-array elements can be formed from only O(M +N) physical sensors. As such, a higher number of degrees-of-freedom (DOFs) is achieved, enabling direction-of-arrival (DOA) estimation of more targets than the number of physical sensors. In this paper, we propose an alternative structure to implement co-prime arrays. A single sparse uniform linear array is used to exploit two or more continuous-wave signals whose frequencies satisfy a co-prime relationship. This extends the co-prime array and filtering to a joint spatio-spectral domain, thereby achieving high flexibility in array structure design to meet system complexity constraints. The DOA estimation is obtained using group sparsity-based compressive sensing techniques. In particular, we use the recently developed complex multitask Bayesian compressive sensing for group sparse signal reconstruction. The achievable number of DOFs is derived for the two-frequency case, and an upper bound of the available DOFs is provided for multi-frequency scenarios. Simulation results demonstrate the effectiveness of the proposed technique and verify the analysis results.