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As an alternative paradigm to the Shannon-Nyquist sampling theorem, compressive sensing enables sparse signals to be acquired by sub-Nyquist analog-to-digital converters thus may launch a revolution in signal collection, transmission and processing. In the practical compressive sensing applications, the sparse signal is always affected by noise and interference, and therefore the recovery performance reduces based on the conventional compressive sensing, especially in the low signal-to-noise scene, the sparse recovery is usually unavailable. In this paper, the influence of noise on recovery performance is analyzed, so as to provide the theoretical basis for the noise folding phenomenon in compressive sensing. From the analysis, we find that the expected noise gain in the random measure process is closely related to the row and column of the measurement matrix. However, only those columns corresponding to the support for the sparse signal contribute to the sparse vector. In the traditional Shannon-Nyquist sampling system, an antialiasing filter is applied before the sampling process, so as to filter the noise beyond the passband of interest. Inspired by the necessity of antialiasing filtering in bandpass signal sampling, we propose a selective measurement scheme, namely adapted compressive sensing, whose measurement matrix can be updated according to the noise information fed back by the processing center. The measurement matrix is specially designed, and the sensing matrix has directivity so that the signal noise can be suppressed. The measurement matrix senses only the spectrum of interest, where the sparse spectrum is most likely to lie. Moreover, we compare the recovery performance of the proposed adaptive scheme with those of the non-adaptive orthogonal matching pursuit algorithm, FOCal underdetermined system solver algorithm, and sarse Bayesian learning algorithm. Extensive numerical experiments show that the proposed scheme has a better improvement in the performance of the sparse signal recovery. From the viewpoint of implementation, the measurement noise should be taken into consideration in the system, and more efficient algorithms will be developed for source pre-estimation at lower signal-to-noise ratio.
[1] Donoho D L 2006 IEEE Trans. Inform. Theory 52 1289
[2] Mishali M, Eldar Y C 2009 IEEE Trans. Signal Process. 57 993
[3] Zhang J C, Fu N, Qiao L Y 2014 Acta Phys. Sin. 63 030701 (in Chinese) [张京超, 付宁, 乔立岩 2014 63 030701]
[4] Wang L Y, Li L, Yan B, Jiang C S, Wang H Y, Bao S L 2010 Chin. Phys. B 19 088106
[5] Zhao S M, Zhuang P 2014 Chin. Phys. B 23 054203
[6] Sun Y L, Tao J X 2014 Chin. Phys. B 23 078703
[7] Wang Z, Wang B Z 2014 Acta Phys. Sin. 63 120202 (in Chinese) [王哲, 王秉中 2014 63 120202]
[8] Yang F Q, Zhang D H, Huang K D 2014 Acta Phys. Sin. 63 058701 (in Chinese) [杨富强, 张定华, 黄魁东 2014 63 058701]
[9] Candes E J, Tao T 2005 IEEE Trans. Inform. Theory 51 4203
[10] Rao B D, Engan K, Cotter S F 2003 IEEE Trans. Signal Process. 51 760
[11] Castro E A, Eldar Y C 2011 IEEE Signal Process. Lett. 18 478
[12] Davenport M A, Laska J N, Treichler J, Baraniuk R G 2012 IEEE Trans. Signal Process. 60 4628
[13] Cotter S F, Rao B D, Engan K, Delgado K K 2005 IEEE Trans. Signal Process. 53 2477
[14] Candes E J 2008 Comptes Rendus Mathematique 346 589
[15] Tropp J A, Gilbert A C 2007 IEEE Trans. Inform. Theory 53 4655
[16] Ji S H, Xue Y, Carin L 2008 IEEE Trans. Signal Process. 56 2346
[17] Davenport M A 2010 Ph. D. Dissertation (Texas: Rice University)
[18] Zhang J D, Zhu D Y, Zhang G 2012 IEEE Trans. Signal Process. 60 1718
[19] Wipf D P, Rao D B 2007 IEEE Trans. Signal Process. 55 3704
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[1] Donoho D L 2006 IEEE Trans. Inform. Theory 52 1289
[2] Mishali M, Eldar Y C 2009 IEEE Trans. Signal Process. 57 993
[3] Zhang J C, Fu N, Qiao L Y 2014 Acta Phys. Sin. 63 030701 (in Chinese) [张京超, 付宁, 乔立岩 2014 63 030701]
[4] Wang L Y, Li L, Yan B, Jiang C S, Wang H Y, Bao S L 2010 Chin. Phys. B 19 088106
[5] Zhao S M, Zhuang P 2014 Chin. Phys. B 23 054203
[6] Sun Y L, Tao J X 2014 Chin. Phys. B 23 078703
[7] Wang Z, Wang B Z 2014 Acta Phys. Sin. 63 120202 (in Chinese) [王哲, 王秉中 2014 63 120202]
[8] Yang F Q, Zhang D H, Huang K D 2014 Acta Phys. Sin. 63 058701 (in Chinese) [杨富强, 张定华, 黄魁东 2014 63 058701]
[9] Candes E J, Tao T 2005 IEEE Trans. Inform. Theory 51 4203
[10] Rao B D, Engan K, Cotter S F 2003 IEEE Trans. Signal Process. 51 760
[11] Castro E A, Eldar Y C 2011 IEEE Signal Process. Lett. 18 478
[12] Davenport M A, Laska J N, Treichler J, Baraniuk R G 2012 IEEE Trans. Signal Process. 60 4628
[13] Cotter S F, Rao B D, Engan K, Delgado K K 2005 IEEE Trans. Signal Process. 53 2477
[14] Candes E J 2008 Comptes Rendus Mathematique 346 589
[15] Tropp J A, Gilbert A C 2007 IEEE Trans. Inform. Theory 53 4655
[16] Ji S H, Xue Y, Carin L 2008 IEEE Trans. Signal Process. 56 2346
[17] Davenport M A 2010 Ph. D. Dissertation (Texas: Rice University)
[18] Zhang J D, Zhu D Y, Zhang G 2012 IEEE Trans. Signal Process. 60 1718
[19] Wipf D P, Rao D B 2007 IEEE Trans. Signal Process. 55 3704
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