认知无线网络中宽带压缩频谱感知关键技术
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摘要
认知无线电是下一代无线网络发展中的关键技术,它是一种解决频谱稀缺、实现频谱动态管理以及提高频谱利用率等问题的有效途径。认知无线电技术具有感知周围环境和自动学习的能力,能够自适应地进行频谱接入,并最终实现频谱共享和频谱管理。作为认知无线电关键技术之一的频谱感知技术,因其基础性作用而成为近年来的研究热点。
传统的窄带频谱感知技术每次只能检测一个信道,若将其用于对划分成多个子频带的宽频带进行检测,显然检测效率太低;为了提高检测效率,宽带频谱感知技术应运而生。长久以来,Nyquist采样定理一直主导着窄带频谱感知的信号采样和编码过程,但在对宽频带信号的感知过程中,Nyquist采样定理的使用势必会使数字信号处理器面临严峻考验;压缩感知理论突破了Nyquist采样定理的限制,提出了一种全新的信号获取方法,为宽带频谱感知技术的实现带来了福音;而后,分布式压缩感知理论的提出进一步扩展了压缩感知理论的应用,它将单信号的压缩采样扩展到了信号群的压缩采样。
本文针对宽带频谱感知技术进行了研究,并将压缩感知理论应用其中,提出了多用户合作的宽带压缩频谱感知模型。具体工作如下:
首先,概述了认知无线电技术中频谱感知技术的研究背景及发展现状,简单介绍了几种常见的单用户频谱感知算法,并由单用户频谱感知技术的局限性展开了对合作频谱感知技术的介绍,着重对合作频谱感知的融合方案进行了综合比较和仿真分析。
其次,引入了压缩感知理论,主要从信号的稀疏表示、编码测量和信号重构三个方面对其进行了介绍。依据算法实现思想的不同,将现有的压缩感知重构算法进行了分类和比较,并着重研究了匹配追踪系列算法。
最后,将压缩感知理论应用于认知无线电网络,并依据宽带频谱感知和合作频谱感知思想研究了宽带压缩频谱感知关键技术,提出了认知无线电网络的基于压缩感知的合作频谱感知和基于分布式压缩感知的合作频谱感知两个模型,并分别对其进行了算法设计和性能仿真。
Cognitive radio (CR), a key technology in the next-generation wireless network, is an effective way to solve the shortage of spectrum, realizing dynamic management of spectrum and enhancing the usage of spectrum. CR technology has a good ability to perceive the surrounding environment and learn by itself, as well as access the spectrum adaptively. As a result, CR realizes spectrum sharing and management of spectrum. As a key technology of CR, spectrum sensing has become a hot research topic in recent years because of its fundamental role.
If we apply the traditional narrow-band spectrum sensing technology which can detect only one channel at a time to detect a wide band that is divided into several sub-band, the detection efficiency will be too low, obviously. In order to improve the detection efficiency, the wideband spectrum sensing technology comes into being. For a long time, Nyquist sampling theorem has been dominated the signal sampling and coding procedures of narrow-band spectrum sensing. While in the sensing process of wide band, the usage of Nyquist sampling theorem is bound to bring severe tests to the digital signal processor. Fortunately, the compressed sensing (CS) theory brings the gospel to it. CS theory breaks through the limitation of Nyquist sampling theorem and puts forward a completely new method of signal acquisition. And then, distributed compressed sensing (DCS) that develops the compression and sampling of signals from single-signal to multi-signal comes up to expand the application of compressed sensing theory further.
This paper does some researches on wideband spectrum sensing technology and applies CS theory on it and then proposes two wideband spectrum compressive sensing models for multiple cooperative users. The specific works are as follows:
Firstly, this paper has an overview of its research background and current development status of spectrum sensing technology in cognitive radio. In addition, it briefly introduces several common single user spectrum sensing algorithms. Then, after analyzing the limitations of single user spectrum sensing technology, the paper launches a statement of cooperative spectrum sensing technology. The comprehensive comparison and simulation analysis of fusion schemes are done as well.
