Ebook: Statistical Signal Processing: Frequency Estimation

Signal processing may broadly be considered to involve the recovery of information from physical observations. The received signal is usually disturbed by thermal, electrical, atmospheric or intentional interferences. Due to the random nature of the signal, statistical techniques play an important role in analyzing the signal. Statistics is also used in the formulation of the appropriate models to describe the behavior of the system, the development of appropriate techniques for estimation of model parameters and the assessment of the model performances. Statistical signal processing basically refers to the analysis of random signals using appropriate statistical techniques. The main aim of this book is to introduce different signal processing models which have been used in analyzing periodic data, and different statistical and computational issues involved in solving them. We discuss in detail the sinusoidal frequency model which has been used extensively in analyzing periodic data occuring in various fields. We have tried to introduce different associated models and higher dimensional statistical signal processing models which have been further discussed in the literature. Different real data sets have been analyzed to illustrate how different models can be used in practice. Several open problems have been indicated for future research.

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Signal processing may broadly be considered to involve the recovery of information from physical observations. The received signal is usually disturbed by thermal, electrical, atmospheric or intentional interferences. Due to the random nature of the signal, statistical techniques play an important role in analyzing the signal. Statistics is also used in the formulation of the appropriate models to describe the behavior of the system, the development of appropriate techniques for estimation of model parameters and the assessment of the model performances. Statistical signal processing basically refers to the analysis of random signals using appropriate statistical techniques. The main aim of this book is to introduce different signal processing models which have been used in analyzing periodic data, and different statistical and computational issues involved in solving them. We discuss in detail the sinusoidal frequency model which has been used extensively in analyzing periodic data occuring in various fields. We have tried to introduce different associated models and higher dimensional statistical signal processing models which have been further discussed in the literature. Different real data sets have been analyzed to illustrate how different models can be used in practice. Several open problems have been indicated for future research. Table of Contents Cover Preface Acknowledgments Contents Abbreviations Symbols 1 Introduction References 2 Notations and Preliminaries 2.1 Prony's Equation 2.2 Undamped Exponential Model 2.3 Sum of Sinusoidal Model 2.4 Linear Prediction 2.5 Matrix Pencil 2.6 Stable Distribution: Results References 3 Estimation of Frequencies 3.1 ALSEs and PEs 3.2 EVLP 3.3 MFBLP 3.4 NSD 3.5 ESPRIT 3.6 TLS-ESPRIT 3.7 Quinn's Method 3.8 IQML 3.9 Modified Prony Algorithm 3.10 Constrained Maximum Likelihood Method 3.11 Expectation Maximization Algorithm 3.12 Sequential Estimators 3.13 Quinn and Fernandes Method 3.14 Amplified Harmonics Method 3.15 Weighted Least Squares Estimators 3.16 Nandi and Kundu Algorithm 3.17 Super Efficient Estimator 3.18 Conclusions References 4 Asymptotic Properties 4.1 Introduction 4.2 Sinusoidal Model with One Component 4.3 Strong Consistency of LSE and ALSE of th 4.3.1 Proof of the Strong Consistency of th"0362th, the LSE of th 4.3.2 Proof of Strong Consistency of th, the ALSE of th 4.4 Asymptotic Distribution of LSE and ALSE of th 4.4.1 Asymptotic Distribution of th"0362th Under Assumption 3.2 4.4.2 Asymptotic Distribution of th"0362th Under Assumption 4.2 4.4.3 Asymptotic Equivalence of LSE th"0362th and ALSE th 4.5 Superefficient Frequency Estimator 4.6 Multiple Sinusoidal Model 4.7 Weighted Least Squares Estimators 4.8 Conclusions References 5 Estimating the Number of Components 5.1 Introduction 5.2 Likelihood Ratio Approach 5.3 Cross Validation Method 5.4 Information Theoretic Criteria 5.4.1 Rao's Method 5.4.2 Sakai's Method 5.4.3 Quinn's Method 5.4.4 Wang's Method 5.4.5 Kundu's Method 5.5 Conclusions References 6 Real Data Example 6.1 Introduction 6.2 ECG Data 6.3 Variable Star Data 6.4 ``uuu'' Data 6.5 Airline Passenger Data 6.6 Conclusions References 7 Multidimensional Models 7.1 Introduction 7.2 2-D Model: Estimation of Frequencies 7.2.1 LSEs 7.2.2 Sequential Method 7.2.3 Periodogram Estimators 7.2.4 Nandi--Prasad--Kundu Algorithm 7.2.5 Noise Space Decomposition Method 7.3 2-D Model: Estimating the Number of Components 7.4 Conclusions References 8 Related Models 8.1 Introduction 8.2 Damped Sinusoidal Model 8.3 Amplitude Modulated Model 8.4 Fundamental Frequency Model 8.4.1 Test for Harmonics 8.5 Generalized Fundamental Frequency Model 8.6 Partially Sinusoidal Frequency Model 8.7 Burst-Type Model 8.8 Discussion/Remarks References Index