OFDM系统下的稀疏信道估计
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摘要
信道估计是OFDM系统中最主要的挑战之一。作为21世纪最主要发现之一的压缩传感理论引领整个信息与信号处理领域的一场革命。它广泛地应用于图像、语音、音乐、无线通信、雷达以及天文数据等领域。本文主要关注压缩传感理论在OFDM系统下的稀疏信道估计方面的应用研究。专注于有效地解决OFDM系统下稀疏信道估计的挑战，本文构造了稀疏信道估计框架，基于这一框架，提出了新的稀疏信道估计方法。具体地，论文的创新性总结如下：
1）OFDM系统下基于阈值的主要抽头检测的稀疏信道估计（M≥Lc p,M为导频的数目Lcp是循环前缀的长度）
大量的传统信道估计算法都是在M≥Lcp情况下基于LS算法，LS算法在估计密集多径信道时性能达到最优。然而，如果信道时稀疏的，LS算法易受到噪声的干扰，导致估计的性能下降。为了克服LS算法的缺陷，提出了一种时域阈值来检测最主要抽头，此阈值由初始阈值滤除信道分量，保留噪声分量得到。由于提出的方法不需要信道的先验统计特性和噪声标准差，因此对于实际的无线通信系统带来很大的好处。理论分析与仿真结果表明提出的方法能在较大的稀疏率范围内获得更好的BER和NMSE估计性能，好的频谱利用效率以及适中的计算复杂度。
2）OFDM系统下基于压缩传感的稀疏信道估计算法(M a)建立基于压缩传感的稀疏信道估计的框架。基于这一框架，提出全新的阈值估计方法。具体地，在最大迭代次数是K max情况下（也就是信道可能出现的主要径数的最大值），采用OMP算法估计信道冲击响应。为了提高估计的性能，估计具有m (m b)在所提出的算法中， m个信道分量的索引对于阈值估计的精度至关重要。为了解决这一问题，引入基之间的相关性阈值来搜寻m个基的索引。
3）高效的基于压缩传感的非采样间隔稀疏信道估计(M a)与采样间隔稀疏信道不同，非采样间隔稀疏信道会造成接收端的能量泄漏。本文推导了观测到的接收端具有不同的过采样因子R的信道冲击响应并发现如果考虑R>1的情况，相比较基带采样泄露情况将会改善。如果考虑R→∞,将不会出现泄露的情况。基于这一点，本文开发了具有高分辨率的测量矩阵来实现高分辨率的信道估计。采用具有次优导频排布与高分辨率的测量矩阵，在压缩传感的框架下实现了利用有限数目的导频(M b)对于基带采样来说，测量矩阵只有Lcp个基，然而，如果考虑过采样的情况，测量矩阵有(R-1) Lcp+1个基（R (R>1)为过采样因子）。过采样时测量矩阵所包含的基向量的个数比基带采样时测量矩阵所包含的基向量的个数高R-1倍。然而，对于K (K <Channel estimation is one of the most important challenges in OFDM system. Asone of the major discoveries in the21th century, compressed sensing (CS) theoryleads a breakthrough in the whole information and signal processing societies. It canbe widely applied in images, audio, music, wireless communications, radar, andastronomical data etc. This thesis focuses on the research of applications ofcompressed sensing in sparse channel estimation in OFDM system. Aiming toeffectively solving the challenges of sparse channel estimation in OFDM system, thiswork constructs a sparse channel estimation framework, based on which, noveleffective sparse channel estimation methods are proposed. Specifically, the noveltiesof the thesis can be summarized as follows:
1) Threshold based most significant taps detection for sparse channel estimationin OFDM system (M≥Lc p,Mis the number of pilots andLc pis the length of cyclicprefix)
Numerous traditional channel estimation methods are initially based on LS in thecase of M≥Lcp,which is actually optimal when channel is rich multipath channel.However, if the channel is sparse, LS method is vulnerable to noise, which leads tothe degradations on the estimation performance. In order to overcome the drawbacksof LS method, a novel effective time domain threshold depending only on theeffective noise standard deviation estimated from the noise coefficients obtained byeliminating the channel coefficients with an initial estimated threshold is proposed todetect the most significant taps (MST). Since the proposed method requires neitherthe prior knowledge of channel statistics nor the noise standard deviation, which willsignificantly benefit the practical wireless communications. Both theoretical analysisand simulation results show that the proposed method can achieve better performancein both bit error rate (BER) and normalized mean square error (NMSE) thantraditional methods within a wide range of sparsity rate, has good spectral efficiencyand moderate computational complexity.
2) A novel CS based sparse channel estimation in OFDM system (M a) The framework of CS based sparse channel estimation method is constructed.Based on the framework, a novel effective threshold is proposed. Specifically, channelimpulse response (CIR) is firstly estimated by OMP with the assumption of maximumiteration number ofK max,which is also the maximum possible number of significanttaps. Then, in order to improve the estimation performance by an effective threshold,partial CIR with m (m b) In the proposed method, the m channel coefficients index is essential for theprecision of the threshold estimation. To solve this problem, the threshold ofcoherence between bases is introduced for searching the indices of the m bases.
3) Efficient and effective CS based non-sample spaced sparse channel estimationin the case ofM a) Unlike the sample spaced sparse channels, the non-sample spaced sparsechannel can cause power leakage at the receiver. We have derived the observed CIR atthe receiver with different oversampling factors R on the estimated CIR and foundthat if the oversampling R>1is considered, the leakage effect will be reducedcompared with the baseband sampling. If R→∞,there will be no leakage effect.Based on this fact, measurement matrix with finer time resolutions is developed forhigh resolution CIR estimation. By employing the measurement matrix with bothsuboptimal pilot arrangement and high resolution, CIR with finer time resolution canbe effectively estimated with limited number of pilots (M b) For the baseband sampling, we only getLc pbases for the measurementmatrix, however, if the oversampling is considered, we get (R1) Lcp+1(In the casewhere R (R1),which is the oversampling factor) bases for the measurementmatrix. R1times higher. When we go back to a K (K <
1）OFDM系统下基于阈值的主要抽头检测的稀疏信道估计（M≥Lc p,M为导频的数目Lcp是循环前缀的长度）
大量的传统信道估计算法都是在M≥Lcp情况下基于LS算法，LS算法在估计密集多径信道时性能达到最优。然而，如果信道时稀疏的，LS算法易受到噪声的干扰，导致估计的性能下降。为了克服LS算法的缺陷，提出了一种时域阈值来检测最主要抽头，此阈值由初始阈值滤除信道分量，保留噪声分量得到。由于提出的方法不需要信道的先验统计特性和噪声标准差，因此对于实际的无线通信系统带来很大的好处。理论分析与仿真结果表明提出的方法能在较大的稀疏率范围内获得更好的BER和NMSE估计性能，好的频谱利用效率以及适中的计算复杂度。
2）OFDM系统下基于压缩传感的稀疏信道估计算法(M
3）高效的基于压缩传感的非采样间隔稀疏信道估计(M
1) Threshold based most significant taps detection for sparse channel estimationin OFDM system (M≥Lc p,Mis the number of pilots andLc pis the length of cyclicprefix)
Numerous traditional channel estimation methods are initially based on LS in thecase of M≥Lcp,which is actually optimal when channel is rich multipath channel.However, if the channel is sparse, LS method is vulnerable to noise, which leads tothe degradations on the estimation performance. In order to overcome the drawbacksof LS method, a novel effective time domain threshold depending only on theeffective noise standard deviation estimated from the noise coefficients obtained byeliminating the channel coefficients with an initial estimated threshold is proposed todetect the most significant taps (MST). Since the proposed method requires neitherthe prior knowledge of channel statistics nor the noise standard deviation, which willsignificantly benefit the practical wireless communications. Both theoretical analysisand simulation results show that the proposed method can achieve better performancein both bit error rate (BER) and normalized mean square error (NMSE) thantraditional methods within a wide range of sparsity rate, has good spectral efficiencyand moderate computational complexity.
