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- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Sprache: Englisch
Separate signals from noise with this valuable introduction to signal processing by applied decomposition
The decomposition of complex signals into the sub-signals, or individual components, is a crucial tool in signal processing. It allows each component of a signal to be analyzed individually, enables the signal to be isolated from noise, and processed in full. Decomposition processes have not always been widely adopted due to the difficult underlying mathematics and complex applications. This text simplifies these obstacles.
Signal Processing: An Applied Decomposition Approach demystifies these tools from a model-based perspective. This offers a mathematically informed, “step-by-step” analysis of the process by breaking down a composite signal/system into its constituent parts, while introducing both fundamental concepts and advanced applications. This comprehensive approach addresses each of the major decomposition techniques, making it an indispensable addition to any library specializing in signal processing.
Signal Processing readers will find:
- Signal decomposition techniques developed from the data-based, spectral-based and model-based perspectives incorporate: statistical approaches (PCA, ICA, Singular Spectrum); spectral approaches (MTM, PHD, MUSIC); and model-based approaches (EXP, LATTICE, SSP)
- In depth discussion of topics includes signal/system estimation and decomposition, time domain and frequency domain techniques, systems theory, modal decompositions, applications and many more
- Numerous figures, examples, and tables illustrating key concepts and algorithms are developed throughout the text
- Includes problem sets, case studies, real-world applications as well as MATLAB notes highlighting applicable commands
Signal Processing is ideal for engineering and scientific professionals, as well as graduate students seeking a focused text on signal/system decomposition with performance metrics and real-world applications.
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Table of Contents
Cover
Table of Contents
Title Page
Copyright
Dedication
About the Author
Preface
References
Acknowledgments
Glossary
About the Companion Website
1 Introduction
1.1 Background
1.2 Spectral Decomposition
1.3 Data Decomposition
1.4 Model-based Decomposition
1.5 Notation and Terminology
1.6 Summary
MATLAB® Notes
References
Problems
Note
2 Random Signals and Systems
2.1 Introduction
2.2 Discrete Random Signals
2.3 Spectral Representation of Random Signals
2.4 Discrete Systems with Random Inputs
2.5 Classical Spectral Estimation
2.6 Case Study: Sinusoids in Noise
2.7 Summary
MATLAB® Notes
References
Problems
Notes
3 Signal Models
3.1 Data-Based Models
3.2 Parametric-Based Models
3.3 State-space Models
3.4 Summary
MATLAB® Notes
References
Problems
Notes
4 Signal Estimation
4.1 Classical Estimation
4.2 Minimum Variance (MV) Estimation
4.3 Maximum A-Posteriori (MAP) Estimation
4.4 Maximum Likelihood (ML) Estimation
4.5 Least-squares (LS) Estimation
4.6 Optimal Signal Estimation
4.7 Projection Theory
4.8 Summary
MATLAB® Notes
References
Problems
Notes
5 Signal Decomposition
5.1 Introduction
5.2 Data-Based Decompositions
5.3 Spectral-Based Decompositions
MULTITAPER METHOD (
MTM
) Spectral Decomposition:
5.4 Model-Based Decomposition
5.5 Case Study: Harmonics in Noise
5.6 Summary
MATLAB® Notes
References
Problems
Notes
6 Model-based Decomposition: Time Domain
6.1 Background: State-space Systems
6.2 Realization Problem
6.3 Realization Decomposition
6.4 Subspace Decomposition: Orthogonal Projections
6.5 Subspace Decomposition: Oblique Projections
6.6 System Order Estimation and Validation
6.7 Case Study: Multichannel Mechanical Systems
6.8 Summary
MATLAB® Notes
References
Problems
Notes
7 Model-Based Decomposition: Frequency Domain
7.1 Introduction
