1、稀疏信號處理簡介“Signal & information processing is an Art ” Petre Stoica成都電子科技大學電子工程學院 萬群2022/9/112一所大學，兩個戰場科學：一個是和其他世界一流大學共同面對的國際學術前沿戰場技術：另一個是為我們國家經濟、社會、國防、發展戰略需要服務的戰場U E S T C2022/9/113內容從幾個問題開始稀疏重建理論幾個例子陣列信號處理的例子實孔徑超分辨、陣列稀疏布陣無線定位的例子MDS、MC信道估計的例子2022/9/114一、從幾個問題開始高斯分布憑什么無所不在？ MMSE是最優的？吝嗇原則：免費的午餐？分辨率受孔徑限

2、制？機器學習：支持向量是稀疏的？什么是多維標度問題?2022/9/115高斯分布：An equation is for eternity The fundamental nature of this distribution and its main properties were derived by Laplace (1781) when Gauss was six years oldThe distribution itself had been found by de Moivre (1733) before Laplace was born2022/9/116Gausss Quest

3、ion (1809)What would be a distribution density f(x ;) for which the maximum likelihood estimate of is the sample meanwe use the modern terminology adopted by the scientific community more than a century later （the method of maximum likelihood was proposed by Fisher in 1921）2022/9/117DERIVATION OF GA

4、USS (1809)Using i.i.d. observations, the maximum likelihood estimate of parameter of location2022/9/118Derivation any real number can be arbitrarily accurately approximated by rational numbers2022/9/119Result Gauss assumed the sample mean due to its computational convenience and derived the Gaussian

5、 law.This line of reasoning is quite the opposite to the modern exposition in textbooks on statistics and signal processing where the LS method is derived from the assumed Gaussianity.2022/9/1110為什么要折衷？性能最優計算最簡單跑題了？2022/9/11111.1高斯分布憑什么無所不在？The role of Gaussian models in signal processing is based o

6、n the optimal property of the Gaussian distribution minimizing Fisher information over the class of distributions with a bounded variance.The central limit theorem (CLT) is not only a unique reason but perhaps it is even not the main reason.2022/9/1112Fisher information2022/9/11131.2 MMSE是最優的？If h i

7、s known to be sparse, can we do even better than the MMSE estimate? And if so, how much better can we do?有偏估計！2022/9/1114NP-Hard ？現代最小二乘(P0) subject to (P1) subject to 2022/9/11151.3吝嗇原則：免費的午餐？多成分混合（合成，正問題）分離各個成分（感知，反問題）2022/9/1116貪婪的譜估計 = 濾波：2022/9/11171.4分辨率受孔徑限制？DFTOOOOOOOO=O O O O O O O OO O O O

8、 O O O OO O O O O O O OO O O O O O O OO O O O O O O OO O O O O O O OO O O O O O O OO O O O O O O OOOOOOOOOX 2022/9/1118BWE2022/9/11191GHz (S) + 1GHz (X) = 10GHz ？L band1 to 2GHzS band2 to 4GHzC band4 to 8GHzX band8 to 12GHzKu band12 to 18GHz2022/9/1120超分辨是一個欠定問題在線測量 + 先驗模型稀疏2022/9/11211.5 機器學習：支持向量是

9、稀疏的？the training samplehyperplane that does the separation-2022/9/1122primal formulation of the problem2022/9/1123convex quadratic programming problem2022/9/1124dual formulation2022/9/1125Great watershedin optimizationIt is not between linearity and nonlinearity, but convexity and non-convexity R. R

10、ockafellar, SIAM Review 19932022/9/11261.6 什么是多維標度問題?測距定位2022/9/1127倒行逆施：解的表示計算最簡單性能最優2022/9/1128子空間分析2022/9/1129矩陣完整性分析Rank = 2, 3節點之間無測量節點之間測量誤差很大計算最簡單性能最優所需測量不多！2022/9/1130He-Wen Wei, Rong Peng, Qun Wan, Zhang-Xin Chen, and Shang-Fu Ye, Multidimensional Scaling Analysis for Passive Moving Target

11、Localization with TDOA and FDOA Measurements, IEEE Transactions on Signal Processing, vol.58 , no.3 , pp.1677-1688, 2010S. Qin, Q. Wan, Z. X. Chen, A Fast Multidimensional Scaling Analysis for Mobile Positioning, IET Signal Processing. Zhang-Xin Chen, He-Wen Wei, Qun Wan, Shang-Fu Ye and Wan-Lin Yan