Secondly, this paper introduces and narrates the theory of CS from three aspects: the sparse representation of signal, coding and measurement as well as reconstruction of signal. Then, the existing CS reconstruction algorithms are classified and compared; matching pursuit (MP) series algorithms are well studied, especially.
Finally, by doing deep search on wideband compressed spectrum sensing key technology in accordance with wideband spectrum sensing and cooperative spectrum sensing applying CS theory to CR networks, this paper puts forward two cooperative spectrum sensing models respectively based on CS and DCS in CR networks. Furthermore, the performances of algorithms based on two models are investigated in simulations.
传统的窄带频谱感知技术每次只能检测一个信道,若将其用于对划分成多个子频带的宽频带进行检测,显然检测效率太低;为了提高检测效率,宽带频谱感知技术应运而生。长久以来,Nyquist采样定理一直主导着窄带频谱感知的信号采样和编码过程,但在对宽频带信号的感知过程中,Nyquist采样定理的使用势必会使数字信号处理器面临严峻考验;压缩感知理论突破了Nyquist采样定理的限制,提出了一种全新的信号获取方法,为宽带频谱感知技术的实现带来了福音;而后,分布式压缩感知理论的提出进一步扩展了压缩感知理论的应用,它将单信号的压缩采样扩展到了信号群的压缩采样。
本文针对宽带频谱感知技术进行了研究,并将压缩感知理论应用其中,提出了多用户合作的宽带压缩频谱感知模型。具体工作如下:
首先,概述了认知无线电技术中频谱感知技术的研究背景及发展现状,简单介绍了几种常见的单用户频谱感知算法,并由单用户频谱感知技术的局限性展开了对合作频谱感知技术的介绍,着重对合作频谱感知的融合方案进行了综合比较和仿真分析。
其次,引入了压缩感知理论,主要从信号的稀疏表示、编码测量和信号重构三个方面对其进行了介绍。依据算法实现思想的不同,将现有的压缩感知重构算法进行了分类和比较,并着重研究了匹配追踪系列算法。
最后,将压缩感知理论应用于认知无线电网络,并依据宽带频谱感知和合作频谱感知思想研究了宽带压缩频谱感知关键技术,提出了认知无线电网络的基于压缩感知的合作频谱感知和基于分布式压缩感知的合作频谱感知两个模型,并分别对其进行了算法设计和性能仿真。
Cognitive radio (CR), a key technology in the next-generation wireless network, is an effective way to solve the shortage of spectrum, realizing dynamic management of spectrum and enhancing the usage of spectrum. CR technology has a good ability to perceive the surrounding environment and learn by itself, as well as access the spectrum adaptively. As a result, CR realizes spectrum sharing and management of spectrum. As a key technology of CR, spectrum sensing has become a hot research topic in recent years because of its fundamental role.
If we apply the traditional narrow-band spectrum sensing technology which can detect only one channel at a time to detect a wide band that is divided into several sub-band, the detection efficiency will be too low, obviously. In order to improve the detection efficiency, the wideband spectrum sensing technology comes into being. For a long time, Nyquist sampling theorem has been dominated the signal sampling and coding procedures of narrow-band spectrum sensing. While in the sensing process of wide band, the usage of Nyquist sampling theorem is bound to bring severe tests to the digital signal processor. Fortunately, the compressed sensing (CS) theory brings the gospel to it. CS theory breaks through the limitation of Nyquist sampling theorem and puts forward a completely new method of signal acquisition. And then, distributed compressed sensing (DCS) that develops the compression and sampling of signals from single-signal to multi-signal comes up to expand the application of compressed sensing theory further.
This paper does some researches on wideband spectrum sensing technology and applies CS theory on it and then proposes two wideband spectrum compressive sensing models for multiple cooperative users. The specific works are as follows:
Firstly, this paper has an overview of its research background and current development status of spectrum sensing technology in cognitive radio. In addition, it briefly introduces several common single user spectrum sensing algorithms. Then, after analyzing the limitations of single user spectrum sensing technology, the paper launches a statement of cooperative spectrum sensing technology. The comprehensive comparison and simulation analysis of fusion schemes are done as well.