7.2 Frequency Response Functions (
FRF
)
7.3 Least-squares Complex Frequency (
LSCF
) Method
7.4 PolyReference Least-Squares Complex Frequency (
pLSCF
) Method
7.5 Maximum Likelihood PolyReference Frequency Domain Estimation (
ML-pLSCF
)
7.6 Case Study: 15-DOF Structure
7.7 Summary
MATLAB® Notes
References
Problems
Notes
8 Performance Analysis
8.1 Statistical Performance Methods
8.2 Physical Performance Metrics
8.3 Case Study: Resonant Modal
MCK
System
8.4 Summary
MATLAB® Notes
References
Note
9 Applications
9.1 Modal Decomposition: Sounding Rocket Flight
9.2 Vibrational Response of a Cylindrical Structure: Identification and Modal Tracking
9.3 Resonant Ultrasound Spectroscopy
9.4 Model-Based Subsystem Decomposition of an 8-Story (8-Mass) Structure
9.5 Data-Based Decomposition: Time-Reversal Processing
References
A Probability and Statistics Overview
A.1 Probability Theory
A.2 Gaussian Random Vectors
A.3 Uncorrelated Transformation: Gaussian Random Vectors
A.4 Toeplitz Correlation Matrices
A.5 Important Processes
References
Note
B Projection Theory
B.1 Projections: Deterministic Spaces
B.2 Projections: Random Spaces
B.3 Projection: Operators
References
Note
C Matrix Decompositions
C.1 Singular Value Decomposition
C.2 QR Decomposition
C.3 LQ Decomposition
References
Index
End User License Agreement
List of Tables
Chapter 2
Table 2.1 Properties of covariance and spectral functions.
Table 2.2 Linear system with random inputs: covariance/spectrum relationship...
Chapter 3
Table 3.1
ARMAX
representation.
Table 3.2
LATTICE
recursions.
Table 3.3 Gauss–Markov representation.
Chapter 4
Table 4.1 Recursive least-squares (scalar) algorithm.
Chapter 5
Table 5.1 MTM LEAKAGE (Window) properties.
Chapter 6
Table 6.1 Ho–Kalman (SVD) realization algorithm.
Table 6.2 Multivariable output error state-space (MOESP) algorithm.
Table 6.3 Numerical algorithm for subspace state-space system identification...
Table 6.4 Deterministic subspace realization: three -mass mechanical syste...
Chapter 7
Table 7.1 Large-scale structure (15-DOF, IR data): modal frequency estimatio...
Chapter 8
Table 8.1 Order estimation methods: statistical tests.
Table 8.2
EMTF
models and validation.
Chapter 9
Table 9.1 MODAL identification/timing statistics: sounding rocket flight se...
Table 9.2 MODAL identification/timing statistics: sounding rocket flight ter...
Table 9.3 Subspace identification: Six (6) modal frequencies.
Table 9.4 Statistical metrics for identification of -mass structural syste...
2
Table B.1 Matrix projections.
List of Illustrations
Chapter 1
Figure 1.1
Deterministic
sinusoidal data: (a) composite signal consisting of...
Figure 1.2
Random
sinusoidal data with disturbances and noise: (a) Composite...
Figure 1.3 Filtering
random
sinusoidal data for decomposition/extraction: (a...
Figure 1.4 Spectral decomposition of noisy ( dB) sinusoidal data: (a) Compo...
Figure 1.5 Spatial spectral decomposition of multiple source data: (a) Multi...
Figure 1.6 Signal processing approach: the model-based “staircase.” [Step 1]...
Figure 1.7 Simple oscillation example. (a) Noisy oscillation (10.54 Hz) in n...
Figure 1.8 Model-based signal processing of a noisy measurement conceptually...
Figure 1.9 Classes of model-based signal processors: (a) Simple measurement ...
Figure 1.10 Model-based signal processing “model” for an RC circuit.
Figure 1.11 The -mass mechanical system under investigation.
Figure 1.12 Decomposition of -mass mechanical system. (a) Composite channel...
Chapter 2
Figure 2.1 Random signal comparison: random signal and spectrum and the dete...
Figure 2.2 Random signal processing: (a) data and spectrum, (b) covariance a...
Figure 2.3 Ensemble of random walk realizations.
Figure 2.4 Sinusoids in noise (...
Chapter 3
Figure 3.1
ARMAX
(2,1,1) simulation: (a) simulated signal, (b) mean propagati...
Figure 3.2 Lattice model.
Chapter 4
Figure 4.1 Processing of a
random
signal and noise: (a) raw data and Fourier...
