压缩传感理论及其在宽带阵列信号参数估计中的应用
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摘要
信号处理的一个重要分支是阵列信号处理,其在通信、雷达、声纳、射电天文等领域有着广泛的应用。阵列信号处理解决的关键问题之一是信号的参数估计,如备受关注的信号波达方向(DOA)估计。如今通信技术日新月异的发展,扩频信号、跳频信号、线性调频(LFM)信号等宽带信号在通信系统中应用越来越广泛,另外,许多信号,如自然界中的地震波等信号本质上就是宽带信号。目前,对于宽带信号波达方向(DOA)估计问题,广泛使用时频分析方法,但此类方法存在时频点选取问题,另外,使用传统乃奎斯特采样,也存在大量数据处理的问题。因此,对宽带信号的阵列信号处理方法的研究显得尤为重要。
继信号的稀疏分解,06年出现了压缩传感这个新的理论,打破了乃奎斯特采样模式,只要信号在某个域具有稀疏性,便可以远少于乃奎斯特采样的采样点恢复重建原始信号。本文对压缩传感理论进行了研究,并将其引入到宽带阵列信号参数估计中来,研究了基于压缩传感理论的宽带LFM信号的频率以及DOA参数估计的方法。主要的工作和创新有：
1.研究了压缩传感理论：基本理论原理及关键,从观测矩阵、信号的稀疏表示以及恢复算法几个方面做了研究。
2.研究了几种不同的基于压缩传感的宽带LFM信号频率和DOA参数的估计算法。由于要估计的目标参数在参数的空间具有稀疏性,如单个信源的DOA,则目标为在角度空间寻找符合目标方位的一个角度,因此在角度域中,目标方位角具有稀疏性,则基于压缩传感实现参数的估计是可能的。根据压缩传感理论的关键,分别构造观测矩阵、信号稀疏表示的过完备原子库,选择恢复算法,最后实现宽带LFM信号频率和DOA参数的估计。计算机仿真实验表明,在低采样情况下,本文算法分辨率优于经典的宽带估计方法(如时频子空间法)。
3.基于压缩传感,实现了单信源和多信源情况下的宽带LFM信号到达角估计。对于多个信源情况下的DOA估计,当各个信源不相关或相关性很小时,仍然可以利用压缩的方法,在低采样率下得到较好的估计效果。
将压缩传感理论引入到宽带阵列信号参数估计中,避免了时频类方法估计性能受时频点选取及交叉项影响的问题,且在低采样情况下可获得好的估计效果,减轻了数据存储与传输的负担。
Array signal processing is one of the important parts of signal analysis and processing, which is widely used in communication, radar, sonar, chronometer and other areas of science and technology. One of the key problems to be solved array signals processing is parameter estimation, like the most attentioned problem:direction of arrival (DOA) of signal. With the fast development of communication technologies, many wide-band signals, such as spread-spectrum signal, frequency-hopping signal and linear frequency modulation (LFM) signal, are used more and more widely in communication systems. In addition, many signals, such as seismology wave is a kind of wide-band signal in nature. Currently, time-frequency method is widely used for DOA of wide-band array signals, but some disadvantages like selection of the correct time-frequency points exist in these methods. In addition, large-scale data processing is also a problem cause by tranditonal Nyquist sampling. So the research on wide-band array signal processing method becomes more and more vital.
After the sparse decomposition of signals, a new theory called compressed sensing was present on 2006, which breaks the classical signal sampling rule and with many new advantages. If the signal is sparse on some domain, it's possible to be reconstructed based on far fewer random samples than required by tranditional Nyquist sampling rate. Based on the reaserches of compressed sensing, this paper applies it to wideband array signal parameters estimation problems, and presents a new way to frequency and DOA estimation of wideband LFM signals. The main work and contributions of the thesis as follows:
1. Theory of compressed sensing was studied totally, from basic theory principle to the key points of compressed sensing, include:measurement matrix, sparse representation of signal and signal recovery algorithm.
2. Several new methods based on compressed sensing are researched to estimate the frequency and DOA parameters of wideband LFM signals in array signal processing. Because the target parameter is sparse in the space of parameters, for example, single DOA of signal, our target is to find the DOA of signal source, so the only one direction is sparse in angle space, so it's possible to find the DOA by compressed sensing method. According to realization of compressed sensing, the measurement matrix, over-complete atom dictionary for signal sparse representation are constructed separately, a kind of recovery algorithm is selected, then the frequency and DOA parameters of LFM signal can be estimated. Experimental simulations show that the resolution is much higher than conventional methods (like STFD) in the case of lower samples situation.
3. Single and multiple source DOA estimation of wideband LFM signal method based on compressed sensing was presented. To multi-source DO A problem, when the sources are disrelated or the relativity of the sources is very small, it's also feasible to use compressed sensing method to get a good estimation result under low samples situation.
Using compressed sensing method to estimate parameters of wide-band array signal which avoid so many disadvantages, like the selection of the correct time-frequency points and performance cross-terms impact in time-frequency method, and large-scale data storage and transmission cause by traditional sampling way.
