What is signal processing? A very classic explanation of signal processing was given by Dr. Don Johnson (former President of IEEE Signal Processing Society): “Signal processing is a ‘stealth technology’ that only the people who actually work in signal processing think about who we are and what we do”. This quote stated the essence of signal processing. Weak signal detection and parameter estimation are some of the most fundamental theories in signal processing.
The theoretical bases of signal detection and parameter estimation are infor
mation theory, whose mathematical bases include probability, random processes, mathematical statistics, linear algebra, matrix analysis, optimization theories and algorithms, etc. The applications of radar and sonar in the 1940s greatly promoted the development of these related theories. At present, these theoretic bases have become a common foundation in the ?elds of communication, radar, sonar, navi gation, automatic control, medicine, celestial observation, seismic exploration, and so on. The objective of the parameter estimation method is to identify signals and systems based on the modeling technology. When the model is in good agreement with the actual situation, the parametric method often has better performance than the nonparametric method. But the parameter estimation method is more sensitive to various model errors, including signal, noise or interference modeling errors, calibration errors of the sensor array (including location error, channel mismatch, mutual coupling), uncertainty of the number of signal sources, and the estimation error caused by a limited number of observations. How to maintain parameter estimation methods good performance in the presence of various model errors has always been the focus and eternal topic of signal processing. In this book, we choose to use “robustness” for this performance measure.
In the years of 1996–1998, 2001, and 2002, I worked 3 times as a Postdoctoral
Fellow and Visiting Professor at the Spectral Analysis Laboratory of the Department of Electrical and Computer Engineering at Florida University in the United States and had fruitful collaborations with Professor Jian Li who was the director of the laboratory and a young IEEE fellow. Between 2004 and 2007, supported by the National Natural Science Foundation of China, Professor Jian Li had spent 2 months every year working at the Tianjin Key Lab for Advanced Signal
Processing, located at Civil Aviation University of China which was led by me. In 1996, she proposed a signal estimation method for sinusoidal signals called RELAX, which is a parameterized cyclic optimization algorithm based on Nonlinear Least Squares (NLS) criterion and RELAXation. Later, we worked together on more than 20 IEEE/IEE publications to expand the applications of RELAX on time delay estimation, radar target imaging, airborne radar ground moving target high resolution imaging, ground penetrating radar, and vehicle cavitation shape control for underwater supercavitation. At the same time, the research team of the Civil Aviation University of China led by me has carried out research on the applications of RELAX in adaptive antijamming of satellite nav igation and airborne weather radar, and published dozens of related papers.
For parameter estimation problems involving multiple overlapping signals in noise or interference, the Nonlinear Least Squares (NLS) method is a common method to solve such problems. In white or colored Gaussian noise or even nonGaussian noise backgrounds, the NLS method has identical or similar esti mation performance as the maximum likelihood method and is more robust. Because it does not estimate the parameters in the noise, the computation load is smaller than that of the maximum likelihood method, but the NLS method cannot avoid the multidimensional search over the signal parameter space so the amount of computation is still very large, and it is dif?cult to guarantee convergence toward the global optimal solution.
This book generalizes the basic ideas of RELAX to solve these problems. Like RELAX, the generalized version also transforms a multiple signal parameter estimation problem into a series of single signal parameter estimation problems using the signal separation estimation method or the cyclic optimization method with special structures. As a result, it can not only greatly reduce the amount of computation, but also have a good global convergence property, and does not need a separate initialization process. The RELAX estimation method is a commonly used multiple signal parameter estimation method, which is insensitive to various model errors so it is robust. By using the word RELAX (meaning relaxation), we imply that the method has superior performance and robustness because it is an ef?cient implementation of NLS estimation method.