Figure 4.2 The estimation problem: source, probabilistic transition mechanis...
Figure 4.3 Trajectory motion compensation using the
WLS
estimator: (a) raw d...
Figure 4.4 Orthogonal projection of the optimal signal estimate.
Figure 4.5 Geometric perspective of least squares: (a) projection of point
Chapter 5
Figure 5.1 Signal decomposition approach: data-based (
PCA
,
ICA
,
SSA
), spectr...
Figure 5.2 Impulse response data from 3-mass
MCK
-system: (a) -scatter plot ...
Figure 5.3 Impulse response data from 3-mass
MCK
-system: (a) extracted (sepa...
Figure 5.4 Estimated (
ICA
) individual response data from three-mass
MCK
-syst...
Figure 5.5
SSA
processing of 3-mass
MCK
-system data: (a) decomposed (reconst...
Figure 5.6 Discrete prolate spheroidal (Slepian) sequences (
DPSS
) for mode...
Figure 5.7 Spectral decomposition of 3-mass MCK response data: (a) determini...
Figure 5.8 Damped exponential method of model-based decomposition for determ...
Figure 5.9 Feed forward lattice (all-zero) structure.
Figure 5.10 Model-based lattice decomposition of sinusoidal (5, 10 Hz) data:...
Figure 5.11 Model-based state-space decomposition MCK (3-mass) data: (a) est...
Figure 5.12 Spectral decomposition of harmonic data: (a) deterministic respo...
Figure 5.13 Damped exponential method of model-based decomposition of harmon...
Figure 5.14 Lattice method of model-based decomposition of harmonic data at ...
Figure 5.15 Model-based state-space decomposition of harmonic data at a
SNR
...
Chapter 6
Figure 6.1 Deterministic realization problem: classical approach and subspac...
Figure 6.2
SVD
realization. (a) estimated singular values of the Hankel matr...
Figure 6.3 Orthogonal projection operator: orthogonal complement operation....
Figure 6.4 Oblique projections: projection of future outputs () onto past...
Figure 6.5 Extraction of future state and shifted state sequences: (a) obliq...
Figure 6.6 The -output mechanical system under investigation.
Figure 6.7 Deterministic impulse response and spectrum of -output mechanica...
Figure 6.8
SVD
realization of deterministic -output mechanical system. (a) ...
Figure 6.9 Pseudo-random response of -output mechanical system ( dB). (a) ...
Figure 6.10
SVD
realization of -output mechanical system for random excitat...
Figure 6.11
SVD
realization of -output mechanical system for random excitat...
Chapter 7
Figure 7.1 Basic frequency domain (
BFD
) method: 3-mass problem in noise ( d...
Figure 7.2 Complex mode indicator function (
CMIF
) for 3-mode/3-output
MCK
-sy...
Figure 7.3 Least-square complex frequency domain (
LSCF
) method for the -mod...
Figure 7.4 PolyReference least-square complex frequency domain method for th...
Figure 7.5 Maximum likelihood with PolyReference least-squares complex frequ...
Figure 7.6 Block diagram of large-scale structure (-DOF) interconnections....
Figure 7.7 Impulse response of large-scale structure (15-DOF): (a) determini...
Figure 7.8 FRF estimation for large-scale structure (15-DOF): phase, FRF, co...
Figure 7.9 Spectral estimation for large-scale structure (15-DOF): (a) SVD P...
Figure 7.10 Least-squares complex frequency estimation (
LSCF
) for large-scal...
Figure 7.11 PolyReference least-squares complex frequency estimation (
pLSCF
)...
Figure 7.12 Maximum likelihood with PolyReference least-squares complex freq...
Chapter 8
Figure 8.1 Electromagnetic test facility: signal processing model.
Figure 8.2 Pulser identification: (a) signal error, (b) order, and (c) white...
Figure 8.3 True/identified system: (a) True power spectrum (16 modes). (b) I...
Figure 8.4 Identified system analysis (batch): (a) identified spectrum. (b) ...
Figure 8.5 Identified system modal validation analysis: (a) modal assurance ...
Chapter 9
Figure 9.1 Sounding rocket experiment: (a) Sounding rocket (Terrier-Malemute...