继信号的稀疏分解,06年出现了压缩传感这个新的理论,打破了乃奎斯特采样模式,只要信号在某个域具有稀疏性,便可以远少于乃奎斯特采样的采样点恢复重建原始信号。本文对压缩传感理论进行了研究,并将其引入到宽带阵列信号参数估计中来,研究了基于压缩传感理论的宽带LFM信号的频率以及DOA参数估计的方法。主要的工作和创新有：
1.研究了压缩传感理论：基本理论原理及关键,从观测矩阵、信号的稀疏表示以及恢复算法几个方面做了研究。
2.研究了几种不同的基于压缩传感的宽带LFM信号频率和DOA参数的估计算法。由于要估计的目标参数在参数的空间具有稀疏性,如单个信源的DOA,则目标为在角度空间寻找符合目标方位的一个角度,因此在角度域中,目标方位角具有稀疏性,则基于压缩传感实现参数的估计是可能的。根据压缩传感理论的关键,分别构造观测矩阵、信号稀疏表示的过完备原子库,选择恢复算法,最后实现宽带LFM信号频率和DOA参数的估计。计算机仿真实验表明,在低采样情况下,本文算法分辨率优于经典的宽带估计方法(如时频子空间法)。
3.基于压缩传感,实现了单信源和多信源情况下的宽带LFM信号到达角估计。对于多个信源情况下的DOA估计,当各个信源不相关或相关性很小时,仍然可以利用压缩的方法,在低采样率下得到较好的估计效果。
将压缩传感理论引入到宽带阵列信号参数估计中,避免了时频类方法估计性能受时频点选取及交叉项影响的问题,且在低采样情况下可获得好的估计效果,减轻了数据存储与传输的负担。
Array signal processing is one of the important parts of signal analysis and processing, which is widely used in communication, radar, sonar, chronometer and other areas of science and technology. One of the key problems to be solved array signals processing is parameter estimation, like the most attentioned problem:direction of arrival (DOA) of signal. With the fast development of communication technologies, many wide-band signals, such as spread-spectrum signal, frequency-hopping signal and linear frequency modulation (LFM) signal, are used more and more widely in communication systems. In addition, many signals, such as seismology wave is a kind of wide-band signal in nature. Currently, time-frequency method is widely used for DOA of wide-band array signals, but some disadvantages like selection of the correct time-frequency points exist in these methods. In addition, large-scale data processing is also a problem cause by tranditonal Nyquist sampling. So the research on wide-band array signal processing method becomes more and more vital.
After the sparse decomposition of signals, a new theory called compressed sensing was present on 2006, which breaks the classical signal sampling rule and with many new advantages. If the signal is sparse on some domain, it's possible to be reconstructed based on far fewer random samples than required by tranditional Nyquist sampling rate. Based on the reaserches of compressed sensing, this paper applies it to wideband array signal parameters estimation problems, and presents a new way to frequency and DOA estimation of wideband LFM signals. The main work and contributions of the thesis as follows:
1. Theory of compressed sensing was studied totally, from basic theory principle to the key points of compressed sensing, include:measurement matrix, sparse representation of signal and signal recovery algorithm.
2. Several new methods based on compressed sensing are researched to estimate the frequency and DOA parameters of wideband LFM signals in array signal processing. Because the target parameter is sparse in the space of parameters, for example, single DOA of signal, our target is to find the DOA of signal source, so the only one direction is sparse in angle space, so it's possible to find the DOA by compressed sensing method. According to realization of compressed sensing, the measurement matrix, over-complete atom dictionary for signal sparse representation are constructed separately, a kind of recovery algorithm is selected, then the frequency and DOA parameters of LFM signal can be estimated. Experimental simulations show that the resolution is much higher than conventional methods (like STFD) in the case of lower samples situation.
3. Single and multiple source DOA estimation of wideband LFM signal method based on compressed sensing was presented. To multi-source DO A problem, when the sources are disrelated or the relativity of the sources is very small, it's also feasible to use compressed sensing method to get a good estimation result under low samples situation.
Using compressed sensing method to estimate parameters of wide-band array signal which avoid so many disadvantages, like the selection of the correct time-frequency points and performance cross-terms impact in time-frequency method, and large-scale data storage and transmission cause by traditional sampling way.
引文
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[2]王永良,陈辉,彭应宁,万群.空间谱估计理论与算法[M].清华大学出版社,2004：253-273.
[3]Haykin S, Reitly J, Keays V, Vertatschitsch E. Some aspects of array signal processing [J]. Proc. IEE. F Feb.1992,139(1):1-42.
[4]Monika Agrawal, Surendra Prasad. DOA estimation of wideband sources using a harmonic source model and uniform linear array [J]. IEEE Trans. SP.1999,47(3):619-629.
[5]Wang H, Kaveh M. Coherent Signal-subspace Processing for the Direction and Estimation of Angle of Arrival of Multiple Wideband Source [J]. IEEE Trans on SSP.1985,33(8):823-831.
