The fourth-order difference co-array construction by expanding and shift nested array: Revisited and improved
Introduction
Direction of arrival (DOA) estimation has received significant attention in the field of radar system, wireless communication, speech processing and so on [1], [2], [3], [4], [5]. According to the parameter identification theory, for a uniform linear array with sensors, smaller than sources can be resolved by the classical DOA estimation techniques. To improve the identifiability, the underdetermined DOA estimation by sparse array has attracted enormous interest in the domain of array signal processing. A variety of sparse arrays exploiting the second-order difference co-array (SODC) have been put forward, where coprime array (CA) [6] and nested array (NA) [7] are the most representative. The extensions and variations of NA and CA are soon developed to increase the degrees of freedom (DOFs) for DOA estimation [8], [9], [10], [11], [12]. Although various ingenious sparse array structures have been created, the DOFs is still limited.
By adopting the fourth-order cumulants and the fourth-order difference co-array (FODC), a remarkable increase of DOFs can be realized [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23]. In [16], the four-level NA (FL-NA) is designed to achieve DOFs by using physical sensors. Through adding an extra sub-array to the original NA, the sparse array extension with the FODC enhancement based on NA (SAFOE-NA) in [17] effectively improves the DOFs and DOA estimation performance. In [18], a two-level NA for the fourth-order cumulants based DOA estimation (2L-FONA) is proposed to provide more DOFs than SAFOE-NA when the number of physical sensors is smaller than 12. By fixing one sparse array while expanding and shifting another sparse array, the expanding and shift scheme (EAS) is proposed [19,20], which can obtain many more DOFs than the aforementioned FODC-based sparse arrays. The similar structure is also considered in the design of a generalized FODC (G-FODC) [21]. Based on FL-NA, the enhanced FL-NA (E-FL-NA) is introduced in [22]. To compress the number of sensors in NA, the compressed NA (CNA) is proposed in [23].
Although the EAS scheme provides a systematic way for the construction of a FODC-based sparse array, it is not optimized and deserves further improvement. This letter mainly focuses on the EAS composed of two NAs (EAS-NA-NA) [19], which is easy to construct and effective for achieving large number of DOFs. The original EAS-NA-NA consists of two NAs: one is the standard NA and the other is the expanded NA. In EAS-NA-NA, the maximum number of DOFs is realized by setting the physical sensor spacing in the expanded NA according to the DOFs in the SODC of the standard NA. As a matter of fact, the sensor spacing in the expanded NA is not optimal, hence the number of DOFs in the FODC of EAS-NA-NA is also not maximum. This letter illustrates that the sensor spacing in the expanded NA can be enlarged to achieve notably increased number of consecutive FODC sensors. As a consequence, the DOFs available for DOA estimation is also remarkably improved. In comparison with other representative sparse arrays, EAS-NA-NA with larger spacing (EAS-NA-NALS) can achieve more DOFs, realizing more resolved sources and better DOA estimation accuracy. Finally, simulation experiments are conducted to verify the performance of EAS-NA-NALS.
Notations: For any two sets and representing sensor positions, the operations on these two sets are listed as follows for a better understanding of the letter [17]where is a constant and represents the integer set. denotes a consecutive set ranging from to .
Section snippets
Signal model
In general, the sensor positions in a sensor array subject to the following setwhere represents the position of a sensor. and denotes the wavelength. The array output for far-field narrowband sources iswhere contains the source signals; represents the Gaussian noise, and is the steering vector for the incident angle of ,
The proposed EAS-NA-NALS
With a total of sensors, the original EAS-NA-NA is deployed as [19]where and are the number of sensors contained in the two subarrays of the first NA, while and are the number of sensors contained in the two subarrays of the second NA; is an integer representing the expanding factor and is the shift factor equaling to . For EAS-NA-NA, and represent the
Compared with other sparse arrays
EAS-NA-NALS is compared with several representative FODC-based sparse arrays, including SAFOE-NA, 2L-FONA, the original EAS-NA-NA, and E-FL-NA. The expressions with regard to the number of consecutive FODC sensors in these works are listed in Table 2, and are denoted as , ,, , and , respectively. should be no smaller than 4 to ensure the construction of E-FL-NA. Obviously, is larger than no matter how many physical sensors are deployed. In Fig. 1, the maximum
Simulation results
In the simulations, the sparse array structures with 7 physical sensors are listed in Table 4. According to the expressions showing the number of consecutive FODC sensors in Table 2, , , , and . The spatial smoothing MUSIC (SS-MUSIC) [7] is employed and the number of DOFs available for DOA estimation equals half the number of consecutive FODC sensors. All the sources are uniformly distributed from to .
In the first simulation, the MUSIC spectrum
Conclusion
Based on the original EAS-NA-NA, the EAS-NA-NALS is proposed to further enlarge the sensor spacing of the expanded NA. The range for the consecutive segment in the FODC of EAS-NA-NALS is determined. Accordingly, the relationship between the number of physical sensors and the number of consecutive FODC sensors for EAS-NA-NALS is expressed in the closed-form. Compared with other representative sparse arrays, the proposed EAS-NA-NALS can achieve a significant increase in the number of consecutive
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
This work was sponsored by National Natural Science Foundation of China under Grants 61901371, 61901372, 61801363 and 61971349. This work was also supported in part by the Natural Science Basic Research Program of Shaanxi under Grants 2020JQ-600 and 2020JQ-599, in part by China Postdoctoral Science Foundation under Grant 2020M683541.
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