基于广义逆的信号重构方法研究
|
摘要
作为把物理世界的模拟量转换为信息世界的数字量的必备手段，信号采样与重构理论及其实现技术长期以来一直是国内外学者的研究热点。随着现代无线通信技术的不断发展，信号的频率越来越高、复杂度不断增加，对采样的频率和精度的要求也越来越高。但由于受元器件技术和集成电路工艺的限制，信号的数字化获取的关键器件ADC无法满足实际需要，这大大限制了传统采样方式的应用。通常情况下，在对射频和微波等高速复杂信号数字化时，首先要采用多级复杂的降频变换，然后再进行采样和重构。这一方面增加了硬件的复杂性，另一方面混频带来的附加频率也容易引起信号失真，降低了采样精度。为了解决以上问题，多通道采样可以低速、高精度的模数转换实现高速复杂信号的采样与重构，因此，发展新的重构方法是多通道采样技术发展的趋势之一。
本文以实际工程中的信号为主要研究对象，在深入分析信号自身特点的基础上，利用多通道周期非匀采样系统对信号进行采样，根据实际需求运用广义逆实现信号的完整重构或者部分重构（镜像频率抑制）。
本文在基于广义逆的信号重构模型、重构方法、镜像频率抑制等多个方面展开研究：
1)从Shannon采样定理出发，对信号空间理论在采样信号重构系统中的作用进行深入的分析，根据周期非均匀采样需要多个采样通道的特点，利用联合子空间理论把采样和重构过程转换成矩阵向量运算，通过求解方程组就可以实现信号的完整重构或者目标信号的提取，并对系统的构成条件进行深入探讨。
2)针对周期非均匀采样的重构，使用最小二乘算法求解矩阵广义逆，为了能使重构的信号能在数字系统中应用，使用了插值滤波器实现信号的完整重构；针对系统误差，提出了一种自适应的迭代校准算法对系统感知矩阵的偏移量进行补偿。
3)针对稀疏信号等特殊信号的重构问题，对稀疏信号的定义、稀疏度的确定、频带的划分等方面进行详细的阐述；通过正交匹配追踪、最小L1范数、多测量向量等算法寻找、确定非零元素的位置参数，由此构成新的方程组通过广义逆和插值器实现稀疏信号的完整重构，并深入分析三种算法实现完整重构的条件；针对系统误差，提出了一种后端反馈式自适应误差补偿系统；通过与传统并行多通道采样方式在重构成功率、采样通道数、信噪比方面的比较可知本文使用的重构方式在对稀疏信号的采样与重构上远远优于传统方式；最后分别以带通信号和时变信号为例，进一步验证系统可以实现稀疏信号的采样与重构。
4)为了在采样过程中实现对镜像频率的抑制，提出了一种基于广义逆的镜像频率抑制系统，利用周期非均匀采样的时间延迟τ为τ=1/4fc时对镜像频率有最大抑制效果的特点，把镜像频率抑制的过程转换为矩阵向量运算，通过广义逆得到抑制后的信号；针对该方法对多带信号抑制效果不足的情况，提出了一种前向固定式误差补偿方式。与传统的镜像抑制方式进行比较表明本文提出的方法在抑制效果方面优于传统方式。
Sampling and construction theory has long been a hotspot in the field ofwireless communication. With the development of wireless technology, thisenables the modulation of narrow-band signals by high carrier frequencies.To demodulate the desired signals, the required sampling rate for the ADCcould often be too high to be attained if the Nyquist sampling theorem is tobe satisfied. Therefore, the multi-level down-conversions are used beforethe RF or microwave signals are sampled. However, this method increasesthe complexity of the hardware and leads to signal distortion. Recently, thehigh-speed, high-resolution sampling and reconstruction technology growto currently one of the hottest topics in the field of signal sampling andprocessing.
The various signals in actual project are the main object of study inthis paper. The periodic non-uniform is used to sample the signals on thebasis of analysis of the characteristics of the various signals in details.Complete reconstruction or partial reconstruction of the signal according to the actual needs by using the generalized inverse.
In this paper, the proposed sampling model based on generalizedinverse, reconstruction method, the image frequency suppression is studied.
1) From the Shannon theory, according to the feature of periodicnon-uniform sampling that it needs multiple channels，the sampling andreconstruction of signals were transformed into matrix and vectoroperations by using theory of union of subspaces, the constructed signalcan be obtained by using generalized inverse, and the condition whichconstitute the proposed system is studied. The oversampling andsubsampling in shift-invariant space are defined;
2) The least square method is used to get the generalized inverse forthe oversampling system in the chapter. The complete reconstruction ofsampled sparse signal is achieved in virtue of interpolations, which caninsure that the signals could be applied in digital system. An adaptativeiterative compensation is employed to compensate the offset of sensingmatrix. The conclusion shows the frame-work presented here is feasible.
3) The subsampling construction is proposed for sampling sparsesignals. The define of sparse signals, the determination of sparsity and division of the band is described in detail; The aim of the algorithm such asorthogonal matching pursuit, minimum L1normal, multiple measurementvectors etc. is to find the unique sparse representation of the sampled sparsesignals, which is the set of indices corresponding to the non-zero elements;the complete reconstruction of sparse signal is achieved in virtue ofinterpolations. The necessary condition of reconstruction is analyzed; aback-end feedback adaptative compensation system is proposed;multi-band signals are taken as an example to prove that the method canachieve the sampling and reconstruction of sparse signals.
