Joint estimation of frequency and direction of arrival under the single-and-parallel spatial-temporal undersampling condition

Abstract

As the application frequency is increasingly high, it becomes difficult to design joint estimators for the frequencies and directions of arrival (DOAs) under the spatial-temporal undersampling condition. Specifically, on one hand, the temporal Nyquist theorem requires that the sampling rate be at least twice the highest frequency, which is unfordable for the existing analog-to-digital converters; on the other hand, the spatial Nyquist theorem also requires that each inter-element spacing be less than or equal to half the wavelength, which inevitably results in severe mutual coupling among sensors. To solve these intractable problems, in this paper, we propose a joint estimator based on a co-prime sparse array. Firstly, based on the combination of this sparse array and the closed-form robust Chinese remainder theorem (CRT), the theoretical model for the proposed frequency and DOA joint estimator is built up. Secondly, at each sensor, a frequency estimate for the source object can be calculated through implementing the closed-form robust CRT on two frequency remainders, which are generated by applying the Tsui spectrum correction to the discrete Fourier transform results of two receiver sequences. Then, averaging these estimates at all sensors yields the final frequency estimate. Lastly, on the basis of frequency estimation, the final DOA estimate can be calculated through implementing the closed-form robust CRT on all phase-difference remainders, which are also derived from the Tsui spectrum correction. It needs to be emphasized that the proposed joint estimator possesses two attractive merits. One merit is that due to the fact that the proposed array allows a high sparsity of element-spacings, both the hardware cost and the mutual coupling among sensors can be considerably reduced; the other merit is that compared with the existing estimators, the proposed joint estimator achieves high estimation precision even in the single-and-parallel undersampling condition (i.e., multi-time undersampling is bypassed in each sensor element, leading to a high data processing efficiency). In particular, this high accuracy attributes to two aspects:1) the Tsui spectum corrector incorporated in the proposed joint estimator can provide high-accuracy remainders for the CRT recovery; 2) the closed-form robust CRT itself is unbiased and thus the CRT recovery brings no extra system errors. Numerical results verify that the proposed joint estimator possesses both strong noise robustness and high estimation accuracy, which presents a vast potential application in several passive sensing fields such as radar and remote sensing.