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1、稀疏信号处理简介“Signal ) for which the maximum likelihood estimate of is the sample meannwe 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）Date6DERIVATION OF GAUSS (1809)nUsing i.i.d. observations, th
2、e maximum likelihood estimate of parameter of locationDate7Derivation nany real number can be arbitrarily accurately approximated by rational numbersDate8Result nGauss assumed the sample mean due to its computational convenience and derived the Gaussian law. nThis line of reasoning is quite the oppo
3、site to the modern exposition in textbooks on statistics and signal processing where the LS method is derived from the assumed Gaussianity. Date9为什么要折衷？n性能最优n计算最简单n跑题了？Date101.1 高斯分布凭什么无所不在？nThe role of Gaussian models in signal processing is based on the optimal property of the Gaussian distributio
4、n minimizing Fisher information over the class of distributions with a bounded variance.nThe central limit theorem (CLT) is not only a unique reason but perhaps it is even not the main reasonDate11Fisher informationDate121.2 MMSE是最优的？If h is known to be sparse, can we do even better than the MMSE es
5、timate? And if so, how much better can we do?有偏估计！Date13NP-Hard ？现代最小二乘(P0) subject to (P1) subject to Date141.3 吝啬原则：免费的午餐？n多成分混合（合成，正问题）n分离各个成分（感知，反问题 ）Date15贪婪的谱估计 = 滤波 ：Date161.4 分辨率受孔径限制？nDFTO O O O O O O O=O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O
6、O O O O O O O O O O O O O O O O O O O OO O O O O O O OX Date17BWEDate181GHz (S) + 1GHz (X) = 10GHz ？L band1 to 2 GHz S band2 to 4 GHz C band4 to 8 GHz X band8 to 12 GHz Ku band12 to 18 GHzDate19超分辨是一个欠定问题n在线测量 + 先验模型n稀疏Date201.5 机器学习：支持向量是稀疏的？nthe training samplenhyperplane that does the separation-
7、Date21primal formulation of the problemDate22convex quadratic programming problemDate23dual formulationDate24Great watershed in optimizationIt is not between linearity and nonlinearity, but convexity and non-convexity R. Rockafellar, SIAM Review 1993Date251.6 什么是多维标度问题?n测距定位Date26倒行逆施：解的表示计算最简单 性能最优
8、Date27子空间分析Date28矩阵完整性分析nRank = 2, 3n节点之间无测量n节点之间测量误差很大计算最简单 性能最优 所需测量不多！Date29nHe-Wen Wei, Rong Peng, Qun Wan, Zhang-Xin Chen, and Shang-Fu Ye, Multidimensional Scaling Analysis for Passive Moving Target Localization with TDOA and FDOA Measurements, IEEE Transactions on Signal Processing, vol.58 ,
9、no.3 , pp.1677-1688, 2010nS. Qin, Q. Wan, Z. X. Chen, A Fast Multidimensional Scaling Analysis for Mobile Positioning, IET Signal Processing. nZhang-Xin Chen, He-Wen Wei, Qun Wan, Shang-Fu Ye and Wan-Lin Yang，A Supplement to Multidimensional Scaling Framework for Mobile Location : A Unified View，IEE
10、E Transactions on Signal Processing, vol. 57, no. 5, pp. 2230-2234, May 2009nHewen Wei, Qun Wan, Shangfu Ye, A Novel Weighted Multidimensional Scaling Analysis for Time-of-Arrival-Based Mobile Location, IEEE Transactions on Signal Processing, Vol.56, No.7, July 2008, pp.3018-3022nHewen Wei, Qun Wan,
11、 Shangfu Ye, Multidimensional scaling based passive emitter localization from range-difference measurements, IET Signal Processing, Volume 2, Issue 4, December 2008 Page(s):415 - 423nZhang-Xin Chen, Qun Wan, He-Wen Wei and Wan-Lin Yang，A Novel Subspace Approach for Hyperbolic Mobile Location, Chines
12、e Journal of Electronics，2009年第3期, pp.569-573nHuang 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, pp.848-849 Date30二、稀疏重建理论n基追踪：Basis Pursuit，贪婪算法n稀疏重建条件：RIPn字典n确定型n随机型n结构+随机型计算最简单 性能最优
13、 所需测量最少Date31Date32CVX: convex optimization March 3, 2008，mcgrantstanford.edu l1_ls large-scale l1-regularized least-squares l1_logreg large-scale l1-regularized logistic regression GGPLAB geometric programming L1-MAGIC convex optimization to Compressed Sensing SparseLab sparse solutions to linear e
14、quations, particularly underdetermined systemsCurrent softwareDate33Date34Date35Date36Date37Date38三、几个例子n阵列信号处理的例子n实孔径超分辨n阵列稀疏布阵n无线定位的例子nMDSnMCn信道估计的例子nSVR Date39稀疏布阵：同阵元数，优化 5.7 dBcompare_ieee_trans_sp_1988_vol.36_no.3_pp372Date40稀疏信道估计cvx_beginvariables h;minimize( norm(S*h-r, 2) + 0.5 * norm(h,1)
15、 );cvx_end1ms 10kbps: 10 1Gbps: 100万 Date41性能比较Date42nYipeng Liu, Qun Wan, Total Variation Minimization and Sparse Constraint Based Robust Beamformer, Electronic Letters nYing Zhang，Qun Wan，Wang Minghui，A Partially Sparse Solution to the Problem of Parameter Estimation of CARD Model, Signal Processi
16、ng, vol.8, No.10, Oct. 2008, pp.2483- 2491 nY. Zhang, B.P. Ng and Q. Wan, Sidelobe suppression for adaptive beamforming with sparse constraint on beam pattern, Electronics Letters, vol.44, no.10, pp.615-616 nYing 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.871-874 nGuo Xiansheng, Wan