4) A new method of the image frequency suppression based ongeneralized inverse is proposed in this chapter. However, the imageattenuation alone is clearly insufficient for band-pass signals. To enhancethis obtainable image attenuation an interference canceled of compensationstructure is proposed. The conclusion shows the frame-work presented hereis better than the traditional method.
本文以实际工程中的信号为主要研究对象，在深入分析信号自身特点的基础上，利用多通道周期非匀采样系统对信号进行采样，根据实际需求运用广义逆实现信号的完整重构或者部分重构（镜像频率抑制）。
本文在基于广义逆的信号重构模型、重构方法、镜像频率抑制等多个方面展开研究：
1)从Shannon采样定理出发，对信号空间理论在采样信号重构系统中的作用进行深入的分析，根据周期非均匀采样需要多个采样通道的特点，利用联合子空间理论把采样和重构过程转换成矩阵向量运算，通过求解方程组就可以实现信号的完整重构或者目标信号的提取，并对系统的构成条件进行深入探讨。
2)针对周期非均匀采样的重构，使用最小二乘算法求解矩阵广义逆，为了能使重构的信号能在数字系统中应用，使用了插值滤波器实现信号的完整重构；针对系统误差，提出了一种自适应的迭代校准算法对系统感知矩阵的偏移量进行补偿。
3)针对稀疏信号等特殊信号的重构问题，对稀疏信号的定义、稀疏度的确定、频带的划分等方面进行详细的阐述；通过正交匹配追踪、最小L1范数、多测量向量等算法寻找、确定非零元素的位置参数，由此构成新的方程组通过广义逆和插值器实现稀疏信号的完整重构，并深入分析三种算法实现完整重构的条件；针对系统误差，提出了一种后端反馈式自适应误差补偿系统；通过与传统并行多通道采样方式在重构成功率、采样通道数、信噪比方面的比较可知本文使用的重构方式在对稀疏信号的采样与重构上远远优于传统方式；最后分别以带通信号和时变信号为例，进一步验证系统可以实现稀疏信号的采样与重构。
4)为了在采样过程中实现对镜像频率的抑制，提出了一种基于广义逆的镜像频率抑制系统，利用周期非均匀采样的时间延迟τ为τ=1/4fc时对镜像频率有最大抑制效果的特点，把镜像频率抑制的过程转换为矩阵向量运算，通过广义逆得到抑制后的信号；针对该方法对多带信号抑制效果不足的情况，提出了一种前向固定式误差补偿方式。与传统的镜像抑制方式进行比较表明本文提出的方法在抑制效果方面优于传统方式。
Sampling and construction theory has long been a hotspot in the field ofwireless communication. With the development of wireless technology, thisenables the modulation of narrow-band signals by high carrier frequencies.To demodulate the desired signals, the required sampling rate for the ADCcould often be too high to be attained if the Nyquist sampling theorem is tobe satisfied. Therefore, the multi-level down-conversions are used beforethe RF or microwave signals are sampled. However, this method increasesthe complexity of the hardware and leads to signal distortion. Recently, thehigh-speed, high-resolution sampling and reconstruction technology growto currently one of the hottest topics in the field of signal sampling andprocessing.
The various signals in actual project are the main object of study inthis paper. The periodic non-uniform is used to sample the signals on thebasis of analysis of the characteristics of the various signals in details.Complete reconstruction or partial reconstruction of the signal according to the actual needs by using the generalized inverse.
In this paper, the proposed sampling model based on generalizedinverse, reconstruction method, the image frequency suppression is studied.
1) From the Shannon theory, according to the feature of periodicnon-uniform sampling that it needs multiple channels，the sampling andreconstruction of signals were transformed into matrix and vectoroperations by using theory of union of subspaces, the constructed signalcan be obtained by using generalized inverse, and the condition whichconstitute the proposed system is studied. The oversampling andsubsampling in shift-invariant space are defined;
2) The least square method is used to get the generalized inverse forthe oversampling system in the chapter. The complete reconstruction ofsampled sparse signal is achieved in virtue of interpolations, which caninsure that the signals could be applied in digital system. An adaptativeiterative compensation is employed to compensate the offset of sensingmatrix. The conclusion shows the frame-work presented here is feasible.
3) The subsampling construction is proposed for sampling sparsesignals. The define of sparse signals, the determination of sparsity and division of the band is described in detail; The aim of the algorithm such asorthogonal matching pursuit, minimum L1normal, multiple measurementvectors etc. is to find the unique sparse representation of the sampled sparsesignals, which is the set of indices corresponding to the non-zero elements;the complete reconstruction of sparse signal is achieved in virtue ofinterpolations. The necessary condition of reconstruction is analyzed; aback-end feedback adaptative compensation system is proposed;multi-band signals are taken as an example to prove that the method canachieve the sampling and reconstruction of sparse signals.
4) A new method of the image frequency suppression based ongeneralized inverse is proposed in this chapter. However, the imageattenuation alone is clearly insufficient for band-pass signals. To enhancethis obtainable image attenuation an interference canceled of compensationstructure is proposed. The conclusion shows the frame-work presented hereis better than the traditional method.