2) A novel CS based sparse channel estimation in OFDM system (M
3) Efficient and effective CS based non-sample spaced sparse channel estimationin the case ofM
引文
[1]Y. Li, L. J. Cimini, Jr., and N. R. Sollenberger,"Robust channel estimation for OFDM systems with rapid dispersive fading channels [J]," IEEE Trans. Commun, vol.46, pp.902-915,1998.
[2]J. K. Cavers,"An analysis of pilot symbol assisted modulation for rayleigh fading channels [J]," IEEE Trans. Veh. Technol, vol.40, pp.686-693,1991.
[3]A. J. Coulson,"Maximum likelihood synchronization for OFDM using a pilot symbol: algorithms [J]," IEEE J. Select. Areas Commun, vol.19, pp.2486-2494,2001.
[4]M. Morelli and U. Mengali,"A comparison of pilot-aided channel estimation methods for OFDM systems [J]," IEEE Trans. Signal Process, vol.49, pp.3065-3073,2001.
[5]I. Barhumi, G. Leus, and M. Moonen,"Optimal training design for MIMO OFDM systems in mobile wireless channels [J]," IEEE Trans. Signal Process, vol.5, pp.1615-1624,2003.
[6]X. Ma, G. B. Giannakis, and S. Ohno,"Optimal training for block transmissions over doubly selective wireless fading channels [J]," IEEE Trans. Signal Process, vol.51, pp.1351-1366,2003.
[7]B. M. S. L. Tong and M. Dong,"Pilot-assisted wireless transmissions:General model, design criteria, and signal processing [J]," IEEE Signal Process. Mag, vol.21, pp.12-25,2004.
[8]Y. Mostofi and D. C. Cox,"ICI mitigation for pilot-aided OFDM mobile systems [J], IEEE Trans. Wireless Commun, vol.4, pp.765-774,2005.
[9]Z. J. Tang, R. C. Cannizzaro, G. Leus, and P. Banelli,"Pilot-assisted time-varying channel estimation for OFDM systems [J]," IEEE Trans. Signal Process, vol.55, pp.2226-2238,2007.
[10]G. X. L. Tong and T. Kailath,"Blind identification and equalization based on second-order statistics:A time domain approach [J]," IEEE Trans. Inform. Theory, vol.40, pp.340-349,1994.
[11]L. Tong and S. Perreau,"Multichannel blind identification:From subspace to maxi-mum likelihood methods [J]," Proc. IEEE, vol.86, pp.1951-1968,1998.
[12]S. Roy and C. Y. Li,"A subspace blind channel estimation method for OFDM systems without cyclic prefix [J]," IEEE Trans. Wireless Commun, vol.1, pp.572-579,2002.
[13]Y. Li,"Simplified channel estimation for OFDM systems with multiple transmit an-tennas [J]," IEEE Trans. Wireless Commun, vol.1, pp.67-75,2002.
[14]P. Stoica and O. Besson,"Training design for frequency offset and frequency-selective channel estimation [J]," IEEE Trans. Commun, vol.51, pp.1910-1917,2003.
[15]S. Coleri, M. Ergen, A. Puri, and A. Bahai,"Channel estimation techniques based on pilot arrangement in OFDM systems [J]," IEEE Trans. Broadcast, vol.48, pp.223-229,2002.
[16]P. Hammarberg, F. Rusek, and O. Edfors,"Iterative receivers with channel estimation for multi-user MIMO-OFDM:complexity and performance [J]," EURASIP Journal on Wireless Commun and Network, vol.75, pp.1-17,2012.
[17]A. Fehske, G. Fettweis, J. Malmodin, and G. Biczok,"The global footprint of mobile communications:The ecological and economic perspective [J]," IEEE Commun. Mag, vol.49, pp.55-62,2011.
[18]C. Despins, F. Labeau, T. L. Ngoc, R. Labelle, M. Cheriet, F. G. C. Thibeault, A. Leon-Garcia, O. Cherkaoui, B. S. Arnaud, J. Mcneill, Y. Lemieux, and M. Lemay,"Leverag-ing green communications for carbon emission reductions:Techniques, testbeds, and emerging carbon footprint standards [J]," IEEE Commun. Mag, vol.49, pp.101-109,2011.
[19]W. U. Bajwa, J. Haupt, A. M. Sayeed, and R. Nowak,"Compressed channel sensing: a new approach to estimating sparse multipath channels [J]," Proc of IEEE, vol.98, pp.1058-1076,2010.
[20]D. Donoho,"Compressed sensing [J]," IEEE Trans. Inf. Theory, vol.52, pp.1289-1306,2006.
[21]E. J. Candes, J. Romberg, and T. Tao,"Robust uncertainty principles:exact signal reconstruction from highly incomplete frequency information [J]," IEEE Trans. Inform. Theory, vol.52, pp.489-509,2006.
[22]C. R. Berger, Z. H. Wang, J. Z. Huang, and S. L. Zhou,"Application of compressive sensing to sparse channel estimation [J]," IEEE Commun Mag, vol.48, pp.164-174,2010.
[23]J. L. J. Cimini,"Analysis and simulation of a digital mobile channel using orthogonal frequency-devision multiplexing [J]," IEEE Trans. Commun, vol.33, pp.665-675,1985.
[241J.A. C. Bingham,"Multicarrier modulation for data transmission:An idea whose time has come [J]," IEEE Commun Mag, vol.28, pp.5-14,1990.
[25]Z. Wang and G. G. Giannakis,"Wireless multicarrier communications:Where Fourier meets Shannon [J]," IEEE Signal Process. Mag, vol.17, pp.29-48,2000.
[26]L. Zheng and D. N. C. Tse,"Diversity and multiplexing:a fundamental tradeoff in multiple-antenna channels [J]," IEEE Trans. Inf. Theory, pp.1073-1096,2003.
[27]W. U. Bajwa, A. M. Sayeed, and R. Nowak,"Sparse multipath channels:Modeling and estimation [A]," in Proc.13th IEEE Digital Signal Process Workshop [C], pp.320-325,2009.
[28]A. A. M. Saleh, A. J. Rustako, and R. S. Roman,"Distributed antennas for indoor radio communications [J]," IEEE Trans. Commun, vol.35, pp.1245-1251,1987.
[29]H. Nishimoto, Y. Ogawa, T. Nishimura, and T. Ohgane,"Measurement-based perfor-mance evaluation of MIMO spatial multiplexing in a multipath-rich indoor environment [J]," IEEE Trans. Antennas Propagat, vol.55, pp.3677-3689,2007.