Secondly, this paper introduces and narrates the theory of CS from three aspects: the sparse representation of signal, coding and measurement as well as reconstruction of signal. Then, the existing CS reconstruction algorithms are classified and compared; matching pursuit (MP) series algorithms are well studied, especially.
Finally, by doing deep search on wideband compressed spectrum sensing key technology in accordance with wideband spectrum sensing and cooperative spectrum sensing applying CS theory to CR networks, this paper puts forward two cooperative spectrum sensing models respectively based on CS and DCS in CR networks. Furthermore, the performances of algorithms based on two models are investigated in simulations.
引文
[1] FCC. Spectrum policy task force report[R]. ET Docket no.02-135, November 2002.
[2] M. A. McHenry. NSF spectrum occupancy measurements project summary[R]. Shared Spectrum Company Report, December 2005.
[3] J. Mitola. Cognitive Radio: An Integrated Agent Architecture for Software Defined Radio[D]. PhD Dissertation, Royal Inst. Techno1.(KTH), Stockholm, Sweden, 2000.
[4] J. Mitola, G. Q. Maguire. Cognitive radio: making software radios more personal[J]. IEEE Personal Communications Magazine, 1999, 6(4):13-18.
[5] http://transition.fcc.gov/sptf/.
[6] FCC. FCC part 15[S]. FCC, December 8, 2003.
[7] FCC. Unlicensed operation in the TV broadcast bands and additional spectrum for unlicensed devices below 900MHz in the 3 GHz band[R]. Notice of proposed rulemaking 04-186, 2004.
[8]周贤伟.认知无线电[M].北京:国防工业出版社, 2008.
[9] C. Cordeiro, K. Challapali, D. Birru. IEEE 802.22: The first worldwide wireless standard based on cognitive radios[C]. Maryland, USA: IEEE DySPAN, November 2005:328-337.
[10] IEEE 802.22 Working Group on Wireless Regional Area Networks[DB/OL]. http://ieee802.org/22/.
[11] DARPA XG Working Group. DARPA XG policy language framework[R]. BBN Technologies, Cambridge, Massachusetts, USA, April 2004.
[12] S. Haykin. Cognitive radio: brain-empowered wireless communications[J]. IEEE Journal on Selected Areas in Communications, 2005, 23(2):201–220.
[13] P. Kolodzy. Next generation communications: Kickoff meeting[R]. DARPA, October, 2001.
[14] I. F. Akyildiz, W. Y. Lee, M. C. Vuran, et al. NeXt generation/dynamic spectrum access/cognitive radio wireless networks: A survey[J]. Computer Networks, September 2006, 50:2127–2159.
[15]普罗基斯.数字通信(第4版)[M].北京:电子工业出版社,2006.
[16] F. F. Digham, M. S. Alouini, M. K. Simon. On the energy detection of unknown signals over fading channels[C]. Piscataway, USA: IEEE ICC 2005, May 2003:3575–3579.
[17] W. A. Gardner. Exploitation of spectral redundancy in cyclostationary signals[J]. IEEE SignalProcessing Magazine, April 1991, 8(2):14-36.
[18]周小飞,张宏纲.认知无线电原理及应用[M].北京:北京邮电大学出版社, 2007.
[19] G. Ganesan, Y. G. Li. Cooperative spectrum sensing in cognitive radio networks[C]. Baltimore, USA: IEEE DySPAN 2005, November 2005:137-143.
[20] Z. Chair, P. K. Varshney. Optimal data fusion in multiple sensor detection systems[J]. IEEE Transactions on Aerospace and Electronic Systems, 1986, 22: 98-101.
[21] J. Unnikrishnan, V. V. Veeravalli. Cooperative spectrum sensing and detection for cognitive radio[C]. Washington. DC, USA: IEEE Global Telecommunications Conference, GLOBECOM'07, November 2007:2972-2976.