Figure 9.2 CREPE simulation (COMSOL) results for chirp (-to- Hz): (a) CREP...
Figure 9.3 Sounding rocket flight test data: (a) Raw (average) flight data. ...
Figure 9.4 CREPE flight test stage spectra: (a) Extracted stage processed (d...
Figure 9.5 Sounding rocket flight test data results: Average power spectra f...
Figure 9.6 Sounding rocket flight test data for second-stage burn-out and te...
Figure 9.7 Sounding rocket flight test spectra for second-stage burn-out and...
Figure 9.8 Cylindrical object (copper pipe) vibrational response experiment:...
Figure 9.9 Multichannel cylindrical object data: (a) pre-processed tri-axial...
Figure 9.10 Cylindrical object multichannel identification results: (a) iden...
Figure 9.11 Identification of the cylindrical object (pipe) power spectral: ...
Figure 9.12 Subspace identification of cylindrical object, -mode model: (a)...
Figure 9.13 Post-processing of modal frequency tracking (ensemble) statistic...
Figure 9.14 Post-processed modal frequency tracking of cylindrical object da...
Figure 9.15 Resonant ultrasound spectroscopy (RUS): Ultrasonic frequency mea...
Figure 9.16 Resonant ultrasound spectroscopy measurement system and acquired...
Figure 9.17 Simulated (MODEL) solid parallelepiped (RPP) structure frequency...
Figure 9.18 Estimated
MIMO
state-space model for synthesized (MODEL) solid p...
Figure 9.19 RUS experimental (DATA) solid parallelepiped (RPP) structure fre...
Figure 9.20 Estimated
MIMO
state-space model ( modes) of the
RUS
experiment...
Figure 9.21 Structural vibrations of -mode mechanical system (SNR= dB): (a...
Figure 9.22 Structural vibration subspace identification -mode mechanical s...
Figure 9.23 -Mass structural stabilization diagram metric: modal identifica...
Figure 9.24 8-Mass structure mode shapes: (a) shape estimates. (b) modal ass...
Figure 9.25 Structural 8-mass modal response simulation: (a) impulse respons...
Figure 9.26 8-Mass structure shaping filter subsystem design (impulse respon...
Figure 9.27 8-Mass structure shaping filter subsystem estimation (impulse re...
Figure 9.28 -Mass structure shaping filter multichannel subsystem identific...
Figure 9.29 -Mass structure multichannel subsystem decomposition: extracted...
Figure 9.30 Time reversal focusing: After reception of scattered field, the ...
Figure 9.31
T/R
focusing PITCH-CATCH-REVERSE sequence.
Figure 9.32
T/R
focusing pitch-catch-reverse iteration sequences: (a) iterat...
Figure 9.33 Eigen-decomposition selective focusing using the singular value ...
Figure 9.34 Eigen-decomposition selective focusing in aluminum: (a) composit...
1
Figure A.1 Probability mass and distribution functions for coin tossing expe...
2
Figure B.1 Orthogonal and oblique projections: (a) orthogonal projection of
Guide
Cover
Table of Contents
Series Page
Title Page
Copyright
Dedication
About the Author
Preface
Acknowledgments
Glossary
About the Companion Website
Begin Reading
A Probability and Statistics Overview
B Projection Theory
C Matrix Decompositions
Index
WILEY END USER LICENSE AGREEMENT
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IEEE Press445 Hoes LanePiscataway, NJ 08854
IEEE Press Editorial BoardSarah Spurgeon, Editor-in-Chief
Moeness AminJón Atli BenediktssonAdam DrobotJames Duncan
Ekram HossainBrian JohnsonHai LiJames LykeJoydeep Mitra
Desineni Subbaram NaiduTony Q. S. QuekBehzad RazaviThomas RobertazziDiomidis Spinellis
Signal Processing
An Applied Decomposition Approach
James Vincent Candy
Lawrence Livermore National LaboratoryUniversity of California, Santa Barbara, USA
Copyright © 2025 by The Institute of Electrical and Electronics Engineers, Inc.All rights reserved.
Published by John Wiley & Sons, Inc., Hoboken, New Jersey.Published simultaneously in Canada.
