压缩感知算法及其在超宽带信道估计中的应用研究
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摘要
在以往的信号处理流程中，对信号的采样所采用的采样准则是奈奎斯特采样定理。定理规定，要精确重构出原始信号所需要的采样速率至少为信号在傅里叶变换域上带宽的两倍。对于小带宽信号而言，这样的采样速率是可以接受的，但是如果是对于带宽较大的信号，这样的采样速率要求是艰难实现的，即使能够实现，系统的处理速夏也会受到很大制约，采样的成本以及后续存储和传输所需要的开销也会急剧增加。随着信息科技的发展，人们所要处理的信号类型更加宽泛，对信号处理系统的能力要求更高，因此如何突破奈奎斯特采样定理的束缚，形成一个低成本高速夏的采样系统，是一个非常现实又亟待解决的问题。通过学者的不断探索，一种新的信号采集、处理方法一压缩感知理论诞生了，该理论将有望解决这些问题。它指出，只要能够找到一个变换域，使得信号在其上是稀疏的或者可压缩的，便可以用一个与该变换域不相关的矩阵将原始信号投影到一个维数更低的空间上，最后通过相关算法重构原始信号。该理论在许多方面部有着巨大的应用前景，脉冲超宽带无线通信系统信道估计就是研究热点之一。本文的主要工作有以下几点：
(1)对基于压缩感知fn范数最小化重构算法的研究：重构算法是压缩感知理论的研究核心之一，基于近似，0范数最小化算法的平滑，0范数(smoothed，0 no丌11)算法是凸优化算法的一种，易于计算，重构效果较好，但是重建速夏以及在有噪声下的信号重构不是很理想。本文在此基础上提出了一种改进算法，并针对无噪和有噪两种情形下算法的性能进行了分析。结果表明，与原算法相比，在无噪情形下，改进算法的运行时间更短，且随着信号长发增大，这种优越性愈加突出，因此更适用于决大规模信号的重构问题；有噪情形下，由于改进算法加入了降噪技术，提高了抗噪能力，即使在低输入信噪比下仍能取得较好的估计效果，扩展了平滑fn范数算法的应用范围。
(2)对压缩感知在超宽带信道估计中的研究：针对超宽带通信系统采样速率要求过高实现困难的问题，利用该信道固有的稀疏特性，本文提出一种改进的基于压缩感知理论的超宽带信道估计模型。该方法属于基于训练序列的信道估计方法，根据接收信号获取信道增益分布特点，构造自适应观测矩阵，利用Dcs理论构建虚拟信道建立多信道联合估计模型，最后利用相关算法得到估计信道。仿真结果显示与传统的压缩感知信道估计法相比，改进算法性能更佳。
Nyquist sampling theorem is the theoretical guidance of traditional signal sampling and processing. It proves that the signal sampling rate is not less than twice the signal bandwidth for precise signal reconstruction. However, with the development of information technology, the demand for information requirements and the ability to process information is also increasing, which brings great challenges to traditional signal processing framework based on the Nyquist theorem on sampling rate and processing speed. At the same time, it also leads to a heavy burden to the subsequent processing and transmission because of excessive sampling points. Recently, a new signal acquisition and processing method - compressed sensing theory is expected to solve these problems. It points out that as long as the signal is compressible or sparse in a transform domain, a measurement matrix which is not related to transform domain can be used to transform the high-dimensional signal to the low-dimensional space, and then reconstruct the original signal with appropriate algorithm. The theory has a great prospect in many ways; channel estimation in ultra-wideband wireless communication systems is one of research focuses. The main work of this paper is as follows:
I) The research of CS reconstruction algorithms based on l0 norm minimization: Signal reconstruction is one of the researches in the theory of compressed sensing. The smooth l0 norm algorithm based on approximate norm minimization is one of the convex optimization algorithms which are easy to calculate and the reconstruction quality is satisfied, but the rate of reconstruction and the noisy signal reconstruction is not very satisfactory. On this basis, an improved algorithm is proposed and the performance analysis is made in noise-free case and in noisy case. The results show that, compared with the original algorithm in the noiseless case, the running time is shorter, and with the signal length increases, this superiority is even more prominent and therefore more applicable to the large-scale signal reconstruction; in noisy situation, the improved is better due to adding a noise reduction technology and it still obtain good estimation effect even at low input SNR, which extends the scope of application of the smoothing norm algorithm.