[30]S. F. Cotter and B. D. Rao,"Sparse channel estimation via matching pursuit with application to equalization [J]," IEEE Trans. Commun, vol.50, pp.374-377,2002.
[31]C. R. Berger, S. Zhou, J. C. Preisig, and P.Willett,"Sparse channel estimation for mul-ticarrier underwater acoustic communication:From subspace methods to compressed sensing [J]," IEEE Trans. Signal Process, vol.58, pp.1708-1721,2010.
[32]P. Maechler, P. Greisen, B. Sporrer, S. Steiner, N. Felber, and A. Burg,"Implemen-tation of greedy algorithms for LTE sparse channel estimation [A]," in Proc.44th Asilomar Conf. Signals, Syst and Comp [C], pp.400-405,2010.
[33]. Recommendation ITU-R M.1225, International Telecommunication Union, Guidelines for evaluation of radio transmission technologies for IMT-2000[M].1997.
[34]V. E. et al,"Channel models for fixed wireless applications [S]," IEEE802.16.3c-01/29r4, pp.1-36,2001.
[35]H. Minn and V. K. Bhargava,"An investigation into time-domain approach for OFDM channel estimation [J]," IEEE Trans. Broadcast, vol.46, pp.240-248,2000.
[36]M. R. Raghavendra and K. Giridhar,"Improving channel estimation in OFDM systems for sparse multipath channels [J]," IEEE Signal Precess. Lett, vol.12, pp.52-55,2005.
[37]J. Oliver, R. Aravind, and K. M. M. Prabhu,"Sparse channel estimation in OFDM system by threshold-based pruning [J]," Electron. Lett, vol.44, pp.830-832,2008.
[38]W. Z. F. Wan and M. Swamy,"Semi-blind most significant tap detection for sparse channel estimation of OFDM systems [J]," IEEE Trans. Circuits Syst. I, Reg. Papers, vol.57, pp.703-713,2010.
[39]M. R. Raghavendra, E. Lior, S. Bhashyam, and K. Giridhar,"Parametric channel esti-mation for pseudo-random tile-allocation in uplink OFDM A [J]," IEEE Trans. Signal Process, vol.55, pp.5370-5381,2007.
[40]S. Rosati, G. E. Corazza, and A. Venelli-Coralli,"OFDM channel estimation based on impulse response decimation:analysis and novel algorithms [J]," IEEE Trans. Com-mun, vol.60, pp.1996-2008,2012.
[41]C. E. Shannon,"Communications in the presence of noise [J]," Proc. IRE, vol.37, pp.10-21,1949.
[42]H. Nyquist,"Certain topics in telegraph transmission theory [J]," AIEE Trans, vol.47, pp.617-644,1928.
[43]R. G. Baraniuk,"Compressive sensing [J]," IEEE Signal Process. Mag, vol.24, pp.118120,124,2007.
[44]F. J. Harris,"On the use of windows for harmonic analysis with the discrete fourier transform [J]," Proc. IEEE, vol.66, pp.51-83,1978.
[45]N. Ahmed, T. Natarajan, and K. R. Rao,"On image processing and a discrete cosine transform [J]," IEEE Trans. Comput, vol. C-23, pp.90-93,1974.
[46]S. Mallat, A wavelet tour of signal processing [M]. Academic Press,2009.
[47]E. J. Candes, Y. C. Elad, D. Needell, and P. Randall,"Compressed sensing with coherent and redundant dictionaries [J]," Appl and Comput. Harmo Anal, vol.31, pp.5973,2011.
[48]H. Huang and A. Makur,"Backtracking-based matching pursuit method for sparse signal reconstruction [J]," IEEE Signal Process. Lett, vol.18, pp.391-394,2011.
[49]J. A. Tropp,"Signal recovery from random measurements via orthogonal matching pursuit [J]," IEEE Trans. Inf. Theory, vol.53, pp.4655-4666,2007.
[50]C. Qi and L. Wu,"Tree-based backward pilot generation for sparse channel estimation [J]," Electron. Lett, vol.48, pp.501-503,2012.
[51]E. J. Candes and M. B. Wakin,"An introduction to compressive sampling [J]," IEEE Signal Process. Mag, vol.25, pp.21-30,2008.
[52]E. J. Candes and T. Tao,"Decoding by linear programming [J]," IEEE Trans. Inform. Theory, vol.51, pp.4203-4215,2005.
[53]S. S. Chen, D. L. Donoho, and M. A. Saunders,"Atomic decomposition by basis pursuit [J]," SIAM J. Scientific Comput, vol.20, pp.33-61,1999.
[54]E. J. Candes and T. Tao,"The dantzig selector:Statistical estimation when p is much larger than n [J]," Ann. Stat., vol.35, pp.2313-2351,2007.
[55]R. Tibshirani,"Regression shrinkage and selection via the lasso [J]," J. R. Stat. Soc. B, vol.58, pp.267-288,1996.
[56]H. Rauhut, K. Schnass, and P. Vandergheynst,"Compressed sensing and redundant dictionaries [J]," IEEE Trans. Inform. Theory, vol.54, pp.2210-2219,2008.
[57]T. Blumensath and M. Davies,"Iterative hard thresholding for sparse approximations [J]," J. Fourier Anal. Appl, vol.14, pp.629-654,2008.
[58]T. Blumensath and M. Davies,"Iterative hard thresholding for compressed sensing [J],' Appl. Comput. Harmonic Anal, vol.27, pp.265-274,2009.
[59]X. He and R. Song,"Pilot pattern optimization for compressed sensing based sparse channel estimation in OFDM systems [A]," in Proc. Conf. Wireless Commun and Signal Process [C],2010.
[60]C. Qi and L. Wu,"Optimized pilot placement for sparse channel estimation in OFDM systems [J]," IEEE Signal Process. Lett, vol.18, pp.749-752,2011.
[61]G. Taubock and F. Hlawatsch,"A compressed sensing technique for OFDM channel estimation in mobile environments:Exploiting channel sparsity for reducing pilots [A],' in Proc. IEEE Int. Conf. Acoust, Speech and Signal Process (ICASSP)[C], pp.2885-2888,2008.
[62]J. Meng, Y. Li, N. Nguyen, W. Yin, and Z. Han,"Compressive sensing based high res-olution channel estimation for OFDM system [J]," IEEE J. Sel. Topics Signal Process, vol.6, pp.15-25,2012.
[63]J. C. Chen, C. K. Wen, and P. Ting,"An efficient pilot design scheme for sparse channel estimation in OFDM systems [J]," IEEE Commun. Lett, vol.17, pp.1352-1355,2013.
[64]G. Taubock and F. Hlawatsch,"Compressed sensing based estimation of doubly se-lective channels using a sparsity-optimized basis expansion [A]," in Proc. Eur. Signal Process. Conf (EUSIPCO)[C],2008.
[65]G. Taubock, F. Hlawatsch, D. Eiwen, and H. Rauhut,"Compressive estimation of dou-bly selective channels in multicarrier systems:Leakage effects and sparsity-enhancing processing [J]," IEEE J. Sel. Topics Signal Process, vol.4, pp.255-271,2010.
[66]F. H. D. Eiwen, G. Taubock and H. G. Feichtinger,"Compressive tracking of doubly se-lective channels in multicarrier systems based on sequential delay-doppler sparsity [A], in Proc. IEEE Int. Conf. Acoust, Speech and Signal Process (ICASSP)[C], pp.2928-2931,2011.