引文
[1]曾浩.宽带无缝采集技术以及在示波器中的应用[D].成都：电子科技大学,2010：1-10
[2] Jaesik Lee et al. A5-b10-GSample/s A/D converter for10-Gb/s opticalreceivers[J]. IEEE Journalof Solid-State Circuits,2004,39(10):1671-1679
[3] Y. Akazawa et al. A400MSPS8-b flash AD conversion LSI[C].1987Solid-State Circuits Conference, Dig.Tech. Pap,98-99
[4] David W.Cline. Noise, Speed, and Power Tradeoffs in Pipeline Analogto Digital Converter[C]. Ph.D.thesis,University of CaliforniaBerkeley,Nov1995
[5] B.Provost and E.Sanchez-Sinencio.A practical self-calibration schemeimplementation for pipeline ADC[J]. IEEE Trans.On Instrument andMeasurement,2004,53(2):448-456
[6]王涛，朱臻，易婷等.一种全差分双通道采样保持的流水线操作AD变换器[J].固体电子学研究与进展，22(1),2002：53-56。
[7] H.Neubauer,T.Desel,and H.Hauer.A successive approximation A/Dconverter with16-bit200ks/s in0.6μm CMOS using self calibrationand low power techiniques[C].2001International Conference onElectronics,Circuits and Systems:859-862
[8] Analog Device Inc.,Analog Device Product AD7655Datasheet,available online at http://ww.analog.com
[9] R. J. Baker, H. W. Li and D. F. Boyce. CMOS Circuit Design, Layout,and Simulation[C]. JOHN WILEY&SONS, INC., ISBN0-7803-3416-7,1997
[10] G. Promitzer,12-bit Low-Power Fully Differential SwitchedCapacitor Noncalibrating Successive Approximation ADC with1Ms/s[J]. IEEE Journal of Solid-State Circuits,2001,36(6):432-437
[11] B.H.Leung and S.Sutarja. Multi-bitΣ-△A/D ConverterIncorporating A Novel Class of Dynamic Element Matching[J].IEEETrans.Circuits Syst.II,1992,39(1):35-51
[12] Y.Geerts, M. S. J. Steyaert, and W.Sansen. A high-performancemulti-bit sigma-delta CMOS ADC[J]. IEEE Journal of Solid-StateCircuits,2000,35(12):1829-1840
[13] Gruget, A., Roger, M. Wide-band multipath A to D converter forcognitive radio applications[C].2010IEEE International MicrowaveWorkshop Series on RF Front-ends for Software Defined and CognitiveRadio Solutions:1-4
[14] Lelandais-Perrault, C., Petrescu, T. Wideband, Bandpass, andVersatile Hybrid Filter Bank A/D Conversion for Software Radio[J].IEEE Transactions on Circuits and Systems I:Regular Papers,2009,56(8):1772-1782
[15] Petraglia A.and Mitra S.K.High-speed A/D conversionincorporating a QMF bank[J].IEEETrans.Instrum.Meas.,1992,41(3):427-431
[16] Vaidyanathan P.P. Multirate Systems and Filter Banks[C].Englewood Cliffs, NJ,Prentice-Hall,1993
[17] J. M. Ingino and B. A. Wooley, A Continuously Calibrated12-b,10-Ms/s,3.3-V A/D Converter[J]. IEEE Journal of Solid-State Circuits,1998,33(12):1920-1931
[18] J.Riikonen. et. al. A10-bit400-MS/s170mw4-times interleavedA/D converter in0.35-μm BiCMOS[C].2005IEEE InternationalSymposium on Circuits and Systems,Vol.5:4622-4625
[19] Y.-P. Lin and P. P. Vaidyanathan, Periodically nonuniformsampling of bandpass signals[J]. IEEE Trans. Circuits Syst. II,1998.45(3):340–351
[20] Chen Meng and Jamal Tuqan. Generalized periodic non-uniformsampling of non-bandlimited signals, Information Theory andApplications Workshop[C]. San Diego, CA, USA,2008,417–421
[21] Ha T. Nguyen and Minh N. Do. Signal reconstruction from aperiodic nonuniform set of samples using H∞optimization[J]. IEEETrans. Signal Proc.,2005,53(5):1731–1738
[22]文静,文玉梅,李平.周期非均匀采样带通信号的采样参数[J].数据采样与处理,2006,21(1):108-112.
[23]文静,文玉梅,李平.周期非均匀采样信号的重建中的优化子频带划分[J].信号处理,2006,22(4):568-572.