II) The research of compressed sensing in UWB Channel estimation: To achieve the difficult problem that the sampling rate requirement is too high in the UWB communication system, an improved ultra-wideband estimation model based on CS is proposed in the paper by utilizing the channel inherent sparse features. Access to the channel gain distribution from the received pilot signal, and by this to construct the adaptive observation matrix, then combined with the distributed compressed sensing technology to build multi-channel joint estimation model, finally the correlation algorithm is used to estimate the channel. Simulation results show that compared with traditional compressed sensing channel estimation method, the pilot transmission power of the modified algorithm is lower and the estimation performance is better at low SNR.
(1)对基于压缩感知fn范数最小化重构算法的研究：重构算法是压缩感知理论的研究核心之一，基于近似，0范数最小化算法的平滑，0范数(smoothed，0 no丌11)算法是凸优化算法的一种，易于计算，重构效果较好，但是重建速夏以及在有噪声下的信号重构不是很理想。本文在此基础上提出了一种改进算法，并针对无噪和有噪两种情形下算法的性能进行了分析。结果表明，与原算法相比，在无噪情形下，改进算法的运行时间更短，且随着信号长发增大，这种优越性愈加突出，因此更适用于决大规模信号的重构问题；有噪情形下，由于改进算法加入了降噪技术，提高了抗噪能力，即使在低输入信噪比下仍能取得较好的估计效果，扩展了平滑fn范数算法的应用范围。
(2)对压缩感知在超宽带信道估计中的研究：针对超宽带通信系统采样速率要求过高实现困难的问题，利用该信道固有的稀疏特性，本文提出一种改进的基于压缩感知理论的超宽带信道估计模型。该方法属于基于训练序列的信道估计方法，根据接收信号获取信道增益分布特点，构造自适应观测矩阵，利用Dcs理论构建虚拟信道建立多信道联合估计模型，最后利用相关算法得到估计信道。仿真结果显示与传统的压缩感知信道估计法相比，改进算法性能更佳。
Nyquist sampling theorem is the theoretical guidance of traditional signal sampling and processing. It proves that the signal sampling rate is not less than twice the signal bandwidth for precise signal reconstruction. However, with the development of information technology, the demand for information requirements and the ability to process information is also increasing, which brings great challenges to traditional signal processing framework based on the Nyquist theorem on sampling rate and processing speed. At the same time, it also leads to a heavy burden to the subsequent processing and transmission because of excessive sampling points. Recently, a new signal acquisition and processing method - compressed sensing theory is expected to solve these problems. It points out that as long as the signal is compressible or sparse in a transform domain, a measurement matrix which is not related to transform domain can be used to transform the high-dimensional signal to the low-dimensional space, and then reconstruct the original signal with appropriate algorithm. The theory has a great prospect in many ways; channel estimation in ultra-wideband wireless communication systems is one of research focuses. The main work of this paper is as follows:
I) The research of CS reconstruction algorithms based on l0 norm minimization: Signal reconstruction is one of the researches in the theory of compressed sensing. The smooth l0 norm algorithm based on approximate norm minimization is one of the convex optimization algorithms which are easy to calculate and the reconstruction quality is satisfied, but the rate of reconstruction and the noisy signal reconstruction is not very satisfactory. On this basis, an improved algorithm is proposed and the performance analysis is made in noise-free case and in noisy case. The results show that, compared with the original algorithm in the noiseless case, the running time is shorter, and with the signal length increases, this superiority is even more prominent and therefore more applicable to the large-scale signal reconstruction; in noisy situation, the improved is better due to adding a noise reduction technology and it still obtain good estimation effect even at low input SNR, which extends the scope of application of the smoothing norm algorithm.