[67]J.J.van de Beek, O. Edfors, M. Sandell, S. K. Wilson, and P. O. Borjesson,"On channel estimation in OFDM systems [A]," in Proc. IEEE Vehicular Technology Conf.(VTC)[C], vol.2, pp.815-819,1995.
[68]R. Negi and J. Cioffi,"Pilot tone selection for channel estimation in a mobile OFDM system [J]," IEEE Trans. Consum Electron, vol.44, pp.1122-1128,1998.
[69]M. Hsieh and C. Wei,"Channel estimation for OFDM systems based on comb-type pilot arrangement in frequency selective fading channels [J]," IEEE Trans. Consumer Electron, vol.44, pp.217-225,1998.
[70]S. G. Kang, Y. M. Ha, and E. K. Joo,"A comparative investigation on channel esti-mation algorithms for OFDM in mobile communications [J]," IEEE Trans. Broadcast, vol.49, pp.142-149,2003.
[71]X. Dong, W. S. Lu, and A. Soong,"Linear interpolation in pilot symbol assisted channel estimation for OFDM [J]," IEEE Trans. Wireless Commun, vol.6, pp.1910-1920,2007.
[72]Y. S. Lee, H. C. Shin, and H. N. Kim,"Channel estimation based on time-domain threshold for OFDM systems [J]," IEEE Trans. Broadcast, vol.55, pp.656-662,2009.
[731Y-Kang, K. Kim, and H. Park,"Efficient DFT-based channel estimation for OFDM system on multipath channels [J]," IET Commun, vol.1, pp.197-202,2007.
[74]J. K. Hwang and R. L. Chung,"Low-complexity algorithm for tap-selective maxi-mum likelihood estimation over sparse multipath channels [A]," in Proc. IEEE Global Telecommunications Conference (GLOBECOM)[C], pp.2857-2862,2007.
[75]G. Gui, Q. Wan, S. Qin, and A. Huang,"Sparse multipath channel estimation using ds algorithm in wideband communication systems [A]," in Proc. Cong. Image and Signal Process (CISP)[C], pp.4450-4453,2010.
[76]S. F. Cotter and B. D. Rao,"Matching pursuit based decision-feedback equalizers [A],' in Proc. IEEE Int. Conf. Acoust. Speech Signal Process (ICASSP)[C], pp.2713-2716,2000.
[77]S. F. Cotter and B. D. Rao,"The adaptive matching pursuit algorithm for estimation and equalization of sparse time-varying channels [A]," in Proc. Asilomar Conf. Signals Syst. Comput., Pacific Grove [C],2000.
[78]W. Li and J. C. Preisig,"Estimation of rapidly time-varying sparse channels [J]," IEEE J. Ocean. Eng, vol.32, pp.927-939,2007.
[79]C.-J. Wu and D. W. Lin,"A group matching pursuit algorithm for sparse channel estimation for OFDM transmission [A]," in Proc. IEEE Int. Conf. Acoust, Speech and Signal Process (ICASSP)[C], pp.429-432,2006.
[80]T. Kang and R. A. Iltis,"Matching pursuit channel estimation for an underwater acous-tic OFDM modem [A]," in Proc. IEEE Int. Conf. Acoust, speech and signal process (ICASSP)[C], pp.5296-5299,2008.
[81]M. A. Khojastepour, K. Gomadam, and X. D. Wang,"Pilot-assisted channel estimation for MIMO OFDM systems using theory of sparse signal recovery [A]," in Proc. IEEE Intern. Conf. Acoust, speech and signal process (ICASSP)[C], pp.2693-2696,2009.
[82]S. Kim,"Angle-domain frequency-selective sparse channel estimation for underwater MIMO-OFDM systems [J]," IEEE Commun. Lett, vol.16, pp.685-687,2012.
[83]W. C. Jakes, Microwave communications [M]. John Wiley,1947.
[84]A. Goldsmith, Wireless communications [M]. Cambridge University Press,2005.
[85]R. Prasad, OFDM for wireless communications systems [M]. Artech House Press,2004.
[86]J. G. Proakis, Digital communications5th edition [M]. McGraw-Hill Book Co.,2007.
[87]B. Farhang-Boroujeny,"OFDM versus filter bank multicarrier [J]," IEEE Signal Pro-cess. Mag, vol.28, pp.92-112,2011.
[88]B. G. Negash and H. Nikookar,"Wavelet-based multicarrier transmission over multi-path wireless channels [J]," Electron. Lett, vol.36, pp.1778-1788,2000.
[89]K-M-Wong, J. F. Wu, T. N. Davidson, Q. Jin, and P. C. Ching,"Performance of wavelet packet-division multiplexing in impulsive and gaussian noise [J]," IEEE Trans. Commun, vol.48, pp.1083-1086,2000.
[901J.-J.v. d. B. S. K. W. O. Edfors, M. Sandell and P. O. Brjesson,"OFDM chan-nel estimation by singular value decomposition [J]," IEEE Trans. Commun, vol.46, pp.931-939,1998.
[91]S. M. Kay, Fundamentals of statistical signal processing:Estimation Theory [M]. Upper Saddle River, NJ:Prentice-Hall,1993.
[92]G. Auer and E. Karipidis,"Pilot aided channel estimation for ofdm:A separated approach for smoothing and interpolation [A]," in Proc. IEEE Int. Conf. Commun (ICC)[C], vol.4, pp.2173-2178,2005.
[931L. L. Scharf, Statistical signal processing:detection, estimation, and time series anal-ysis [M]. Addison-Wesley,1991.
[94]P. Stoica and A. Nehorai,"Music, maximum likelihood, and Cramer-Rao bound [J]," IEEE Trans. Acoust, Speech and Signal Process, vol.37, pp.720-741,1989.
[95]L. Deneire, P. Vandenameele, L. van der Perre, B. Gyselinckx, and M. Engels,"A low-complexity ML channel estimator for OFDM [J]," IEEE Trans. Commun, vol.51, pp.135-140,2003.
[96]E. J. Candes and T. Tao,"Stable signal recovery from incomplete and inaccurate information [J]," Commun. Pure Appl. Math, vol.59, pp.1207-1233,2005.
[97]E. J. Candes,"The restricted isometry property and its implications for compressed sensing [J]," C. R. Acad. Sci. Paris, Ser. I, vol.346, pp.589-592,2008.
[98]R. G. Baraniuk, M. Davenport, R. A. DeVore, and M. B. Wakin,"A simple proof of the restricted isometry property for random matrices [J]," Constr. Approx, vol.28, pp.253263,2008.
[99]M. A. Davenport and M. B. Wakin,"Analysis of orthogonal matching pursuit approach via restricted isometry property [J]," IEEE Trans. Inf. Theory, vol.56, pp.4395-4401,2010.
[100]D. Donoho, M. Elad, and V. Temlyakov,"Stable recovery of sparse overcomplete rep-resentations in the presence of noise [J]," IEEE Trans. Inf. Theory, vol.52, pp.6-18,2006.
[101]I. E. Nesterov, A. Nemirovskii, and Y. Nesterov,"Interior-point polynominal algorithms in convex programming [J]," Philadelphia, PA:SIAM,1994.
