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Adaptive beamforming is widely used in the fields such as radar, sonar, wireless communication to estimate the parameters of the signal of interest (SOI) at the output of a sensor array by data-adaptive spatial filtering and interference suppression. The standard Capon beamformer (SCB) is a typical adaptive beamforming approach which provides a superior performance by minimizing the array output power while simultaneously maintaining the array response under the assumption of distortionless direction of arrival (DOA). However, the advantages in performance of SCB are obtainable only when the number of snapshots available for the sample covariance matrix estimation is large enough and the direction of the SOI is known accurately. When applied to practical situations where the aforementioned two requirements are not satisfied, SCB will suffer high sidelobe levels and performance degradation in the parameter estimates due to lack of measurements and mismatch in the steering vector.A sparsity-constrained Capon beamformer (SCCB) arises to alleviate these problems. Unlike SCB, the constraint in SCCB is composed of two parts: the original array output power constraint part and the sparse constraint part (?1 norm constraint, encouraging sparse distribution in the array responses). However, if the sparse constraint in SCCB is set too large compared with the array output power constraint part, the responses in the directions of interferences will be influenced, and a tradeoff between the ability to reduce the sidelobe levels and the ability to reject the interferences must be made. Thus, based on the SCCB, a new robust Capon beamformer utilizing a weighted sparse constraint is proposed in this paper. In the proposed method, the sparse constraint part is replaced by a weighted sparse constraint, which is applied only to the sidelobe regions of the beampattern. By doing so, the number of the non-zero elements in the sidelobe response is minimized, resulting in an enhanced mainlobe region and suppressed sidelobe ones.In sparse recovery, the sparse constraint (the l1 norm constraint) does not necessarily enforce democratic penalization, which means that larger coefficients are penalized more heavily than smaller coefficients. Based on such a consideration, a weighting matrix can be constructed to put larger weights in the interferences directions to discourage their responses, and put smaller weights to maintain the responses in the remaining parts of the sidelobe regions. In this paper, the weighting matrix is obtained by utilizing the orthogonality between the signal subspace and the noise subspace. Since the steering vectors corresponding to the interferences and the SOI span the same space as the signal subspace, the inner products between the steering vectors in the interference directions and the noise subspace will produce zeroes ideally. By taking the reciprocals of these inner products, large values will yield in the interference directions while small values are obtained in other directions in the sidelobe regions. Using these values as the weights to the sparse constraint, a beampattern with deeper nulls, lower sidelobes, and better robustness to steering vector mismatch is obtainable as compared with SCB and SCCB. Besides, the output SINR is also effectively improved. Numerical simulations and a water-tank experiment are conducted to demonstrate the effectiveness of the proposed method.
[1] Haykin S 1985 Array Signal Processing (New Jersey: Prentice-Hall) pp15-77
[2] Harry L, Van T 2002 Detection, Estimation, and Modulation Theory, Part IV, Optimum Array Processing (New York: Wiley) pp728-751
[3] Capon J 1969 Proc. IEEE 57 1408
[4] Liu J, Gershman A B, Luo Z Q, Kon M W 2003 IEEE Sign. Process. Lett. 10 331
[5] Cox H 1973 J. Acoust. Soc. Am 54 771
[6] Carlson B D 1988 IEEE Trans. Aerosp. Electron. Syst. 24 397
[7] Besson O, Vincent F 2005 IEEE Trans. Sign. Process. 53 452
[8] Feldman D D, Griffiths L J 1994 IEEE Trans. Sign. Process. 42 867
[9] Chang L, Yeh C C 1992 IEEE Trans. Antenna Propag. 40 1336