II) The research of compressed sensing in UWB Channel estimation: To achieve the difficult problem that the sampling rate requirement is too high in the UWB communication system, an improved ultra-wideband estimation model based on CS is proposed in the paper by utilizing the channel inherent sparse features. Access to the channel gain distribution from the received pilot signal, and by this to construct the adaptive observation matrix, then combined with the distributed compressed sensing technology to build multi-channel joint estimation model, finally the correlation algorithm is used to estimate the channel. Simulation results show that compared with traditional compressed sensing channel estimation method, the pilot transmission power of the modified algorithm is lower and the estimation performance is better at low SNR.
引文
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[2] E. Candes and T. Tao,“Near optimal signal recovery from random projections: Universal encoding strategies“, IEEE Trans. on Information Theory, vol.52, no.12, pp.5406-5425, 2006.
[3]E. Candes and J. Romberg,“Quantitative robust uncertainty principles and optimally sparse decompositions”, Foundations of Computer Math, vol.6, no.2, pp. 227-254, 2006.
[4] E. Candes,“Compressive sampling”, Proceedings of the International Congress of Mathematicians, Madrid, Spain, 2006.
[5] E. Candes, J. Romberg, T. Tao,“Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information”, IEEE Trans. on Information Theory, vol.52,no.2,pp.489-509,2006.
[6] D. L. Donoho, Y. Tsaig,“Extensions of compressed sensing”, Signal Processing, vol.86, no.3, pp.533-548, 2006.
[7] B. Kashin,“The widths of certain finite dimensional sets and classes of smooth functions”, Izv Akad Nauk SSSR, vol.41, no.2, pp.334-351,1977.
[8] W Bajwa, J Haupt, A Sayeed,“Compressive wireless sensing”, Proceedings of the fifth International Conference on Information Processing in Sensor Networks, IPSN 2006 New York: Association for Computing Machinery, pp.134-142, 2006.
[9] M. Lustig, D. L Donoho, J M Pauly,“Rapid MR imaging with compressed sensing and randomly under-sampled 3D-FT trajectories”, Proceedings of the 14th, Annual Meeting of ISMRM, Seattle, WA,2006.
[10] J. Laska, S. Kirolos, Y. Massoud,“Random sampling for analog-to-information conversion of wideband signals“, Processing of the IEEE Dallas Circuits and Systems Workshop (DCAS 2006), Dallas, Texas, 2006.
[11] R, Baraniuk, P. Steeghs,“Compressive radar imaging”, Proceedings of the Radar Conference. Washington D.C., USA: IEEE,pp. 128-133, 2007.
[12] M. D. Benedetto, T. Kaiser, A. F. Molish,“UWB communication systems: a comprehensive overview”, New York, USA: Hindawi Publishing Corporation, 2006.
[13] R. C. Qiu, R. A. Scholtz, X. Shen,“Guest editorial special section on ultra wideband wireless communications: a new horizon”, IEEE Trans on Technology, vol.54, no.5, pp.1525-1527, 2005.
[14] Faranak Nekoogar, , pp.2-10, 2007.
[15] E J Candes, M B Wakin,“An introduction to compressive sampling”, IEEE Signal Processing Magazine, March, pp.21-23. 2008.
[16] R. Berinde and P. Indyk.”Sparse recovery using sparse random matrices”, 2008.
[17] E. Candes, J. Romberg and T. Tao,“Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information”, IEEE Transaction Theory,vol.52,no.2, pp.489-505, 2006.
[18] R Baraniuk,“A lecture on compressive sensing”, IEEE Signal Processing Magazine, vol.24. no.4, pp.118-121, 2007.
[19] E. Cand, J. Ridgelets,“theory and applications”, Stanford:Stanford University, 1998.
[20] E Candes, D L Donoho,“Curvelets”, USA: Department of Statistics, Stanford University. 1999.
[21] E L Pennec, S Mallat,“Image compression with geometrical wavelets”, Proc. of IEEE International Conference on Image Processing, ICIP. 2000[C]. Vancouver, BC: IEEE Computer Society, pp. 661-664,2000.
[22] M. Figueiredo, R. Nowak and S. Wright,“Gradient Projection for sparse reconstruction: Application to compressed sensing and other inverse Problems”, IEEE Journal of Selected Topics in Signal Processing, vol.1, no.4, pp.586-597, 2007.