[102]J. A. Tropp,"Greed is good:Algorithmic results for sparse approximation [J]," IEEE Trans. Inf. Theory, vol.50, pp.2231-2242,2004.
[103]J.A. Tropp,"Computational methods for sparse solution of linear inverse problems [J]," Proc of IEEE, vol.98, pp.948-958,2010.
[104]Z. Ben-Haim and Y. C. Eldar,"The Cramer-Rao bound for estimating a sparse pa-rameter vector [J]," IEEE Trans. Signal Process, vol.58, pp.3384-3389,2010.
[105]Z. Ben-Haim, Y. C. Eldar, and M. Elad,"Coherence-based performance guarantees for estimating a sparse vector under random noise [J]," IEEE Trans. Signal Process, vol.58, pp.5030-5043,2010.
[106]D. Donoho and I. M. Johnstone,"Ideal spatial adaptation by wavelet shrinkage [J]," Biometrika, vol.81, pp.425-455,1994.
[107]D. Donoho,"De-noising by soft thresholding [J]," IEEE Trans. Inf. Theory, vol.41, pp.613-627,1995.
[108]D. Donoho, Y. Tsaig, I. Drori, and J. C. Starck,"Sparse solution of underdetermined systems of linear equations by stagewise orthogonal matching pursuit [J]," IEEE Trans. Inf. Theory, vol.58, pp.1094-1121,2012.
[109]D. Needell and R. Vershynin,"Uniform uncertainty principle and signal recovery via regularized orthogonal matching pursuit [J]," Found. Comput. Math, vol.9, pp.317-334,2009.
[110]D. Needell and J. A. Tropp,"CoSaMP:Iterative signal recovery from incomplete and inaccurate samples [J]," Appl and comput. harmo anal, vol.26, pp.301-321,2009.
[111]W. Dai and O. Milenkovic,"Subspace pursuit for compressive sensing signal recon-struction [J]," IEEE Trans. Inf. Theory, vol.55, pp.2230-2249,2009.
[112]E. J. Candes and T. Tao,"Near optimal signal recovery from random projections: Universal encoding strategies?[J]," IEEE Trans. Inform. Theory, vol.52, pp.5406-5425,2006.
[113]W. U. Bajwa, G. R. J. Haupt, and R. Nowak,"Compressed channel sensing [A]," in Proc.42nd Conf. Information Science and Systems [C], pp.5-10,2008.
[114]A. M. Abd-Elfattah, S. H. Amal, and D. M. Ziedan,"Efficiency of Bayes estimator for rayleigh distribution," Statistics on the Internet,2006.
[1151K.Peter,"Phase preserving denoising of images [A]," in Proc. Conf. Australian Pattern Recognition Society [C], pp.212-217,1999.
[116]L. Najjar,"Sparsity level-aware threshold-based channel structure detection in OFDM systems [J]," Electron. Lett, vol.48, pp.495-496,2012.
[117]Z. P. W. Yi, L. Lihua and L. Zemin,"Optimal threshold for channel estimation in MIMO-OFDM system [A]," in Proc. IEEE Int. Conf. Commun (ICC)[C], pp.4376-4380,2008.
[118]S. Sarvotham, D. Bron, and R. G. Baraniuk,"Sudocodes-fast measurement and reconstruction of sparse signals [A]," in Proc. IEEE Int. Symp. Inf. Theory [C],2006.
[119]H. Xie, G. Adrieux, Y. Wang, J. F. Diouris, and S. Feng,"A novel effective compressed sensing based sparse channel estimation in OFDM system [A]," in Proc. IEEE Int. Conf. Signal Process, Commun and Comput (ICSPCC)[C], pp.1-6,2013.
[120]K. M. Nasr, J. P. Cosmas, M. Bard, and J. Gledhill,"Performance of an echo can-celler and channel estimator for on-channel repeaters in DVB-T/H networks [J]," IEEE Trans. Broadcast, vol.53, pp.609-618,2007.
[121]D. Hu, X. Wang, and L. He,"A new sparse channel estimation and tracking method for time-varying ofdm systems [J]," IEEE Trans. Veh. Tech, vol.62, pp.4648-4653,2013.
[122]C. Heegard and S. B. Wicker, Turbo coding [M]. Kluwer Academic Publishers,1999.
[2]J. K. Cavers,"An analysis of pilot symbol assisted modulation for rayleigh fading channels [J]," IEEE Trans. Veh. Technol, vol.40, pp.686-693,1991.
[3]A. J. Coulson,"Maximum likelihood synchronization for OFDM using a pilot symbol: algorithms [J]," IEEE J. Select. Areas Commun, vol.19, pp.2486-2494,2001.
[4]M. Morelli and U. Mengali,"A comparison of pilot-aided channel estimation methods for OFDM systems [J]," IEEE Trans. Signal Process, vol.49, pp.3065-3073,2001.
[5]I. Barhumi, G. Leus, and M. Moonen,"Optimal training design for MIMO OFDM systems in mobile wireless channels [J]," IEEE Trans. Signal Process, vol.5, pp.1615-1624,2003.
[6]X. Ma, G. B. Giannakis, and S. Ohno,"Optimal training for block transmissions over doubly selective wireless fading channels [J]," IEEE Trans. Signal Process, vol.51, pp.1351-1366,2003.
[7]B. M. S. L. Tong and M. Dong,"Pilot-assisted wireless transmissions:General model, design criteria, and signal processing [J]," IEEE Signal Process. Mag, vol.21, pp.12-25,2004.
[8]Y. Mostofi and D. C. Cox,"ICI mitigation for pilot-aided OFDM mobile systems [J], IEEE Trans. Wireless Commun, vol.4, pp.765-774,2005.
[9]Z. J. Tang, R. C. Cannizzaro, G. Leus, and P. Banelli,"Pilot-assisted time-varying channel estimation for OFDM systems [J]," IEEE Trans. Signal Process, vol.55, pp.2226-2238,2007.
[10]G. X. L. Tong and T. Kailath,"Blind identification and equalization based on second-order statistics:A time domain approach [J]," IEEE Trans. Inform. Theory, vol.40, pp.340-349,1994.
[11]L. Tong and S. Perreau,"Multichannel blind identification:From subspace to maxi-mum likelihood methods [J]," Proc. IEEE, vol.86, pp.1951-1968,1998.
[12]S. Roy and C. Y. Li,"A subspace blind channel estimation method for OFDM systems without cyclic prefix [J]," IEEE Trans. Wireless Commun, vol.1, pp.572-579,2002.
[13]Y. Li,"Simplified channel estimation for OFDM systems with multiple transmit an-tennas [J]," IEEE Trans. Wireless Commun, vol.1, pp.67-75,2002.
[14]P. Stoica and O. Besson,"Training design for frequency offset and frequency-selective channel estimation [J]," IEEE Trans. Commun, vol.51, pp.1910-1917,2003.
[15]S. Coleri, M. Ergen, A. Puri, and A. Bahai,"Channel estimation techniques based on pilot arrangement in OFDM systems [J]," IEEE Trans. Broadcast, vol.48, pp.223-229,2002.
[16]P. Hammarberg, F. Rusek, and O. Edfors,"Iterative receivers with channel estimation for multi-user MIMO-OFDM:complexity and performance [J]," EURASIP Journal on Wireless Commun and Network, vol.75, pp.1-17,2012.