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“In Christ and through his blood we have redemption and forgiveness for our sins”(Ephesians: 1:7).
About the Author
James Vincent Candy is the Chief Scientist for Engineering, a Distinguished Member of the Technical Staff and founder/former Director of the Center for Advanced Signal & Image Sciences (CASIS) at the Lawrence Livermore National Laboratory. Educationally, he received his BSEE degree from the University of Cincinnati and his MSE and PhD degrees in Electrical Engineering from the University of Florida. He is a registered professional Control System Engineer in the state of California. He is an Adjunct Full-Professor at the University of California, Santa Barbara. Dr. Candy is a Life-Fellow of the IEEE “for contributions to model-based ocean acoustic signal processing” and a 25Year-Fellow of the Acoustical Society of America (ASA) “for contributions to model-based acoustic signal processing.” He was elected as a Life Member (Fellow) at the University of Cambridge (Clare Hall College). He is a member of Eta Kappa Nu and Phi Kappa Phi honorary societies. Dr. Candy received the IEEE Distinguished Technical Achievement Award for the “development of model-based signal processing in ocean acoustics.” He is an IEEE Distinguished Lecturer for oceanic signal processing. Dr. Candy has been awarded the Interdisciplinary Helmholtz-Rayleigh Silver Medal in Signal Processing/Underwater Acoustics by the Acoustical Society of America. He received the R&D100 award for his innovative invention in radiation threat detection. He has published over 250 journal articles, book chapters, and technical reports as well as written six texts in signal processing, “Signal Processing: the Model-Based Approach,” (McGraw-Hill, 1986) and “Signal Processing: the Modern Approach,” (McGraw-Hill, 1988), “Model-Based Signal Processing,” (Wiley/IEEE Press, 2006), “Bayesian Signal Processing: Classical, Modern and Particle Filtering” (Wiley/IEEE Press, 2009), “Bayesian Signal Processing: Classical, Modern and Particle Filtering, 2nd Ed.” (Wiley/IEEE Press, 2016), and “Model-Based Processing: An Applied Subspace Identification Approach” (Wiley/IEEE Press, 2019). He was the IEEE Chair of the Technical Committee on “Signal and Image Processing” and was the Chair of the ASA Technical Committee on “Signal Processing in Acoustics.” He was elected to the Administrative Committee of IEEE OES. His research interests include Bayesian learning, signal decomposition, estimation, identification, spatial estimation, signal and image processing, array signal processing, nonlinear signal processing, structural signal processing, tomography, sonar/radar processing and biomedical applications.
Dr. Candy is considered a subject matter expert in signal processing, estimation and detection with a career extending over 48 years. He is an IEEE Distinguished Lecturer and has presented over 50 short courses in statistical signal processing at universities, conferences, businesses, laboratories, etc.
Preface
This text is focused on the decomposition of complex signals into independent sub-signals or equivalently components enabling subsequent analysis and design. Decomposing a signal into its constituent components not only enables an effective mechanism for analysis, but also provides a means to process each individual component more effectively, while mitigating disturbances and noise [1–4]. Decomposition techniques have evolved historically as the Fourier decomposition of a signal and/or system into sinusoidal components for subsequent frequency domain analysis, to statistical decompositions in classifications, to modal decompositions in structural analysis, to wavelet decompositions providing a novel domain for subsequent analysis of variable signals [1, 5–7]. This text is focused on such a decomposition essentially decoupling signals and their underlying systems into more manageable independent sub-signals or subsystems enabling effective analysis and subsequent designs [8]. What motivates this approach are the large number of available numerical methods capable of performing such a decomposition in a robust and efficient manner [9, 10]. Perhaps the most prominent method is the singular value decomposition (SVD), well known for its robust methodology of detecting the rank of a matrix that has evolved as the “go-to” solution ranging from eigen-decompositions to provide reduced rank filtering of noisy data [9–15]. This method has been applied to many problems with great success ranging from the usual tracking problem in radar/sonar, to decomposing images into subimages in optical processing, to decomposing transient speech signals in voice recognition, to medical application in cardiac as well as neurological signals [16–18]. In any case this method is the premier workhorse in signal processing applications that will carefully be interwoven throughout each approach developed within this text.