[22] D. L. Donoho. Compressed sensing[J]. IEEE Transactions on Information Theory, April 2006, 52(4):1289-1306.
[23] Y. Tsaig, D. L. Donoho. Extensions of compressed sensing [J]. Signal Processing, March 2006, 86(3):549-571.
[24] E. J. Candès, T. Tao. Near-optimal signal recovery from random projections: Universal encoding strategies[J]. IEEE Transactions on Information Theory, 2006, 52(12):5406-5425.
[25] E. J. Candès, J. Romberg, T. Tao. Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information[J]. IEEE Transactions on Information Theory, 2006, 52(2):489-509.
[26]石光明,刘丹华等.压缩感知理论及其研究进展[J].电子学报, 2009, 37(5):1070-1081.
[27] E. J. Candès, T. Tao. Near optimal signal recovery from random projections: Universal encoding strategies[J]. IEEE Transactions on Information Theory, 2006, 52(12):5406-5425.
[28] G. Peyré. Best basis compressed sensing[J]. Lecture Notes in Computer Science, 2007, 4485:80-91.
[29] V. Temlyakov. Nonlinear methods of approximation[J]. Foundations of Comp. Math, 2003, 3(1):33- 07.
[30] D. L. Donoho, X. Huo. Uncertainty principles and ideal atomic decompositions[J]. IEEE Transactions on Information Theory, 2001, 47(7):2845-2862.
[31] A. Feuer, A. Nemirovsky. On sparse representation in pairs of bases[J]. IEEE Transactions on Information Theory, 2003, 49(6):1579-1581.
[32] J. A. Tropp. Greed is good: Algorithmic results for sparse approximation[J]. IEEE Transactions on Information Theory, 2004, 50(10):2231-2242.
[33] E. J. Candès. The restricted isometry property and its implications for compressed sensing[J]. Academie des Sciences, 2008, 346(I):598-592.
[34] E. J. Candès, T. Tao. Decoding by linear programming[J]. IEEE Transactions on Information Theory, 2005, 51(12):4203-4215.
[35] R Baraniuk. A lecture on compressive sensing[J]. IEEE Signal Processing Magazine, July 2007, 24(4):118-121.
[36] D. L. Donoho, Y Tsaig. Extensions of compressed sensing[J]. Signal Processing, 2006, 86(3):533-548.
[37] S. S. Chen, D. L. Donoho, M. A. Saunders. Atomic decomposition by basis pursuit[J]. SIAM Review, 2001, 43(1):129-159.
[38] D. Needell, J. A. Tropp. CoSaMP: Iterative signal recovery from incomplete and inaccurate samples[J]. Applied Computational Harmonic Analysis, 2009, 26(3):301-321.
[39] S. G. Mallat. Matching pursuits with time-frequency dictionaries[J]. IEEE Transactions on Signal Processing, December 1993, 41(12):3397-3415.
[40] J. A. Tropp, A. C. Gilbert. Signal Recovery from random measurements via orthogonal matching pursuit[J]. IEEE Transactions on Information Theory, 53(12), December 2007:4655-4666.
[41] D. L. Donoho, Y. Tsaig , I. Drori, et al. Sparse solution of underdetermined linear equations by stagewise orthogonal matching pursuit[R]. Technical Report, 2006.
[42] D. Needell and R. Vershynin. Signal recovery from incomplete and inaccurate measurements via regularized orthogonal matching pursuit[J]. IEEE J. Sel. Top. Signal Process, April 2010, 4(2):310-316.
[43] D. Needell, R. Vershynin. Uniform uncertainty principle and signal recovery via regularized orthogonal matching pursuit[J]. Found. Comput. Math, 2009, 9(3):317-334.
[44] W. Dai, O. Milenkovic. Subspace pursuit for compressive sensing signal reconstruction[J]. IEEE Transactions on Information Theory, May 2009, 55(5):2230-2249.