[17]A. Fehske, G. Fettweis, J. Malmodin, and G. Biczok,"The global footprint of mobile communications:The ecological and economic perspective [J]," IEEE Commun. Mag, vol.49, pp.55-62,2011.
[18]C. Despins, F. Labeau, T. L. Ngoc, R. Labelle, M. Cheriet, F. G. C. Thibeault, A. Leon-Garcia, O. Cherkaoui, B. S. Arnaud, J. Mcneill, Y. Lemieux, and M. Lemay,"Leverag-ing green communications for carbon emission reductions:Techniques, testbeds, and emerging carbon footprint standards [J]," IEEE Commun. Mag, vol.49, pp.101-109,2011.
[19]W. U. Bajwa, J. Haupt, A. M. Sayeed, and R. Nowak,"Compressed channel sensing: a new approach to estimating sparse multipath channels [J]," Proc of IEEE, vol.98, pp.1058-1076,2010.
[20]D. Donoho,"Compressed sensing [J]," IEEE Trans. Inf. Theory, vol.52, pp.1289-1306,2006.
[21]E. J. Candes, J. Romberg, and T. Tao,"Robust uncertainty principles:exact signal reconstruction from highly incomplete frequency information [J]," IEEE Trans. Inform. Theory, vol.52, pp.489-509,2006.
[22]C. R. Berger, Z. H. Wang, J. Z. Huang, and S. L. Zhou,"Application of compressive sensing to sparse channel estimation [J]," IEEE Commun Mag, vol.48, pp.164-174,2010.
[23]J. L. J. Cimini,"Analysis and simulation of a digital mobile channel using orthogonal frequency-devision multiplexing [J]," IEEE Trans. Commun, vol.33, pp.665-675,1985.
[241J.A. C. Bingham,"Multicarrier modulation for data transmission:An idea whose time has come [J]," IEEE Commun Mag, vol.28, pp.5-14,1990.
[25]Z. Wang and G. G. Giannakis,"Wireless multicarrier communications:Where Fourier meets Shannon [J]," IEEE Signal Process. Mag, vol.17, pp.29-48,2000.
[26]L. Zheng and D. N. C. Tse,"Diversity and multiplexing:a fundamental tradeoff in multiple-antenna channels [J]," IEEE Trans. Inf. Theory, pp.1073-1096,2003.
[27]W. U. Bajwa, A. M. Sayeed, and R. Nowak,"Sparse multipath channels:Modeling and estimation [A]," in Proc.13th IEEE Digital Signal Process Workshop [C], pp.320-325,2009.
[28]A. A. M. Saleh, A. J. Rustako, and R. S. Roman,"Distributed antennas for indoor radio communications [J]," IEEE Trans. Commun, vol.35, pp.1245-1251,1987.
[29]H. Nishimoto, Y. Ogawa, T. Nishimura, and T. Ohgane,"Measurement-based perfor-mance evaluation of MIMO spatial multiplexing in a multipath-rich indoor environment [J]," IEEE Trans. Antennas Propagat, vol.55, pp.3677-3689,2007.
[30]S. F. Cotter and B. D. Rao,"Sparse channel estimation via matching pursuit with application to equalization [J]," IEEE Trans. Commun, vol.50, pp.374-377,2002.
[31]C. R. Berger, S. Zhou, J. C. Preisig, and P.Willett,"Sparse channel estimation for mul-ticarrier underwater acoustic communication:From subspace methods to compressed sensing [J]," IEEE Trans. Signal Process, vol.58, pp.1708-1721,2010.
[32]P. Maechler, P. Greisen, B. Sporrer, S. Steiner, N. Felber, and A. Burg,"Implemen-tation of greedy algorithms for LTE sparse channel estimation [A]," in Proc.44th Asilomar Conf. Signals, Syst and Comp [C], pp.400-405,2010.
[33]. Recommendation ITU-R M.1225, International Telecommunication Union, Guidelines for evaluation of radio transmission technologies for IMT-2000[M].1997.
[34]V. E. et al,"Channel models for fixed wireless applications [S]," IEEE802.16.3c-01/29r4, pp.1-36,2001.
[35]H. Minn and V. K. Bhargava,"An investigation into time-domain approach for OFDM channel estimation [J]," IEEE Trans. Broadcast, vol.46, pp.240-248,2000.
[36]M. R. Raghavendra and K. Giridhar,"Improving channel estimation in OFDM systems for sparse multipath channels [J]," IEEE Signal Precess. Lett, vol.12, pp.52-55,2005.
[37]J. Oliver, R. Aravind, and K. M. M. Prabhu,"Sparse channel estimation in OFDM system by threshold-based pruning [J]," Electron. Lett, vol.44, pp.830-832,2008.
[38]W. Z. F. Wan and M. Swamy,"Semi-blind most significant tap detection for sparse channel estimation of OFDM systems [J]," IEEE Trans. Circuits Syst. I, Reg. Papers, vol.57, pp.703-713,2010.
[39]M. R. Raghavendra, E. Lior, S. Bhashyam, and K. Giridhar,"Parametric channel esti-mation for pseudo-random tile-allocation in uplink OFDM A [J]," IEEE Trans. Signal Process, vol.55, pp.5370-5381,2007.
[40]S. Rosati, G. E. Corazza, and A. Venelli-Coralli,"OFDM channel estimation based on impulse response decimation:analysis and novel algorithms [J]," IEEE Trans. Com-mun, vol.60, pp.1996-2008,2012.
[41]C. E. Shannon,"Communications in the presence of noise [J]," Proc. IRE, vol.37, pp.10-21,1949.
[42]H. Nyquist,"Certain topics in telegraph transmission theory [J]," AIEE Trans, vol.47, pp.617-644,1928.
[43]R. G. Baraniuk,"Compressive sensing [J]," IEEE Signal Process. Mag, vol.24, pp.118120,124,2007.
[44]F. J. Harris,"On the use of windows for harmonic analysis with the discrete fourier transform [J]," Proc. IEEE, vol.66, pp.51-83,1978.
[45]N. Ahmed, T. Natarajan, and K. R. Rao,"On image processing and a discrete cosine transform [J]," IEEE Trans. Comput, vol. C-23, pp.90-93,1974.
[46]S. Mallat, A wavelet tour of signal processing [M]. Academic Press,2009.
[47]E. J. Candes, Y. C. Elad, D. Needell, and P. Randall,"Compressed sensing with coherent and redundant dictionaries [J]," Appl and Comput. Harmo Anal, vol.31, pp.5973,2011.
[48]H. Huang and A. Makur,"Backtracking-based matching pursuit method for sparse signal reconstruction [J]," IEEE Signal Process. Lett, vol.18, pp.391-394,2011.
[49]J. A. Tropp,"Signal recovery from random measurements via orthogonal matching pursuit [J]," IEEE Trans. Inf. Theory, vol.53, pp.4655-4666,2007.
[50]C. Qi and L. Wu,"Tree-based backward pilot generation for sparse channel estimation [J]," Electron. Lett, vol.48, pp.501-503,2012.
[51]E. J. Candes and M. B. Wakin,"An introduction to compressive sampling [J]," IEEE Signal Process. Mag, vol.25, pp.21-30,2008.