12、g，A Supplement to Multidimensional Scaling Framework for Mobile Location : A Unified View，IEEE Transactions on Signal Processing, vol. 57, no. 5, pp. 2230-2234, May 2009Hewen Wei, Qun Wan, Shangfu Ye, A Novel Weighted Multidimensional Scaling Analysis for Time-of-Arrival-Based Mobile Location, IEEE

13、Transactions on Signal Processing, Vol.56, No.7, July 2008, pp.3018-3022Hewen Wei, Qun Wan, Shangfu Ye, Multidimensional scaling based passive emitter localization from range-difference measurements, IET Signal Processing, Volume 2, Issue 4, December 2008 Page(s):415 - 423Zhang-Xin Chen, Qun Wan, He

14、-Wen Wei and Wan-Lin Yang，A Novel Subspace Approach for Hyperbolic Mobile Location, Chinese Journal of Electronics，2009年第3期, pp.569-573Huang Ji Yan, Wan Qun, Comments on The Cramer-Rao Bounds of Hybrid TOA/RSS and TDOA/RSS Location Estimation Schemes, IEEE Comm. Letters, Vol.11 , Issue 11, Nov. 2007

15、, pp.848-849 2022/9/1131二、稀疏重建理論基追蹤：Basis Pursuit，貪婪算法稀疏重建條件：RIP字典確定型隨機型結構+隨機型計算最簡單性能最優所需測量最少2022/9/11322022/9/1133CVX: convex optimizationMarch 3, 2008， l1_ls large-scale l1-regularized least-squaresl1_logreglarge-scale l1-regularized logistic regressionGGPLAB geometric programmi

16、ngL1-MAGIC convex optimization to Compressed SensingSparseLab sparse solutions to linear equations, particularly underdetermined systemsCurrent software2022/9/11342022/9/11352022/9/11362022/9/11372022/9/11382022/9/1139三、幾個例子陣列信號處理的例子實孔徑超分辨陣列稀疏布陣無線定位的例子MDSMC信道估計的例子SVR2022/9/1140稀疏布陣：同陣元數，優化 5.7 dBcom

17、pare_ieee_trans_sp_1988_vol.36_no.3_pp3722022/9/1141稀疏信道估計 cvx_begin variables h; minimize( norm(S*h-r, 2) + 0.5 * norm(h,1) );cvx_end1ms10kbps: 101Gbps: 100萬 2022/9/1142性能比較2022/9/1143Yipeng Liu, Qun Wan, Total Variation Minimization and Sparse Constraint Based Robust Beamformer, Electronic Letters

18、Ying Zhang，Qun Wan，Wang Minghui，A Partially Sparse Solution to the Problem of Parameter Estimation of CARD Model, Signal Processing, vol.8, No.10, Oct. 2008, pp.2483-2491 Y. Zhang, B.P. Ng and Q. Wan, Sidelobe suppression for adaptive beamforming with sparse constraint on beam pattern, Electronics L

19、etters, vol.44, no.10, pp.615-616 Ying Zhang, Qun Wan, H.P. Zhao, W.L. Yang, Support Vector Regression for Basis Selection in Laplacian Noise Environment, IEEE Signal Processing Letters, Vol. 14, Issue 11, Nov. 2007, pp.871-874Guo Xiansheng, Wan Qun, Wu Bin, Yang Wanlin, Parameters localization of c

20、oherently distributed sources based on sparse signal representation, IET Radar, Sonar & Navigation, vol.1, issue 4, Aug. 2007, pp.261-265Rui Ming Yang, Qun Wan, and Wan Lin Yang, Greedy Approach to Sparse Multi-path Channel Estimation Using Sensing Dictionary, International Journal of Adaptive Contr

21、ol and Signal ProcessingGuo, Xiansheng, Wan, Qun; Chang, Chunqi; Lam, Edmund Y., Source localization using a sparse representation framework to achieve superresolution, Multidimensional Systems and Signal Processing, v 21, n 4, p 391-402, December 2010Yipeng Liu, Qun Wan andXiaoli Chu, A robust beamformer based on weighted sparse constraint, Progress In Electromagnetics Research Letters, Vol. 16, 53-60, 2010. Yipeng Liu, Qun Wan, “Total difference based partial sparse LCMV beamformer,” Progress In Electro