[24] Y. Lu and M. N. Do. A geometrical approach to sampling signalswith finite rate of innovation[C]. In IEEE Int.Conf. Acoustic, Speechand Signal Proc., Montreal, Canada, May2004:40-43
[25] C. E. Shannon. Communication in the presence of noise[J]. Proc.IRE,1949,37:10-21
[26] M. Unser. Sampling-50years after Shannon[J]. IEEE Proc.,2000,88(4):569-587
[27] Black W.C., Hodges D.A. Time interleaved converter arrays[J].IEEE Journal of Solid-State Circuits,1980,15(6):1022–1029
[28] C Schiller, P Byrne. A4GHz8-bit ADC system[J]. IEEE Journal ofSolid-state Circuit,1992,12(11):1781-1789
[29] M T McTigue, P J Byrne. An8-Giga sample-per-second,8-bit dataacquisition system for a sampling digital oscilloscope[J]. HP Journal,1993,10:11-23
[30]王永刚,王砚方.低成本100MSPS数据采样系统研制[J].核电子学与探铡技术,1996,15(3):194-196
[31]张昊，师奕兵，王志刚．时间交替ADC系统通道时钟延迟的多频正弦拟合算法[J]．电子测量与仪器学报,2010，24(3):244-249
[32]邓琳,吕幼新,王洪.并行采样系统通道失配误差测量及校正[J].电子科技大学学报,2006,35(3):313-316
[33]刘进军，吕幼新，王洪.并行ADC采样系统的时间误差测量与校正[J].电子科技大学学报,2005,34(6):736-738
[34]张清洪，吕幼新,王洪，刘霖.多片A DC并行采样系统的误差时域测量与校正[J].电讯技术,2005,3:189-193
[35]林茂六,吴芝路,王金龙,任广辉.高速AD C及由其构成的并行／交替式数据采样系统非线性研究[J].电子学报,2000,2:72-74
[36] W. Namgoong. Finite-length synthesis filters for non-uniformlytime-interleaved analog-to-digital converter[C].2002IEEEInternational Symposium on Circuits and Systems,4:815-818
[37]刘治宇,林茂六,许洪光.一种基于混合滤波器组的高速高分辨率ADC系统研究[J].电子学报,2003,9:1404-1406
[38] A. Kohlenber. Exact interpolation of band-limited functions[J].J.Appl. Phys,1953.24(12):1432-1436
[39] H S Black．Modulation Theory．New York：D Van NostrandCompany，Inc，1953
[40] Tao R, Li Z B, Wang Y. Spectral analysis and reconstruction forperiodic nonuniformly sampled signals in fractional Fourier domain[J].IEEE Trans Signal Proc.2007,55(7):3541—3547
[41] C. Herley and P. W. Wong. Minimum rate sampling of signals witharbitrary frequency support[C]. IEEE Int. Conf. Image Processing.Lausanne, Switzerland,1996
[42] C. Meng and J. Tuqan, Difference sampling theorems for a class ofnon-bandlimited signals[J]. in Proc. ISIT,2006:123–127
[43] J. Tuqan, Periodic non uniform sampling of non ban-dlimitedsignals[J]. in Proc. ICASSP,2006:384–387.
[44] Ha T. Nguyen and Minh N. Do. Signal reconstruction from aperiodic nonuniform set of samples using H∞optimization[J]. IEEETrans. Signal Proc.,2005,53(5):1731–1738
[45] I. Maravic, M. Vetterli and K. Ramchandran. Channel estimationand synchronization with sub-Nyquist sampling and application toUWB systems[C]. in Proc. IEEE ISCAS,2004,5:381-384
[46]张峰,陶然,王越.分数阶Fourier域带限信号多通道采样定理[J].中国科学E辑:技术科学,2008,38(11):1874-1885
[47]李炳照，陶然，王越.分数阶Fourier域上非均匀采样信号的频谱重构研究[J].电子学报,2008,36(6):122-125
[48] M.T.Figus. high-speed AD conversion incorporating[J]. IEEEJournal of Solid-State Circuit,1992,41(3):427-431
[49] S. Kirolos, J. Laska, R. G. Baraniuk. Analog-to-informationconversion via random demodulation[J]. IEEE Dallas CircuitsandSystems Workshop,2006:21-25
[50] J. N. Laska, S. Kirolos, R. G. Baraniuk. Theory and implementationof an analog-to-information converter using random demodulation[J].IEEE Int. Symp.on Circuits and Systems (ISCAS).2007,5:1959–1962
[51] T. Ragheb, S. Kirolos. Implementation models foranalog-to-information conversion via random sampling[J]. MidwestSymposium on Circuits and Systems,2007,8:102-108
[52] D. L. Donoho. Compressed sensing[J]. IEEE Trans. on Inf. Theory,2006,52(4):1289–1306
[53] Emamnuel J. Candès. Compressive sampling[C]. The InternationalCongress of Mathematicians, Madrid, Spain,2006,1-20
[54] Emmanuel J. Candès. An introduction to compressive sampling[J].IEEE Signal Processing Magazine,2008,25(3):21-30
[55] L Rebollo-Neira, D. Lowe, Optimized orthogonal matching pursuitapproach[J]. IEEE Signal Processing Letters,2002,9(4):137-140
[56] Shane F. Cotter. Sparse Solutions to Linear Inverse Problems withMultiple Measurement Vectors[J]. IEEE Tramscations on SignalProcessing,2005,53(7):2477–2489