[23] E. Candes and T. Tao,“Near optimal signal recovery from random projections: Universal encoding strategies”, IEEE Transaction on Information Theory, vol.52,no.12, pp.5406-5425,2006.
[24] G Peyr,“Best Basis compressed sensing“, Lecture Notes in Computer Science,vol.4, no.8,pp.80-91, 2007.
[25] Zhang Chunmei, Yin Zhongke, Chen Xiangdong, Xiao Mingxia“Signal overcomplete representation and sparse decomposition based on redundant dictionaries”, Chinese Science Bulletin, vol.50, no.23, pp.2672-2677.
[26] E. Candes,“The restricted isometry property and its implications for compressed sensing”, Academies des sciences, vol.346, no.1, pp.589-592, 2006.
[27] E. Candes and J. Romberg,“Sparsity and incoherence in compressive sampling,”Inverse Problems, vol.23, no.3, pp.969-985, 2007.
[28] R. Tibshirani,“Regression shrinkage and selection via the Lasso”, Journal of Royal Statistical Society, Series B., vol.58, no.1, pp.267-288, 1996.
[29] E. Candes and T. Tao,“The Danzig selector: Statistical estimation when p is much larger than n”, Ann.Stat.,vol.35, no.6, pp.2313-2351, Dec.2007.
[30] Y.L., Chao,“Optimal integration time for UWB transmitted reference correlation recevers”, The 38th Asilomar Conference on Signals, Systems and Computers, vol.1, no.7, pp.647-651, 2004.
[31]D. L. Donoho, X. Huo,“Uncertainty principles and ideal atomic decompositions”, IEEE Translation Information Theory,vol.47, no.7, pp.2845-2862, 2001.
[32]T. T. Do, T.D. Trany, L. Gan,“Fast compressive sampling with structurally random matrices”, Proceedings of the IEEE International Conference on Acoustics, Speech and Signal Processing. Washing D.C., USA: IEEE, pp.3369-3372, 2008.
[33]W. Bajwa, J. Haupt, G. Raz, S. J. Wright and R. D. Nowak,“Toeplitz-structured compressed sensing matrices”, Proceedings of the IEEE Workshop on Statistical Signal Processing, Washington D.C., USA:IEEE, pp.294-298, 2007.
[34]F. Sebert, Y. M. Zou, L. Ying,“Toeplitz block matrices in compressed sensing and their applications in imaging”, Proceedings of International Conference on Technology and Applications in Biomedicine. Washington D.C.USA: IEEE, pp.47-50, 2008.
[35] E. Candes,“Emmanuel. Signal recovery from random projections”, Proceedings of SPIE-IS and T Electronic Imaging-Computational Imaging III 2005,Peter D:SPIE, pp.76-86, 2005.
[36] J. A. Tropp, Anna, C. Gilbert,“Signal recovery from partial information via orthogonal matching pursuit,”IEEE Trans. Information Theory, vol. 53, no. 12, pp. 4655-4666, 2007.
[37] M. Schmidt, G. Fung, R. Rosales,“Fast optimization methods for L1 regularization: A comparative study and two new approaches,”Machine Learning: ECML, vol. 47, no.1, pp. 286-297, 2007.
[38] M. A. T. Figueiredo, R. D. Nowa, S. J. Wright,“Gradient projection for sparse reconstruction: application to compressed sensing and other inverse problems,”IEEE Journal on Selected Topics in Signal Processing, vol. 1, no. 4, pp. 586-597, 2007.
[39] S. Mallat, Z. Zhang,“Matching Pursuit with time-frequency dictionaries”, IEEE Translation Signal Processing, vol.41, no.3, pp.397-415, 1993.
[40] G. Cormode, S. Muthukrishnan,“Towards an algorithmic theory of compressed sensing”, DIMACS Tech.Report , vol.20, no.5, pp.5-25, 2005.
[41] A Mark ,Davenport, B Michael, Wakin. Analysis of orthogonal matching pursuit using the restricted isometry property,IEEE transactions on information theory.vol.56.No.9,Sepetember 2010
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