[45] T. T. Do, L. Gan, N. Nguyen, et al. Sparsity adaptive matching pursuit algorithm for practical compressed sensing[A]. Signals, Systems and Computers[C]. Pacific Grove, CA: 42nd Asilomar Conference on, 2008:581-587.
[46] S. S. Chen, D. L. Donoho, M. A. Saunders. Atomic decomposition by basis pursuit[J]. SIAM Journal on Scientific Computing, 1998, 20(1):33-61.
[47] S. J. Kim, K. Koh, M. Lustig, et al. An interior-point method for large-scale -regularized least squares[J]. IEEE Journal on Selected Topics in Signal Processing, 2007, 4(1):606-617. ?1
[48] M. A. T. Figueiredo, R. D. Nowak, S. J. Wright. Gradient projection for sparse reconstruction: Application to compressed sensing and other inverse problems[J]. Journal of Selected Topics in Signal Processing: Special Issue on Convex Optimization Methods for Signal Processing, 2007, 1(4):586-598.
[49] I. Daubechies, M .Defrise, C. De Mol. An iterative thresholding algorithm for linear inverse problems with a sparsity constraint[J]. Comm. Pure Appl. Math., 2004, 57(11):1413-1457.
[50] A. C. Gilbert, S. Guha, P. Indyk, et al. Near-optimal sparse Fourier representations via sampling [A]. Proceedings of the Annual ACM Symposium on Theory of Computing[C]. Montreal, Que, Canada: Association for Computing Machinery, 2002:152-161.
[51] A. C. Gilbert, S. Muthukrishnan, M. J. Strauss. Improved time bounds for near-optimal sparse Fourier representation[A]. Proceedings of SPIE, Wavelets XI[C]. Bellingham, WA: International Society for Optical Engineering, 2005, 5914:1-15.
[52] A. C. Gilbert, M. J. Strauss, J. A. Tropp, et al. Algorithmic linear dimension reduction in the norm for sparse vectors[C]. Pacific Grove, CA: 44th Annu. Allerton Conf. Communication, Control, and Computing, 2006. ?1
[53] A. C. Gilbert, M. J. Strauss, J. A. Tropp, et al. One sketch for all: Fast algorithms for compressed sensing [A]. Proceedings of the 39th Annual ACM Symposium on Theory of Computing[C]. New York: Association for Computing Machiner, 2007:237-246.
[54] D. Baron, M. F. Duarte, S. Sarvotham, et al. An information-theoretic approach to distributed compressed sensing[C]. Monticello, IL: 43rd Allerton Conf. Communication, Control, and Computing, September. 2005.
[55] D. Baron, M. F. Duarte, S. Sarvotham, et al. Distributed compressed sensing of jointly sparse signals[C]. Pacific Grove, CA: 39th Asilomar Conf. Signals, Systems and Computers, November 2005.
[56] D. Baron, M. B. Wakin, M. F. Duarte, et al. Distributed compressed sensing[DB/OL]. http://dsp. rice.edu/cs/DCS112005.pdf.
[2] M. A. McHenry. NSF spectrum occupancy measurements project summary[R]. Shared Spectrum Company Report, December 2005.
[3] J. Mitola. Cognitive Radio: An Integrated Agent Architecture for Software Defined Radio[D]. PhD Dissertation, Royal Inst. Techno1.(KTH), Stockholm, Sweden, 2000.
[4] J. Mitola, G. Q. Maguire. Cognitive radio: making software radios more personal[J]. IEEE Personal Communications Magazine, 1999, 6(4):13-18.
[5] http://transition.fcc.gov/sptf/.
[6] FCC. FCC part 15[S]. FCC, December 8, 2003.
[7] FCC. Unlicensed operation in the TV broadcast bands and additional spectrum for unlicensed devices below 900MHz in the 3 GHz band[R]. Notice of proposed rulemaking 04-186, 2004.
[8]周贤伟.认知无线电[M].北京:国防工业出版社, 2008.
