成都电子科技大学博士研究生稀疏信号处理讲义

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1.4 分辨率受孔径限制？
DFT
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内容
从几个问题开始
稀疏重建理论 几个例子

阵列信号处理的例子

实孔径超分辨、阵列稀疏布阵 MDS、MC
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无线定位的例子


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信道估计的例子
一、从几个问题开始
高斯分布凭什么无所不在？ MMSE是最优的？
吝啬原则：免费的午餐？
but convexity and non-convexity
— R. Rockafellar, SIAM Review 1993
x ay bxy cx dy e 0
2 2
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1.6 什么是多维标度问题?
测距定位
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成都电子科技大学电子工程学院 万群
“Signal & information processing is · · · an Art ”
— Petre Stoica
一所大学，两个战场
科学：一个是和其他世界一流大学共
同面对的国际学术前沿战场
技术：另一个是为我们国家经济、社
会、国防、发展战略需要服务的战场
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convex quadratic programming problem
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dual formulation
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Great watershed in optimization
It is not between
linearity and nonlinearity,
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Gauss’s Question (1809)
What would be a distribution density
f(x ;θ) for which
the maximum likelihood estimate of θ is the sample mean
we use the modern terminology adopted by the
分辨率受孔径限制？
机器学习：支持向量是稀疏的？
什么是多维标度问题?
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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 old The distribution itself had been found by de Moivre (1733) before Laplace was born
likelihood estimate of parameter of location
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Derivation
any real number can be arbitrarily accurately
approximated by rational numbers
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2
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贪婪的谱估计 = 滤波：
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1

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有偏估计！
If h is known to be sparse, can we do even better than the MMSE estimate? And if so, how much better
can we do?
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NP-Hard ？现代最小二乘
(P0)
min x
Result
Gauss assumed the sample mean due to its
computational convenience and derived the Gaussian 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.
The central limit theorem (CLT) is not only a
unique reason but perhaps it is even not the main reason.
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Fisher information
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1.2 MMSE是最优的？
scientific community more than a century later （the method of maximum likelihood was proposed by Fisher in 1921）
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DERIVATION OF GAUSS (1809)
Using i.i.d. observations, the maximum
March 3, 2008，mcgrant@ large-scale l1-regularized least-squares large-scale l1-regularized logistic regression geometric programming convex optimization to Compressed Sensing sparse solutions to linear equations, particularly underdetermined systems
在线测量 +
先验模型
稀疏
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1.5 机器学习：支持向量是稀疏的？
the training sample
hyperplane that does the separation
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primal formulation of the problem
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BWE
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1GHz (S) + 1GHz (X) = 10GHz ？
L band S band C band
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1 to 2 GHz 2 to 4 GHz 4 to 8 GHz 8 to 12 GHz 12 to 18 GHz
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X band Ku band
超分辨是一个欠定问题
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二、稀疏重建理论
基追踪：Basis Pursuit，贪婪算法 稀疏重建条件：RIP 字典

计算最简单 性能最优 所需测量最少
确定型 随机型

结构+随机型
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Current software
CVX:

convex optimization
计算最简单 性能最优
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子空间分析
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矩阵完整性分析
Rank = 2, 3 节点之间无测量 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！
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He-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 , no.3 , pp.1677-1688, 2010 S. 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 Yang，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 2009 Hewen 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.3018-3022 Hewen 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 - 423 Zhang-Xin Chen, Qun Wan, He-Wen Wei and Wan-Lin Yang，A Novel Subspace Approach for Hyperbolic Mobile Location, Chinese Journal of Electronics，2009年第3期, pp.569-573 Huang 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