[52]E. J. Candes and T. Tao,"Decoding by linear programming [J]," IEEE Trans. Inform. Theory, vol.51, pp.4203-4215,2005.
[53]S. S. Chen, D. L. Donoho, and M. A. Saunders,"Atomic decomposition by basis pursuit [J]," SIAM J. Scientific Comput, vol.20, pp.33-61,1999.
[54]E. J. Candes and T. Tao,"The dantzig selector:Statistical estimation when p is much larger than n [J]," Ann. Stat., vol.35, pp.2313-2351,2007.
[55]R. Tibshirani,"Regression shrinkage and selection via the lasso [J]," J. R. Stat. Soc. B, vol.58, pp.267-288,1996.
[56]H. Rauhut, K. Schnass, and P. Vandergheynst,"Compressed sensing and redundant dictionaries [J]," IEEE Trans. Inform. Theory, vol.54, pp.2210-2219,2008.
[57]T. Blumensath and M. Davies,"Iterative hard thresholding for sparse approximations [J]," J. Fourier Anal. Appl, vol.14, pp.629-654,2008.
[58]T. Blumensath and M. Davies,"Iterative hard thresholding for compressed sensing [J],' Appl. Comput. Harmonic Anal, vol.27, pp.265-274,2009.
[59]X. He and R. Song,"Pilot pattern optimization for compressed sensing based sparse channel estimation in OFDM systems [A]," in Proc. Conf. Wireless Commun and Signal Process [C],2010.
[60]C. Qi and L. Wu,"Optimized pilot placement for sparse channel estimation in OFDM systems [J]," IEEE Signal Process. Lett, vol.18, pp.749-752,2011.
[61]G. Taubock and F. Hlawatsch,"A compressed sensing technique for OFDM channel estimation in mobile environments:Exploiting channel sparsity for reducing pilots [A],' in Proc. IEEE Int. Conf. Acoust, Speech and Signal Process (ICASSP)[C], pp.2885-2888,2008.
[62]J. Meng, Y. Li, N. Nguyen, W. Yin, and Z. Han,"Compressive sensing based high res-olution channel estimation for OFDM system [J]," IEEE J. Sel. Topics Signal Process, vol.6, pp.15-25,2012.
[63]J. C. Chen, C. K. Wen, and P. Ting,"An efficient pilot design scheme for sparse channel estimation in OFDM systems [J]," IEEE Commun. Lett, vol.17, pp.1352-1355,2013.
[64]G. Taubock and F. Hlawatsch,"Compressed sensing based estimation of doubly se-lective channels using a sparsity-optimized basis expansion [A]," in Proc. Eur. Signal Process. Conf (EUSIPCO)[C],2008.
[65]G. Taubock, F. Hlawatsch, D. Eiwen, and H. Rauhut,"Compressive estimation of dou-bly selective channels in multicarrier systems:Leakage effects and sparsity-enhancing processing [J]," IEEE J. Sel. Topics Signal Process, vol.4, pp.255-271,2010.
[66]F. H. D. Eiwen, G. Taubock and H. G. Feichtinger,"Compressive tracking of doubly se-lective channels in multicarrier systems based on sequential delay-doppler sparsity [A], in Proc. IEEE Int. Conf. Acoust, Speech and Signal Process (ICASSP)[C], pp.2928-2931,2011.
[67]J.J.van de Beek, O. Edfors, M. Sandell, S. K. Wilson, and P. O. Borjesson,"On channel estimation in OFDM systems [A]," in Proc. IEEE Vehicular Technology Conf.(VTC)[C], vol.2, pp.815-819,1995.
[68]R. Negi and J. Cioffi,"Pilot tone selection for channel estimation in a mobile OFDM system [J]," IEEE Trans. Consum Electron, vol.44, pp.1122-1128,1998.
[69]M. Hsieh and C. Wei,"Channel estimation for OFDM systems based on comb-type pilot arrangement in frequency selective fading channels [J]," IEEE Trans. Consumer Electron, vol.44, pp.217-225,1998.
[70]S. G. Kang, Y. M. Ha, and E. K. Joo,"A comparative investigation on channel esti-mation algorithms for OFDM in mobile communications [J]," IEEE Trans. Broadcast, vol.49, pp.142-149,2003.
[71]X. Dong, W. S. Lu, and A. Soong,"Linear interpolation in pilot symbol assisted channel estimation for OFDM [J]," IEEE Trans. Wireless Commun, vol.6, pp.1910-1920,2007.
[72]Y. S. Lee, H. C. Shin, and H. N. Kim,"Channel estimation based on time-domain threshold for OFDM systems [J]," IEEE Trans. Broadcast, vol.55, pp.656-662,2009.
[731Y-Kang, K. Kim, and H. Park,"Efficient DFT-based channel estimation for OFDM system on multipath channels [J]," IET Commun, vol.1, pp.197-202,2007.
[74]J. K. Hwang and R. L. Chung,"Low-complexity algorithm for tap-selective maxi-mum likelihood estimation over sparse multipath channels [A]," in Proc. IEEE Global Telecommunications Conference (GLOBECOM)[C], pp.2857-2862,2007.
[75]G. Gui, Q. Wan, S. Qin, and A. Huang,"Sparse multipath channel estimation using ds algorithm in wideband communication systems [A]," in Proc. Cong. Image and Signal Process (CISP)[C], pp.4450-4453,2010.
[76]S. F. Cotter and B. D. Rao,"Matching pursuit based decision-feedback equalizers [A],' in Proc. IEEE Int. Conf. Acoust. Speech Signal Process (ICASSP)[C], pp.2713-2716,2000.
[77]S. F. Cotter and B. D. Rao,"The adaptive matching pursuit algorithm for estimation and equalization of sparse time-varying channels [A]," in Proc. Asilomar Conf. Signals Syst. Comput., Pacific Grove [C],2000.
[78]W. Li and J. C. Preisig,"Estimation of rapidly time-varying sparse channels [J]," IEEE J. Ocean. Eng, vol.32, pp.927-939,2007.
[79]C.-J. Wu and D. W. Lin,"A group matching pursuit algorithm for sparse channel estimation for OFDM transmission [A]," in Proc. IEEE Int. Conf. Acoust, Speech and Signal Process (ICASSP)[C], pp.429-432,2006.
[80]T. Kang and R. A. Iltis,"Matching pursuit channel estimation for an underwater acous-tic OFDM modem [A]," in Proc. IEEE Int. Conf. Acoust, speech and signal process (ICASSP)[C], pp.5296-5299,2008.
[81]M. A. Khojastepour, K. Gomadam, and X. D. Wang,"Pilot-assisted channel estimation for MIMO OFDM systems using theory of sparse signal recovery [A]," in Proc. IEEE Intern. Conf. Acoust, speech and signal process (ICASSP)[C], pp.2693-2696,2009.
[82]S. Kim,"Angle-domain frequency-selective sparse channel estimation for underwater MIMO-OFDM systems [J]," IEEE Commun. Lett, vol.16, pp.685-687,2012.
[83]W. C. Jakes, Microwave communications [M]. John Wiley,1947.
[84]A. Goldsmith, Wireless communications [M]. Cambridge University Press,2005.
[85]R. Prasad, OFDM for wireless communications systems [M]. Artech House Press,2004.
[86]J. G. Proakis, Digital communications5th edition [M]. McGraw-Hill Book Co.,2007.