[9] C. Cordeiro, K. Challapali, D. Birru. IEEE 802.22: The first worldwide wireless standard based on cognitive radios[C]. Maryland, USA: IEEE DySPAN, November 2005:328-337.
[10] IEEE 802.22 Working Group on Wireless Regional Area Networks[DB/OL]. http://ieee802.org/22/.
[11] DARPA XG Working Group. DARPA XG policy language framework[R]. BBN Technologies, Cambridge, Massachusetts, USA, April 2004.
[12] S. Haykin. Cognitive radio: brain-empowered wireless communications[J]. IEEE Journal on Selected Areas in Communications, 2005, 23(2):201–220.
[13] P. Kolodzy. Next generation communications: Kickoff meeting[R]. DARPA, October, 2001.
[14] I. F. Akyildiz, W. Y. Lee, M. C. Vuran, et al. NeXt generation/dynamic spectrum access/cognitive radio wireless networks: A survey[J]. Computer Networks, September 2006, 50:2127–2159.
[15]普罗基斯.数字通信(第4版)[M].北京:电子工业出版社,2006.
[16] F. F. Digham, M. S. Alouini, M. K. Simon. On the energy detection of unknown signals over fading channels[C]. Piscataway, USA: IEEE ICC 2005, May 2003:3575–3579.
[17] W. A. Gardner. Exploitation of spectral redundancy in cyclostationary signals[J]. IEEE SignalProcessing Magazine, April 1991, 8(2):14-36.
[18]周小飞,张宏纲.认知无线电原理及应用[M].北京:北京邮电大学出版社, 2007.
[19] G. Ganesan, Y. G. Li. Cooperative spectrum sensing in cognitive radio networks[C]. Baltimore, USA: IEEE DySPAN 2005, November 2005:137-143.
[20] Z. Chair, P. K. Varshney. Optimal data fusion in multiple sensor detection systems[J]. IEEE Transactions on Aerospace and Electronic Systems, 1986, 22: 98-101.
[21] J. Unnikrishnan, V. V. Veeravalli. Cooperative spectrum sensing and detection for cognitive radio[C]. Washington. DC, USA: IEEE Global Telecommunications Conference, GLOBECOM'07, November 2007:2972-2976.
[22] D. L. Donoho. Compressed sensing[J]. IEEE Transactions on Information Theory, April 2006, 52(4):1289-1306.
[23] Y. Tsaig, D. L. Donoho. Extensions of compressed sensing [J]. Signal Processing, March 2006, 86(3):549-571.
[24] E. J. Candès, T. Tao. Near-optimal signal recovery from random projections: Universal encoding strategies[J]. IEEE Transactions on Information Theory, 2006, 52(12):5406-5425.
[25] E. J. Candès, J. Romberg, T. Tao. Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information[J]. IEEE Transactions on Information Theory, 2006, 52(2):489-509.
[26]石光明,刘丹华等.压缩感知理论及其研究进展[J].电子学报, 2009, 37(5):1070-1081.
[27] E. J. Candès, T. Tao. Near optimal signal recovery from random projections: Universal encoding strategies[J]. IEEE Transactions on Information Theory, 2006, 52(12):5406-5425.
[28] G. Peyré. Best basis compressed sensing[J]. Lecture Notes in Computer Science, 2007, 4485:80-91.
[29] V. Temlyakov. Nonlinear methods of approximation[J]. Foundations of Comp. Math, 2003, 3(1):33- 07.
[30] D. L. Donoho, X. Huo. Uncertainty principles and ideal atomic decompositions[J]. IEEE Transactions on Information Theory, 2001, 47(7):2845-2862.
[31] A. Feuer, A. Nemirovsky. On sparse representation in pairs of bases[J]. IEEE Transactions on Information Theory, 2003, 49(6):1579-1581.
[32] J. A. Tropp. Greed is good: Algorithmic results for sparse approximation[J]. IEEE Transactions on Information Theory, 2004, 50(10):2231-2242.