[10] Li J, Stoica P, Wang Z 2003 IEEE Trans. Sign. Process. 51 1702
[11] Li J, Stoica P, Wang Z 2004 IEEE Trans. Sign. Process. 52 2407
[12] Vorobyov S A, Gershman A B, Luo Z Q 2003 IEEE Trans. Sign. Process. 51 313
[13] Huang C, Sun D J, Zhang D L, Teng T T 2014 Acta Phys. Sin. 63 188401 (in Chinese) [黄聪, 孙大军, 张殿伦, 滕婷婷 2014 63 188401]
[14] Wang Y, Wu W F, Fan Z, Liang G L 2014 Acta Phys. Sin. 63 154303 (in Chinese) [王燕, 吴文峰, 范展, 梁国龙 2014 63 154303]
[15] Wang F, Balakrishnan V, Zhou P Y 2003 IEEE Trans. Sign. Process. 51 1172
[16] Liang G L, Ma W, Fan Z, Wang Y L 2013 Acta Phys. Sin. 62 144302 (in Chinese) [梁国龙, 马巍, 范展, 王逸林 2013 62 144302]
[17] Xiao D, Cai H K, Zheng H Y 2015 Chin. Phys. B 24 060505
[18] Sun Y L, Tao J X 2014 Chin. Phys. B 23 078703
[19] Zhang Y, Ng B P, Wan Q 2008 IEEE Sign. Process. Lett. 44 615
[20] Liu Y P, Wan Q 2010 Prog. Electromagn. Res. Lett. 16 53
[21] Li J, Stoica P 2006 Robust Adaptive Beamforming (New York: Wiley) pp1-94
[22] Rao B D, Engan K, Cotter S F, Palmer J, Delado K K 2003 IEEE Trans. Sign. Process. 51 760
[23] Liu Y P 2011 Ph. D. Dissertation (Chengdu: University of Electronic Science and Technology of China) (in Chinese) [刘翼鹏 2011 博士学位论文(成都: 电子科技大学)]
[24] Chen S S, Donoho D L, Saunders M A 2001 SIAM Review 43 129
[25] Cands E J, Wakin M B, Boyd S P 2007 J. Fourier Anal. Appl. 14 877
[26] Zheng C, Li G, Zhang H, Wang X 2011 Proc. IEEE Int. Conf. Acoust., Speech, Signal Process. Prague, Czech Republic, May 22-27, 2011 p2856
[27] Wang Y L, Chen H, Peng Y N, Wan Q 2004 Spatial Spectrum Estimation Theory and Algorithms (Beijing:Tsinghua University Press) pp54-55 (in Chinese) [王永良, 陈辉, 彭应宁, 万群 2004 空间谱估计理论与算法 (北京:清华大学出版社) 第54-55页]
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[1] Haykin S 1985 Array Signal Processing (New Jersey: Prentice-Hall) pp15-77
[2] Harry L, Van T 2002 Detection, Estimation, and Modulation Theory, Part IV, Optimum Array Processing (New York: Wiley) pp728-751
[3] Capon J 1969 Proc. IEEE 57 1408
[4] Liu J, Gershman A B, Luo Z Q, Kon M W 2003 IEEE Sign. Process. Lett. 10 331
[5] Cox H 1973 J. Acoust. Soc. Am 54 771
[6] Carlson B D 1988 IEEE Trans. Aerosp. Electron. Syst. 24 397
[7] Besson O, Vincent F 2005 IEEE Trans. Sign. Process. 53 452
[8] Feldman D D, Griffiths L J 1994 IEEE Trans. Sign. Process. 42 867
[9] Chang L, Yeh C C 1992 IEEE Trans. Antenna Propag. 40 1336
[10] Li J, Stoica P, Wang Z 2003 IEEE Trans. Sign. Process. 51 1702
[11] Li J, Stoica P, Wang Z 2004 IEEE Trans. Sign. Process. 52 2407
[12] Vorobyov S A, Gershman A B, Luo Z Q 2003 IEEE Trans. Sign. Process. 51 313
[13] Huang C, Sun D J, Zhang D L, Teng T T 2014 Acta Phys. Sin. 63 188401 (in Chinese) [黄聪, 孙大军, 张殿伦, 滕婷婷 2014 63 188401]
[14] Wang Y, Wu W F, Fan Z, Liang G L 2014 Acta Phys. Sin. 63 154303 (in Chinese) [王燕, 吴文峰, 范展, 梁国龙 2014 63 154303]
[15] Wang F, Balakrishnan V, Zhou P Y 2003 IEEE Trans. Sign. Process. 51 1172
[16] Liang G L, Ma W, Fan Z, Wang Y L 2013 Acta Phys. Sin. 62 144302 (in Chinese) [梁国龙, 马巍, 范展, 王逸林 2013 62 144302]
[17] Xiao D, Cai H K, Zheng H Y 2015 Chin. Phys. B 24 060505
[18] Sun Y L, Tao J X 2014 Chin. Phys. B 23 078703
[19] Zhang Y, Ng B P, Wan Q 2008 IEEE Sign. Process. Lett. 44 615
[20] Liu Y P, Wan Q 2010 Prog. Electromagn. Res. Lett. 16 53
[21] Li J, Stoica P 2006 Robust Adaptive Beamforming (New York: Wiley) pp1-94
[22] Rao B D, Engan K, Cotter S F, Palmer J, Delado K K 2003 IEEE Trans. Sign. Process. 51 760
[23] Liu Y P 2011 Ph. D. Dissertation (Chengdu: University of Electronic Science and Technology of China) (in Chinese) [刘翼鹏 2011 博士学位论文(成都: 电子科技大学)]
[24] Chen S S, Donoho D L, Saunders M A 2001 SIAM Review 43 129
[25] Cands E J, Wakin M B, Boyd S P 2007 J. Fourier Anal. Appl. 14 877
[26] Zheng C, Li G, Zhang H, Wang X 2011 Proc. IEEE Int. Conf. Acoust., Speech, Signal Process. Prague, Czech Republic, May 22-27, 2011 p2856
[27] Wang Y L, Chen H, Peng Y N, Wan Q 2004 Spatial Spectrum Estimation Theory and Algorithms (Beijing:Tsinghua University Press) pp54-55 (in Chinese) [王永良, 陈辉, 彭应宁, 万群 2004 空间谱估计理论与算法 (北京:清华大学出版社) 第54-55页]
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