[57] D. L. Donoho and M. Elad, Maximal sparsity representation via L1minimization[J]. Proc. Natl. Acad. Sci.,2003,100(3):2197–2202
[58] J. Chen and X. Huo, Theoretical results on sparse representations ofmultiple-measurement vectors[J]. IEEE Trans. Signal Processing,2006,54(12):4634-4643
[59] R. Venkataramani and Y. Bresler, Further results on spectrum blindsampling of2d signals[C]. Proc. IEEE Int. Conf. on Image Proc.,1998,2:752–756
[60]刘巍,赵春晖.基于2D-DCT等效矩阵的压缩成像[J].仪器仪表学报，2010,31(7):1576-1582
[61]练秋生高彦严陈书贞.基于两步迭代收缩法和复数小波的压缩传感图像重构[J].仪器仪表学报,2009,31(7):1426-1431
[62] Sebastian Hoyos, Brian M. Sadler. Ultra-wideband analog-to-digitalconversion via signal expansion[J]. IEEE Transactions on VehicularTechnology.2005,54(5):1609-1622
[63] M. Mishali and Y. C. Eldar, Blind multiband signal reconstruction:Compressed sensing for analog signals[J]. IEEE Trans. SignalProcess.,2009,57(3):993–1009
[64] Stephen A. Jantzi. Quadrature bandpass delta-sigma modulation fordigital radio[J]. IEEE journal of solid-state circuit,1997,32(12):1935-1950
[65] Upisa Lerstaveesin, A Complex Image Rejection Circuit With SignDetection Only[J]. IEEE journal of solid-state circuits,2006,41(12):2693-2702
[66]向新.软件无线电原理与技术[M].西安:西安电子科技大学出版社,2008:50-62
[67] T. Hentschel, M. Henker, and G. Fettweis. The digital front-end ofsoftware radio terminals[J]. IEEE Personal Communications.1999,6(4):40-46
[68] A. Haghighat. A review on essentials and technical challenges ofsoftware defined radio[C]. Proc. IEEE Conf. Military Communication,2002,10(1):377-382
[69] Ching-Hsiang Tseng and Sun-Chung Chou. Direct Downconversionof Multiband RF Signals Using Bandpass Sampling[J]. IEEEtransaction on wireless communications,2006,5(1):72-76
[70] Ching-Hsiang Tseng. A Universal Formula for the CompleteBandpass Sampling Requirements of Nonlinear Systems[J]. IEEEtransaction on signal processing,2009,57(10):3869-3879
[71] Junghwa Bae and Jinwoo Park. An Efficient Algorithm forBand-pass Sampling of Multiple RF Signals[J]. IEEE signal processingletters,2006,13(4):193-196
[72] Shuyang Yu and Xiangyang Wang. Bandpass Sampling of One RFSignal over Multiple RF Signals with Contiguous Spectrums[J]. IEEEsignal processing letters,2009,16(1):14-17
[73] Mohamed Béchir DADI and Ridha BOUALLEGUE. Non-UniformSampling Schemes for RF Bandpass Sampling Receivers[C].5thInternational Conference: Sciences of Electronic Technologies ofInformation and Telecommunications, TUNISIA,2009:1-6
[74] P L Butzer. A survey of the Whittaker-Shannon sampling theoremand some of its extensions[J]. Journal of Mathematical Research andExposition,1983,3(1):185-212
[75] A. Papoulis. Generalized sampling expansion[J]. IEEETrans.Circuits Syst.1977,24:652-654
[76] G. G. Walter. A sampling theorem for wavelet subspaces[J]. IEEETrans. Inf. Theory.1992,38:881–884
[77] Y. C. Eldar, M. Unser. Non-ideal sampling and interpolation fromnoisy observations in shift-invariant spaces[J]. IEEE Trans. SignalProcess,2006,54(7):2636–2651
[78] Yue Ming LU. A theory for sampling from a union of subspace[J].IEEE Transactions on Signal Processing,2008,56(6):2334-2345
[79]孙文昌,周性伟.向量小波子空间上的采样定理[J].科学通报,1999,44(3):262-265
[80]杨守志,程正兴.杨建伟r重小波子空间上的Shannon型均匀和非均匀采样定理[J].工程数学学报,2003,5,20(2):1-6
[81]卜长江,罗跃生.矩阵论[M].哈尔滨:哈尔滨工程大学出版社,2007:1-20
[82]李红燕,樊庆瑞.一类奇异矩阵的广义逆证明[J].上海工程技术大学学报,2010,8(4):22-27
[83]曹卫平,马吉溥.闭算子的M-P广义逆[J].淮海工学院学报,1997,6(3):1-6
[84]邓斌.外区域上的Dirichlet-neumam算子对角化和广义逆的正则逆表示[D].华东师范大学,2008:1-20
[85]罗浚溢,田书林,王志刚等.基于正交匹配追踪的周期非均匀采样研究[J].仪器仪表学报,2010,31(12):2661-2667
[86] P. L. Butzer. A survey of the Whittaker–Shannon sampling thetheorem and some of its extensions[J]. J. Math. Res. Exposit,1983,3:185-212.
[87] V. A. Kotel’nikov. On the transmission capacity of "ether" and wirein electrocommunications[J]. Izd. Red. Upr. Svyazzi RKKA (Moscow),1932.