[6]Belouchrani A, Amin M.G. Blind source separation based on time-frequency signal representations [J]. IEEE Trans. on SP,1998,46(11):2888-2897.
[7]Belouchrani A, Amin M.G. Time-frequency MUSIC [J]. IEEE Signal Processing Lett,1999,6(5): 109-110.
[8]Wang H, Kaveh M. Coherent Signal-subspace Processing for the Direction and Estimation of Angle of Arrival of Multiple Wideband Source [J]. IEEE Trans on SSP.1985,33(8):823-831.
[9]Hung H, Kaveh M. Focusing matrices for coherent signal-subspace processing [J]. IEEE. on ASSP, 1998,36(8):1272-1281.
[10]Valaee S, Champagne B. Localization of wideband signals using least-squares and total least-squares approaches [J]. IEEE. on SP,1999,47(5):1213-1222.
[11]Valaee S, Kabal P. Wideband array processing using a two-sided correlation transformation [J]. IEEE. on SP,1995,43(1):160-172.
[12]李平安,孙进才,俞卞章.基于插值圆阵的宽带信号二维空间普估计.电子科学学刊.1999,21(1)：136-140.
[13]Valaee S, Kabal P. Wideband array processing using a two-sided correlation transformation [J]. IEEE. on SP,1995,43(1):160-172
[14]Doron M.A, Doron, E. and Weiss, A.J. Coherent wideband processing for arbitrary array geometry. IEEE Trans. Signal Process.1993,41(1):414-417.
[15]B. Friedlander and A. J. Weiss. Direction finding for wide-band signals using an interpolated array. IEEE Transactions on Signal Processing,1993,41(4):1618-1634
[16]J.C.Chen, R.E.Hudson, and K. Yao. Maximum-likelihood source localization and unknown sensor location estimation for wideband signals in the near-field. IEEE Trans. Signal Process.2002, 50(8):1843-1854.
[17]雷中定,黄绣坤,张树京.一种基于神经网络的宽带相干源波达方向估计方法[J].北方交通大学学报,1997,21(1)：28-33.
[18]黄可生.一种宽带阵列信号DOA估计算法[J].航天电子对抗,2005,21(2)：38-40.
[19]Belouchrani A, Amin M.G. Time-frequency MUSIC [J]. IEEE Signal Processing Lett,1999,6(5): 109-110.
[20]Belouchrani A, Amin M.G Blind source separation based on time-frequency signal representations [J]. IEEE Trans. on SP,1998,46(11):2888-2897.
[21]Gershman A.B, Amin M.G. Wideband direction of arrival estimation of multiple Chirp signals using spatial time-frequency distributions [J]. IEEE Signal Processing Lett,2000,7(6):152-155.
[22]Mallat S, Zhang Z. Matching pursuits with time-frequency dictionaries[J]. IEEE Trans Signal Process.1993,41(12):3397-3415.
[23]David Donoho. Compressed sensing[J]. IEEE Trans. on Information Theory.2006,52(4):1289-1306.
[24]Ma J W. Compressed sensing by inverse scale space and curvelet thresholding. Applied Mathematics and Compution,2008,206(2):980-988.
[25]http://dsp.rice.edu/cs
[26]http://www.math.ucla.edu/-tao/preprints/sparse.html.
[27]http://www.acm.caltech.edu/～jtropp/index.html.
[28]http://www.acm.caltech.edu/l1 magic/.
[29]Jason Laska, Sami Kirolos, etc. Theory and implementation of an analog-to-information converter using random demodulation[A]. Proceedings of the IEEE Int. Symp. on Circuits and Systems (ISCAS)[C]. Piscataway:Institute of Electrical and Electronics Engineers Inc.2007,1959-1962.
[30]张春梅,尹忠科,肖明霞.基于冗余字典的信号超完备表示与稀疏分解.科学通报.2006,51(6)：628-633
[31]Temlyakov V. Nonlinear Methods of Approximation. IMI Research Reports, Dept of Mathematics, University of South Carolina.2001,01-09
[32]Chen S, Donoho D, Saunders M. Atomic decomposition by basis pursuit [J]. SIAM Journal on Scientific Computing,1999,20:33-61
[33]E. Candes and T. Tao. Near optimal signal recovery from random projections:universal encoding strategies. IEEE Trans. Info. Theory.2006,52(12):5406-5425.
[34]M. Elad, J.-L. Starck, P. Querre, D.L. Donoho. Simultaneous cartoon and texture image inpainting using morphological component analysis (MCA). Journal on Applied and Computational Harmonic Analysis, Vol.19, pp.340-358, November 2005
[35]Emmanuel Candes and Michael B WAKIN. An Introduction to compressive sampling [J]. IEEE Signal Processing Magazine, MARCH 2008; 21-30.
[36]Rachlin, Y. and Baron, D. The secrecy of compressed sensing measurements. Communication, Control, and Computing,2008 46th Annual Allerton Conference on Digital Object Identifier, 2008;813-817.
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