[33] E. J. Candès. The restricted isometry property and its implications for compressed sensing[J]. Academie des Sciences, 2008, 346(I):598-592.
[34] E. J. Candès, T. Tao. Decoding by linear programming[J]. IEEE Transactions on Information Theory, 2005, 51(12):4203-4215.
[35] R Baraniuk. A lecture on compressive sensing[J]. IEEE Signal Processing Magazine, July 2007, 24(4):118-121.
[36] D. L. Donoho, Y Tsaig. Extensions of compressed sensing[J]. Signal Processing, 2006, 86(3):533-548.
[37] S. S. Chen, D. L. Donoho, M. A. Saunders. Atomic decomposition by basis pursuit[J]. SIAM Review, 2001, 43(1):129-159.
[38] D. Needell, J. A. Tropp. CoSaMP: Iterative signal recovery from incomplete and inaccurate samples[J]. Applied Computational Harmonic Analysis, 2009, 26(3):301-321.
[39] S. G. Mallat. Matching pursuits with time-frequency dictionaries[J]. IEEE Transactions on Signal Processing, December 1993, 41(12):3397-3415.
[40] J. A. Tropp, A. C. Gilbert. Signal Recovery from random measurements via orthogonal matching pursuit[J]. IEEE Transactions on Information Theory, 53(12), December 2007:4655-4666.
[41] D. L. Donoho, Y. Tsaig , I. Drori, et al. Sparse solution of underdetermined linear equations by stagewise orthogonal matching pursuit[R]. Technical Report, 2006.
[42] D. Needell and R. Vershynin. Signal recovery from incomplete and inaccurate measurements via regularized orthogonal matching pursuit[J]. IEEE J. Sel. Top. Signal Process, April 2010, 4(2):310-316.
[43] D. Needell, R. Vershynin. Uniform uncertainty principle and signal recovery via regularized orthogonal matching pursuit[J]. Found. Comput. Math, 2009, 9(3):317-334.
[44] W. Dai, O. Milenkovic. Subspace pursuit for compressive sensing signal reconstruction[J]. IEEE Transactions on Information Theory, May 2009, 55(5):2230-2249.
[45] T. T. Do, L. Gan, N. Nguyen, et al. Sparsity adaptive matching pursuit algorithm for practical compressed sensing[A]. Signals, Systems and Computers[C]. Pacific Grove, CA: 42nd Asilomar Conference on, 2008:581-587.
[46] S. S. Chen, D. L. Donoho, M. A. Saunders. Atomic decomposition by basis pursuit[J]. SIAM Journal on Scientific Computing, 1998, 20(1):33-61.
[47] S. J. Kim, K. Koh, M. Lustig, et al. An interior-point method for large-scale -regularized least squares[J]. IEEE Journal on Selected Topics in Signal Processing, 2007, 4(1):606-617. ?1
[48] M. A. T. Figueiredo, R. D. Nowak, S. J. Wright. Gradient projection for sparse reconstruction: Application to compressed sensing and other inverse problems[J]. Journal of Selected Topics in Signal Processing: Special Issue on Convex Optimization Methods for Signal Processing, 2007, 1(4):586-598.
[49] I. Daubechies, M .Defrise, C. De Mol. An iterative thresholding algorithm for linear inverse problems with a sparsity constraint[J]. Comm. Pure Appl. Math., 2004, 57(11):1413-1457.
[50] A. C. Gilbert, S. Guha, P. Indyk, et al. Near-optimal sparse Fourier representations via sampling [A]. Proceedings of the Annual ACM Symposium on Theory of Computing[C]. Montreal, Que, Canada: Association for Computing Machinery, 2002:152-161.
[51] A. C. Gilbert, S. Muthukrishnan, M. J. Strauss. Improved time bounds for near-optimal sparse Fourier representation[A]. Proceedings of SPIE, Wavelets XI[C]. Bellingham, WA: International Society for Optical Engineering, 2005, 5914:1-15.
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