For instance, acoustical response data encompasses a wide variety of signal classes ranging from sharp transients similar to seismic events or speech to muffled sinusoids in oceans or structural vibrations [6, 17]. Distinct resonant frequency peaks evolving from the measured material vibrations of resonant ultrasound, to step-frequency excitations of vibrating systems enable the evaluation of structural properties critical for the detection of potential failures [19, 20]. Control systems have often depended on somehow decomposing the underlying complex system into more manageable subsystems decoupled for both analysis and design of effective controllers [10, 21]. The response of a large bridge or building with complex and coupled substructures also depends on their decomposition into independent subsystems with their superposed responses producing the resulting measurement data. Biological signals also exhibit this complexity while they evolve from the well-understood heart beat (EKG) to the complex evoked responses of the neurological process (EEG) [18]. In all of these applications, the analysis, design, detection, and information extraction depend heavily on one significant connection–the decomposition of the signal or system into independent subsystems that can easily be evaluated and modified to achieve a specific objective (estimation, detection, control, modification, etc.) [22]. This decomposition is well known mathematically based on simple linear algebraic techniques such as diagonalizing a complex matrix by performing an eigen-decomposition [9, 11]. For mechanical systems producing complicated vibrational responses, this decomposition is called a modal decomposition where the independent modes of the system determined its measured response, while in controls, the overall plant or system is decomposed mathematically into a decoupled set of subsystems for both centralized or decentralized designs [23–25]. This decomposition usually evolves again in the form of an eigen-decomposition or the more robust SVD revealing the underlying modes dominating the composite system signal or frequency response enabling the extraction of prominent features for such applications as estimation, identification, detection, and classification [22, 26, 27]. In any case, the underlying thread that is woven throughout the “signal space” is that of the modal description of the measured data in one form or another [28]. Mechanical (control) engineers have long known about these decomposition approaches as well as signal processors in terms of modal decompositions in which a system is decomposed for analysis and design and then reconstructed through transformations (similarity) to return to the physical coordinates governing its operation [23, 25, 29].
Perhaps the mathematical notions of orthogonality and independence are two concepts that imply “decomposition” especially in the signal processing jargon. The application of techniques to decompose a signal in terms of its individual constituent components enables not only a better understanding both from a mathematical perspective, but also from a physical perspective for analysis and design [4, 5, 30–32]. It is this perspective that can lead to the extraction of crucial information directly from uncertain measured data sets–the primary objective of signal decomposition [33]. Applying this approach leads to immediate and effective as well as significantly robust techniques with a multitude of applications in tracking, estimation, detection, classification, and many other applications [26, 27, 29]. In fact, the classification of various signals depends heavily on the ability to rank the importance of features and particularly their inherent independence as in a multitude of pattern recognition applications [27]. The detection problem is quite challenging in a highly reverberant environment, but can be attacked more readily when signal decomposition techniques are applied leading to the design of decentralized rather than centralized detection solutions [34].
The essential engine that drives the solution to the signal decomposition problem is the SVD of the underlying “data matrix” that evolves directly from measurements or simulation [35, 36]. This matrix appears in a variety of forms relating the “data” directly to the underlying physical system creating it. For example, pure measurement data is produced by sensors through an acquisition system directly or can take the form of correlation estimates, frequency spectra, wavelet estimates, image pixels, and power spectra [14–18]. These data are typically placed into the data matrix that is processed by the algorithm designed to extract the desired information. The matrix can be arranged into various structures depending on the generating physical system or model. For instance, in system identification, that is, constructing a parametric model from noisy data, a Hankel or Toeplitz matrix or both are operated on by the SVD to extract the underlying model, while spectral data can be arranged into a frequency response matrix enabling the extraction of resonant frequencies [36, 37]. In any case, it is the application of the SVD or its counterparts to these matrices that enable the signal decomposition to be successfully performed [37].