[88] Richard Baraniuk. A Lecture on Compressive Sensing[J]. IEEESignal Processing Magazine,2007,6:1-7
[89] G.Golub. An analysis of the total least squares problem. Society forIndustrial and Applied[J]. Mathematics Journal on Numerical Analysis,1980,17:883-893
[90] Stephen A. Jantzi. Kenneth W. Martin. Quadrature Bandpass ΔΣModulation for Digital Radio[J]. IEEE Journal of solid-statecircuits,1997,32(12):1935-1951
[91] W. Feixue, Y. Shaowei, and G. Guirong. Mixer-free digitalquadrature demodulation based on second-order sampling[J], Electron.Lett.,1998,34:854-856
[92]罗浚溢，田书林，王志刚.二阶交替采样实现射频信号下变频[C].2009信息技术与应用会议,北京，2009:56-62
[2] Jaesik Lee et al. A5-b10-GSample/s A/D converter for10-Gb/s opticalreceivers[J]. IEEE Journalof Solid-State Circuits,2004,39(10):1671-1679
[3] Y. Akazawa et al. A400MSPS8-b flash AD conversion LSI[C].1987Solid-State Circuits Conference, Dig.Tech. Pap,98-99
[4] David W.Cline. Noise, Speed, and Power Tradeoffs in Pipeline Analogto Digital Converter[C]. Ph.D.thesis,University of CaliforniaBerkeley,Nov1995
[5] B.Provost and E.Sanchez-Sinencio.A practical self-calibration schemeimplementation for pipeline ADC[J]. IEEE Trans.On Instrument andMeasurement,2004,53(2):448-456
[6]王涛，朱臻，易婷等.一种全差分双通道采样保持的流水线操作AD变换器[J].固体电子学研究与进展，22(1),2002：53-56。
[7] H.Neubauer,T.Desel,and H.Hauer.A successive approximation A/Dconverter with16-bit200ks/s in0.6μm CMOS using self calibrationand low power techiniques[C].2001International Conference onElectronics,Circuits and Systems:859-862
[8] Analog Device Inc.,Analog Device Product AD7655Datasheet,available online at http://ww.analog.com
[9] R. J. Baker, H. W. Li and D. F. Boyce. CMOS Circuit Design, Layout,and Simulation[C]. JOHN WILEY&SONS, INC., ISBN0-7803-3416-7,1997
[10] G. Promitzer,12-bit Low-Power Fully Differential SwitchedCapacitor Noncalibrating Successive Approximation ADC with1Ms/s[J]. IEEE Journal of Solid-State Circuits,2001,36(6):432-437
[11] B.H.Leung and S.Sutarja. Multi-bitΣ-△A/D ConverterIncorporating A Novel Class of Dynamic Element Matching[J].IEEETrans.Circuits Syst.II,1992,39(1):35-51
[12] Y.Geerts, M. S. J. Steyaert, and W.Sansen. A high-performancemulti-bit sigma-delta CMOS ADC[J]. IEEE Journal of Solid-StateCircuits,2000,35(12):1829-1840
[13] Gruget, A., Roger, M. Wide-band multipath A to D converter forcognitive radio applications[C].2010IEEE International MicrowaveWorkshop Series on RF Front-ends for Software Defined and CognitiveRadio Solutions:1-4
[14] Lelandais-Perrault, C., Petrescu, T. Wideband, Bandpass, andVersatile Hybrid Filter Bank A/D Conversion for Software Radio[J].IEEE Transactions on Circuits and Systems I:Regular Papers,2009,56(8):1772-1782
[15] Petraglia A.and Mitra S.K.High-speed A/D conversionincorporating a QMF bank[J].IEEETrans.Instrum.Meas.,1992,41(3):427-431
[16] Vaidyanathan P.P. Multirate Systems and Filter Banks[C].Englewood Cliffs, NJ,Prentice-Hall,1993
[17] J. M. Ingino and B. A. Wooley, A Continuously Calibrated12-b,10-Ms/s,3.3-V A/D Converter[J]. IEEE Journal of Solid-State Circuits,1998,33(12):1920-1931
[18] J.Riikonen. et. al. A10-bit400-MS/s170mw4-times interleavedA/D converter in0.35-μm BiCMOS[C].2005IEEE InternationalSymposium on Circuits and Systems,Vol.5:4622-4625
[19] Y.-P. Lin and P. P. Vaidyanathan, Periodically nonuniformsampling of bandpass signals[J]. IEEE Trans. Circuits Syst. II,1998.45(3):340–351
[20] Chen Meng and Jamal Tuqan. Generalized periodic non-uniformsampling of non-bandlimited signals, Information Theory andApplications Workshop[C]. San Diego, CA, USA,2008,417–421
[21] Ha T. Nguyen and Minh N. Do. Signal reconstruction from aperiodic nonuniform set of samples using H∞optimization[J]. IEEETrans. Signal Proc.,2005,53(5):1731–1738
[22]文静,文玉梅,李平.周期非均匀采样带通信号的采样参数[J].数据采样与处理,2006,21(1):108-112.
[23]文静,文玉梅,李平.周期非均匀采样信号的重建中的优化子频带划分[J].信号处理,2006,22(4):568-572.