This book has seven chapters. Chapter 1 introduces the fundamentals of parameter estimation, including the basic principles of maximum likelihood esti mation, Bayes estimation, linear minimum mean squared error estimation, the standards of evaluating the performance of an estimator, and the compact and general SlepianBangs formula used for estimating the vector parameters’ CramerRao Bound (CRB) under Gaussian background. In Chap. 2, a general representative data model for multiple signal parameter estimation is proposed, and we introduce the least squares method, a wellknown method of another class of parameter estimation, which includes the basic theories of linear and Nonlinear Least Squares (NLS) estimations. The direct solution method and the cyclic opti mization method of NLS are introduced, and from that foundation we give the basic principles and implementation frames of the RELAX method and point out the relationship between RELAX and the Matching Pursuit (MP) greedy algorithm,
which is a very popular method for compressed sensing applications at the present time. Chapter 3 introduces the application of RELAX on line spectrum estimation, including onedimensional and twodimensional hybrid spectrum estimations, exponential attenuation, and arbitrary envelope sinusoidal signal parameter esti mation. Chapter 4 introduces the applications of RELAX on time delay estimation problems, including the general time delay estimation method WRELAX, the estimation methods when the cost functions are highly oscillatory or when the time delay intervals are very close to each other, and using multiple looks data for time delay estimation in a colored noise background. Chapter 5 introduces the appli cation of RELAX on Direction of Arrival (DOA) estimation problems, including the DOA and waveform estimation for narrowband and wideband signal sources. Chapter 6 discusses the application of RELAX in the ?eld of radar target imaging, including Synthetic Aperture Radar (SAR) autofocusing and semiparametric imaging, curvilinear SAR autofocusing and threedimensional imaging, Inverse Synthetic Aperture Radar (ISAR) imaging, and maneuvering target ISAR imaging. Chapter 7 briefly introduces the typical applications of RELAX in other aspects, including airborne moving target detection, airborne radar ground moving target high range resolution imaging, airborne weather radar, ground penetrating radar, adaptive antijamming for satellite navigation, cavitation shape control for under water supercavitation vehicle, sparse array signal processing for compressive sensing, biomedical signal processing, and so on.
I’d like to thank Professor Jian Li at University of Florida for giving me many
opportunities to work and study in her laboratory. During my work at her lab, I was lucky to get some guidance and help from Professor Petre Stoica from Uppsala University in Sweden who is a world famous signal processing expert. I’d also like to thank Professor Jian Li and my colleagues from the Spectrum Analysis Lab (Dr. Zhengshe Liu, Dr. Guoqin Liu, Dr. Xi Li, Dr. Zhaoqiang Bi, Dr. Nanzhi Jiang, Dr. Jianhua Liu, and Mr. Kunlong Gu) for longterm collaboration and exchange. Some contents of this book are the results of our collaborated research over many years. During the process of writing this book, Professor Jian Li has provided the newest RELAX research results generated in her lab. In addition, Professor Jian Li strongly recommended that we differentiate between the RELAX approach and the Matching Pursuit (MP) related greedy algorithms used for compressive sensing which has been a very active research topic at present. In her communication with me, she commented as follows: RELAX was ?rst published in 1996, well before the emergence of “compressive sensing” and Matching Pursuit (MP) related algo rithms, including MP, Orthogonal MP (OMP), Compressive Sensing MP (CoSeMP), Least Squares MP, etc. These are the socalled greedy methods in compressive sensing literature, which have been used in many diverse applications due to the compressive sensing topic being a hot topic for over a decade. Yet, MP is just CLEAN, but RELAX is Super CLEAN, and hence should outperform most of these MP variations in most applications. I have adopted her suggestion and more details on this topic will be discussed in Chap. 2.
I’d like to thank Professor Zheng Bao (the author’s doctoral dissertation advisor) and Professor Mengdao Xing at Xidian University’s National Key Laboratory of Radar Signal Processing for sharing their research results in the ?eld of ISAR imaging of maneuvering targets. More details can be found in Chap. 6.
I’d like to thank my former master student, Dr. Guangli Wang, who wrote
Sect. 7.6, in which he describes how he skillfully used the RELAX algorithm learned at our laboratory to handle the multielectrode recording signal of retinal neuron activity during his doctor’s study at Shanghai Jiao Tong University.
I’d like to thank my three coauthors who are teachers and researchers from the Tianjin Key Laboratory for Advanced Signal Processing at Civil Aviation University of China: Qiongqiong Jia, Lei Yang, and Qing Feng. There are also a large number of teachers and graduate students who have provided help in writing this book. Teachers include Professor Han Ping, Professor Hai Li, Dr. Guimin Jia, Dr. Xiaoguang Lu, Professor Wenyi Wang, Dr. Dan Lu, Dr. Weikun He, Associate Professor Tieqiao Hu, Professor Zhigang Su, and Lecturer Lu Wang. Ph.D. students include Yan Bi and Lina Bao, and master students include Chenchen Wu, Hao Zhang, Wen Ren, Zhihua Niu, Wei Zhu, Lei Zhan, Chenxi Ma, Chao Liu, Ruihua Zhang, Lei Chen, Anfei Zhao, Juan Liu, and Jiayi Li.
This book has been funded by the National Natural Science Foundation of China under grants 61471363 and 61231017.
Renbiao Wu The ?rst draft was written on the Chinese National Day of 2016 at Civil Aviation University of China, Tianjin, China
The ?nal version was completed during the 2017 Chinese Spring
Festival holidays in the beautiful Zhuhai city