[87]B. Farhang-Boroujeny,"OFDM versus filter bank multicarrier [J]," IEEE Signal Pro-cess. Mag, vol.28, pp.92-112,2011.
[88]B. G. Negash and H. Nikookar,"Wavelet-based multicarrier transmission over multi-path wireless channels [J]," Electron. Lett, vol.36, pp.1778-1788,2000.
[89]K-M-Wong, J. F. Wu, T. N. Davidson, Q. Jin, and P. C. Ching,"Performance of wavelet packet-division multiplexing in impulsive and gaussian noise [J]," IEEE Trans. Commun, vol.48, pp.1083-1086,2000.
[901J.-J.v. d. B. S. K. W. O. Edfors, M. Sandell and P. O. Brjesson,"OFDM chan-nel estimation by singular value decomposition [J]," IEEE Trans. Commun, vol.46, pp.931-939,1998.
[91]S. M. Kay, Fundamentals of statistical signal processing:Estimation Theory [M]. Upper Saddle River, NJ:Prentice-Hall,1993.
[92]G. Auer and E. Karipidis,"Pilot aided channel estimation for ofdm:A separated approach for smoothing and interpolation [A]," in Proc. IEEE Int. Conf. Commun (ICC)[C], vol.4, pp.2173-2178,2005.
[931L. L. Scharf, Statistical signal processing:detection, estimation, and time series anal-ysis [M]. Addison-Wesley,1991.
[94]P. Stoica and A. Nehorai,"Music, maximum likelihood, and Cramer-Rao bound [J]," IEEE Trans. Acoust, Speech and Signal Process, vol.37, pp.720-741,1989.
[95]L. Deneire, P. Vandenameele, L. van der Perre, B. Gyselinckx, and M. Engels,"A low-complexity ML channel estimator for OFDM [J]," IEEE Trans. Commun, vol.51, pp.135-140,2003.
[96]E. J. Candes and T. Tao,"Stable signal recovery from incomplete and inaccurate information [J]," Commun. Pure Appl. Math, vol.59, pp.1207-1233,2005.
[97]E. J. Candes,"The restricted isometry property and its implications for compressed sensing [J]," C. R. Acad. Sci. Paris, Ser. I, vol.346, pp.589-592,2008.
[98]R. G. Baraniuk, M. Davenport, R. A. DeVore, and M. B. Wakin,"A simple proof of the restricted isometry property for random matrices [J]," Constr. Approx, vol.28, pp.253263,2008.
[99]M. A. Davenport and M. B. Wakin,"Analysis of orthogonal matching pursuit approach via restricted isometry property [J]," IEEE Trans. Inf. Theory, vol.56, pp.4395-4401,2010.
[100]D. Donoho, M. Elad, and V. Temlyakov,"Stable recovery of sparse overcomplete rep-resentations in the presence of noise [J]," IEEE Trans. Inf. Theory, vol.52, pp.6-18,2006.
[101]I. E. Nesterov, A. Nemirovskii, and Y. Nesterov,"Interior-point polynominal algorithms in convex programming [J]," Philadelphia, PA:SIAM,1994.
[102]J. A. Tropp,"Greed is good:Algorithmic results for sparse approximation [J]," IEEE Trans. Inf. Theory, vol.50, pp.2231-2242,2004.
[103]J.A. Tropp,"Computational methods for sparse solution of linear inverse problems [J]," Proc of IEEE, vol.98, pp.948-958,2010.
[104]Z. Ben-Haim and Y. C. Eldar,"The Cramer-Rao bound for estimating a sparse pa-rameter vector [J]," IEEE Trans. Signal Process, vol.58, pp.3384-3389,2010.
[105]Z. Ben-Haim, Y. C. Eldar, and M. Elad,"Coherence-based performance guarantees for estimating a sparse vector under random noise [J]," IEEE Trans. Signal Process, vol.58, pp.5030-5043,2010.
[106]D. Donoho and I. M. Johnstone,"Ideal spatial adaptation by wavelet shrinkage [J]," Biometrika, vol.81, pp.425-455,1994.
[107]D. Donoho,"De-noising by soft thresholding [J]," IEEE Trans. Inf. Theory, vol.41, pp.613-627,1995.
[108]D. Donoho, Y. Tsaig, I. Drori, and J. C. Starck,"Sparse solution of underdetermined systems of linear equations by stagewise orthogonal matching pursuit [J]," IEEE Trans. Inf. Theory, vol.58, pp.1094-1121,2012.
[109]D. Needell and R. Vershynin,"Uniform uncertainty principle and signal recovery via regularized orthogonal matching pursuit [J]," Found. Comput. Math, vol.9, pp.317-334,2009.
[110]D. Needell and J. A. Tropp,"CoSaMP:Iterative signal recovery from incomplete and inaccurate samples [J]," Appl and comput. harmo anal, vol.26, pp.301-321,2009.
[111]W. Dai and O. Milenkovic,"Subspace pursuit for compressive sensing signal recon-struction [J]," IEEE Trans. Inf. Theory, vol.55, pp.2230-2249,2009.
[112]E. J. Candes and T. Tao,"Near optimal signal recovery from random projections: Universal encoding strategies?[J]," IEEE Trans. Inform. Theory, vol.52, pp.5406-5425,2006.
[113]W. U. Bajwa, G. R. J. Haupt, and R. Nowak,"Compressed channel sensing [A]," in Proc.42nd Conf. Information Science and Systems [C], pp.5-10,2008.
[114]A. M. Abd-Elfattah, S. H. Amal, and D. M. Ziedan,"Efficiency of Bayes estimator for rayleigh distribution," Statistics on the Internet,2006.
[1151K.Peter,"Phase preserving denoising of images [A]," in Proc. Conf. Australian Pattern Recognition Society [C], pp.212-217,1999.
[116]L. Najjar,"Sparsity level-aware threshold-based channel structure detection in OFDM systems [J]," Electron. Lett, vol.48, pp.495-496,2012.
[117]Z. P. W. Yi, L. Lihua and L. Zemin,"Optimal threshold for channel estimation in MIMO-OFDM system [A]," in Proc. IEEE Int. Conf. Commun (ICC)[C], pp.4376-4380,2008.
[118]S. Sarvotham, D. Bron, and R. G. Baraniuk,"Sudocodes-fast measurement and reconstruction of sparse signals [A]," in Proc. IEEE Int. Symp. Inf. Theory [C],2006.
[119]H. Xie, G. Adrieux, Y. Wang, J. F. Diouris, and S. Feng,"A novel effective compressed sensing based sparse channel estimation in OFDM system [A]," in Proc. IEEE Int. Conf. Signal Process, Commun and Comput (ICSPCC)[C], pp.1-6,2013.
[120]K. M. Nasr, J. P. Cosmas, M. Bard, and J. Gledhill,"Performance of an echo can-celler and channel estimator for on-channel repeaters in DVB-T/H networks [J]," IEEE Trans. Broadcast, vol.53, pp.609-618,2007.
[121]D. Hu, X. Wang, and L. He,"A new sparse channel estimation and tracking method for time-varying ofdm systems [J]," IEEE Trans. Veh. Tech, vol.62, pp.4648-4653,2013.
[122]C. Heegard and S. B. Wicker, Turbo coding [M]. Kluwer Academic Publishers,1999.