[24] Y. Lu and M. N. Do. A geometrical approach to sampling signalswith finite rate of innovation[C]. In IEEE Int.Conf. Acoustic, Speechand Signal Proc., Montreal, Canada, May2004:40-43
[25] C. E. Shannon. Communication in the presence of noise[J]. Proc.IRE,1949,37:10-21
[26] M. Unser. Sampling-50years after Shannon[J]. IEEE Proc.,2000,88(4):569-587
[27] Black W.C., Hodges D.A. Time interleaved converter arrays[J].IEEE Journal of Solid-State Circuits,1980,15(6):1022–1029
[28] C Schiller, P Byrne. A4GHz8-bit ADC system[J]. IEEE Journal ofSolid-state Circuit,1992,12(11):1781-1789
[29] M T McTigue, P J Byrne. An8-Giga sample-per-second,8-bit dataacquisition system for a sampling digital oscilloscope[J]. HP Journal,1993,10:11-23
[30]王永刚,王砚方.低成本100MSPS数据采样系统研制[J].核电子学与探铡技术,1996,15(3):194-196
[31]张昊，师奕兵，王志刚．时间交替ADC系统通道时钟延迟的多频正弦拟合算法[J]．电子测量与仪器学报,2010，24(3):244-249
[32]邓琳,吕幼新,王洪.并行采样系统通道失配误差测量及校正[J].电子科技大学学报,2006,35(3):313-316
[33]刘进军，吕幼新，王洪.并行ADC采样系统的时间误差测量与校正[J].电子科技大学学报,2005,34(6):736-738
[34]张清洪，吕幼新,王洪，刘霖.多片A DC并行采样系统的误差时域测量与校正[J].电讯技术,2005,3:189-193
[35]林茂六,吴芝路,王金龙,任广辉.高速AD C及由其构成的并行／交替式数据采样系统非线性研究[J].电子学报,2000,2:72-74
[36] W. Namgoong. Finite-length synthesis filters for non-uniformlytime-interleaved analog-to-digital converter[C].2002IEEEInternational Symposium on Circuits and Systems,4:815-818
[37]刘治宇,林茂六,许洪光.一种基于混合滤波器组的高速高分辨率ADC系统研究[J].电子学报,2003,9:1404-1406
[38] A. Kohlenber. Exact interpolation of band-limited functions[J].J.Appl. Phys,1953.24(12):1432-1436
[39] H S Black．Modulation Theory．New York：D Van NostrandCompany，Inc，1953
[40] Tao R, Li Z B, Wang Y. Spectral analysis and reconstruction forperiodic nonuniformly sampled signals in fractional Fourier domain[J].IEEE Trans Signal Proc.2007,55(7):3541—3547
[41] C. Herley and P. W. Wong. Minimum rate sampling of signals witharbitrary frequency support[C]. IEEE Int. Conf. Image Processing.Lausanne, Switzerland,1996
[42] C. Meng and J. Tuqan, Difference sampling theorems for a class ofnon-bandlimited signals[J]. in Proc. ISIT,2006:123–127
[43] J. Tuqan, Periodic non uniform sampling of non ban-dlimitedsignals[J]. in Proc. ICASSP,2006:384–387.
[44] Ha T. Nguyen and Minh N. Do. Signal reconstruction from aperiodic nonuniform set of samples using H∞optimization[J]. IEEETrans. Signal Proc.,2005,53(5):1731–1738
[45] I. Maravic, M. Vetterli and K. Ramchandran. Channel estimationand synchronization with sub-Nyquist sampling and application toUWB systems[C]. in Proc. IEEE ISCAS,2004,5:381-384
[46]张峰,陶然,王越.分数阶Fourier域带限信号多通道采样定理[J].中国科学E辑:技术科学,2008,38(11):1874-1885
[47]李炳照，陶然，王越.分数阶Fourier域上非均匀采样信号的频谱重构研究[J].电子学报,2008,36(6):122-125
[48] M.T.Figus. high-speed AD conversion incorporating[J]. IEEEJournal of Solid-State Circuit,1992,41(3):427-431
[49] S. Kirolos, J. Laska, R. G. Baraniuk. Analog-to-informationconversion via random demodulation[J]. IEEE Dallas CircuitsandSystems Workshop,2006:21-25
[50] J. N. Laska, S. Kirolos, R. G. Baraniuk. Theory and implementationof an analog-to-information converter using random demodulation[J].IEEE Int. Symp.on Circuits and Systems (ISCAS).2007,5:1959–1962
[51] T. Ragheb, S. Kirolos. Implementation models foranalog-to-information conversion via random sampling[J]. MidwestSymposium on Circuits and Systems,2007,8:102-108
[52] D. L. Donoho. Compressed sensing[J]. IEEE Trans. on Inf. Theory,2006,52(4):1289–1306
[53] Emamnuel J. Candès. Compressive sampling[C]. The InternationalCongress of Mathematicians, Madrid, Spain,2006,1-20
[54] Emmanuel J. Candès. An introduction to compressive sampling[J].IEEE Signal Processing Magazine,2008,25(3):21-30
[55] L Rebollo-Neira, D. Lowe, Optimized orthogonal matching pursuitapproach[J]. IEEE Signal Processing Letters,2002,9(4):137-140
[56] Shane F. Cotter. Sparse Solutions to Linear Inverse Problems withMultiple Measurement Vectors[J]. IEEE Tramscations on SignalProcessing,2005,53(7):2477–2489
[57] D. L. Donoho and M. Elad, Maximal sparsity representation via L1minimization[J]. Proc. Natl. Acad. Sci.,2003,100(3):2197–2202
[58] J. Chen and X. Huo, Theoretical results on sparse representations ofmultiple-measurement vectors[J]. IEEE Trans. Signal Processing,2006,54(12):4634-4643
[59] R. Venkataramani and Y. Bresler, Further results on spectrum blindsampling of2d signals[C]. Proc. IEEE Int. Conf. on Image Proc.,1998,2:752–756
[60]刘巍,赵春晖.基于2D-DCT等效矩阵的压缩成像[J].仪器仪表学报，2010,31(7):1576-1582
[61]练秋生高彦严陈书贞.基于两步迭代收缩法和复数小波的压缩传感图像重构[J].仪器仪表学报,2009,31(7):1426-1431
[62] Sebastian Hoyos, Brian M. Sadler. Ultra-wideband analog-to-digitalconversion via signal expansion[J]. IEEE Transactions on VehicularTechnology.2005,54(5):1609-1622
[63] M. Mishali and Y. C. Eldar, Blind multiband signal reconstruction:Compressed sensing for analog signals[J]. IEEE Trans. SignalProcess.,2009,57(3):993–1009
[64] Stephen A. Jantzi. Quadrature bandpass delta-sigma modulation fordigital radio[J]. IEEE journal of solid-state circuit,1997,32(12):1935-1950
[65] Upisa Lerstaveesin, A Complex Image Rejection Circuit With SignDetection Only[J]. IEEE journal of solid-state circuits,2006,41(12):2693-2702
[66]向新.软件无线电原理与技术[M].西安:西安电子科技大学出版社,2008:50-62
[67] T. Hentschel, M. Henker, and G. Fettweis. The digital front-end ofsoftware radio terminals[J]. IEEE Personal Communications.1999,6(4):40-46
[68] A. Haghighat. A review on essentials and technical challenges ofsoftware defined radio[C]. Proc. IEEE Conf. Military Communication,2002,10(1):377-382
[69] Ching-Hsiang Tseng and Sun-Chung Chou. Direct Downconversionof Multiband RF Signals Using Bandpass Sampling[J]. IEEEtransaction on wireless communications,2006,5(1):72-76
[70] Ching-Hsiang Tseng. A Universal Formula for the CompleteBandpass Sampling Requirements of Nonlinear Systems[J]. IEEEtransaction on signal processing,2009,57(10):3869-3879
[71] Junghwa Bae and Jinwoo Park. An Efficient Algorithm forBand-pass Sampling of Multiple RF Signals[J]. IEEE signal processingletters,2006,13(4):193-196
[72] Shuyang Yu and Xiangyang Wang. Bandpass Sampling of One RFSignal over Multiple RF Signals with Contiguous Spectrums[J]. IEEEsignal processing letters,2009,16(1):14-17
[73] Mohamed Béchir DADI and Ridha BOUALLEGUE. Non-UniformSampling Schemes for RF Bandpass Sampling Receivers[C].5thInternational Conference: Sciences of Electronic Technologies ofInformation and Telecommunications, TUNISIA,2009:1-6
[74] P L Butzer. A survey of the Whittaker-Shannon sampling theoremand some of its extensions[J]. Journal of Mathematical Research andExposition,1983,3(1):185-212
[75] A. Papoulis. Generalized sampling expansion[J]. IEEETrans.Circuits Syst.1977,24:652-654
[76] G. G. Walter. A sampling theorem for wavelet subspaces[J]. IEEETrans. Inf. Theory.1992,38:881–884
[77] Y. C. Eldar, M. Unser. Non-ideal sampling and interpolation fromnoisy observations in shift-invariant spaces[J]. IEEE Trans. SignalProcess,2006,54(7):2636–2651
[78] Yue Ming LU. A theory for sampling from a union of subspace[J].IEEE Transactions on Signal Processing,2008,56(6):2334-2345
[79]孙文昌,周性伟.向量小波子空间上的采样定理[J].科学通报,1999,44(3):262-265
[80]杨守志,程正兴.杨建伟r重小波子空间上的Shannon型均匀和非均匀采样定理[J].工程数学学报,2003,5,20(2):1-6
[81]卜长江,罗跃生.矩阵论[M].哈尔滨:哈尔滨工程大学出版社,2007:1-20
[82]李红燕,樊庆瑞.一类奇异矩阵的广义逆证明[J].上海工程技术大学学报,2010,8(4):22-27
[83]曹卫平,马吉溥.闭算子的M-P广义逆[J].淮海工学院学报,1997,6(3):1-6
[84]邓斌.外区域上的Dirichlet-neumam算子对角化和广义逆的正则逆表示[D].华东师范大学,2008:1-20
[85]罗浚溢,田书林,王志刚等.基于正交匹配追踪的周期非均匀采样研究[J].仪器仪表学报,2010,31(12):2661-2667
[86] P. L. Butzer. A survey of the Whittaker–Shannon sampling thetheorem and some of its extensions[J]. J. Math. Res. Exposit,1983,3:185-212.
[87] V. A. Kotel’nikov. On the transmission capacity of "ether" and wirein electrocommunications[J]. Izd. Red. Upr. Svyazzi RKKA (Moscow),1932.
[88] Richard Baraniuk. A Lecture on Compressive Sensing[J]. IEEESignal Processing Magazine,2007,6:1-7
[89] G.Golub. An analysis of the total least squares problem. Society forIndustrial and Applied[J]. Mathematics Journal on Numerical Analysis,1980,17:883-893
[90] Stephen A. Jantzi. Kenneth W. Martin. Quadrature Bandpass ΔΣModulation for Digital Radio[J]. IEEE Journal of solid-statecircuits,1997,32(12):1935-1951
[91] W. Feixue, Y. Shaowei, and G. Guirong. Mixer-free digitalquadrature demodulation based on second-order sampling[J], Electron.Lett.,1998,34:854-856
[92]罗浚溢，田书林，王志刚.二阶交替采样实现射频信号下变频[C].2009信息技术与应用会议,北京，2009:56-62