Processing structural system data is characterized not just by the usual uncertainties created by environmental disturbances and noise, but also by their underlying complexities resulting from multichannel measurements and coupled responses [6]. State-space techniques have demonstrated the capability of unraveling these complexities due to their physics-based designs while mitigating these uncertainties primarily because of their robust performance all related to one primary linear algebraic methodology–the matrix SVD [17, 36]. This approach has successfully been applied to solving a wide variety of signal processing problems ranging from simple signal enhancement [4, 17, 36], to multichannel spectral estimation and array signal processing [22], anomaly detection and to even the more challenging problem of extracting a coupled structural response from noisy vibrational measurements [26]. The SVD provides the essential engine that drives these state-space processors especially when extracting the structural response of an uncertain vibrational system. In this text, the decomposition approach to signal processing is motivated by the decomposition of response data from a so-called “structural system model” [25]. Such a system is typically a civil engineered structure like a building, bridge, or machinery or even decomposing structured material all induced to vibrate revealing their internal structural properties in the form of modal responses [17, 20]. Similar to modal control systems of system theory, the modes are individual features of the structure [8, 21, 23, 25]. It is the task of the signal processor or the structural engineer or the analyst to extract this critical information from a vibrating structure to design a stable, efficient structure. Much effort has been accomplished in “operational modal analysis” (OMA) to provide a mechanical system decomposed (mathematically) into a decoupled set of subsystems for decentralized control [6, 17, 23–25]. A wealth of techniques has been developed to perform the extraction task for exciting structures, while the design task has not been explained as strongly, but evolves from the control systems perspective [25]. The processing perspective of this text is to focus on the extraction of modal information to analyze complex systems providing the necessary information to evaluate its performance. This approach can lead directly to the evaluation of other mechanical systems such as motors, automobiles, and self-driving vehicles that could potentially fail due to the unforeseen or unpredictable response to vibrations, disturbances, and noise [22, 26].
The objective of this text is aimed at simplifying the analysis of complex signals (systems) by performing a signal decomposition into constituent components that combine to capture the uncertain measured response(s). This approach, although well documented, has not been sufficiently provided to the signal processing and other technical communities because it is somewhat veiled by the complexity of the underlying mathematics along with complex applications. This material is strongly motivated, mathematically, by the modal decomposition methodology evolving from control systems and more specifically the structural and civil engineering technical communities incorporating signal processing applications produced within [6, 17, 36]. It has been this pigeon-holed effect that has confined this information and preventing a more general approach to be made available to the overall technical communities. This model-based decomposition approach is presented primarily as an advanced undergraduate or first-year graduate text with applications and data in both the time and frequency domain. These data and subsequent algorithms (in MATLAB®) have not been made readily available by the communities in a single text, while some attempts have been made but are unfortunately embedded within specialized application areas (structures, materials, etc.) [6, 16, 17, 20]. In this text, the model-based signal decomposition approach is presented that is directly motivated by deterministic as well as random signals leading to fundamental transformation techniques that are more easily comprehended within this framework.
The model-based perspective taken in this text is to introduce the concept of signal decomposition by demonstrating how each of the major approaches decomposes a complex (composite) signal or system into a set of its independent constituent components. Starting with the well-known Fourier decomposition followed by the SVD applied to both time series (signals) and frequency spectra and subsequently the statistical approach based on principal/independent component theory are incorporated to elucidate the concepts of dimensionality reduction as well as simplification and processing leading to a full exposure of this approach [27]. In the introduction, a simple sinusoidal system is simply developed to elucidate the approach initially through its Fourier decomposition followed by the SVD representation terminating with the underlying principal component decomposition. This system, although simple, is rich enough to motivate the entire text demonstrating not only how a signal can be decomposed effectively, but also a system into subsystems leading to eventual analysis and design.
This proposed text is designed primarily as a graduate-level text; however, it will prove useful to practicing signal processing professionals and scientists, since a wide variety of case studies are included to demonstrate the applicability of the decomposition approach to real-world problems. The prerequisite for such a text is a melding of undergraduate work in linear algebra (especially matrix decomposition methods), random processes, linear systems, and some basic digital signal processing. It is somewhat unique in the sense that many texts cover some of its topics in piecemeal fashion, but limit exposure to frequency domain approaches especially without the notion of signal/spectral decompositions. The underlying approach of this text is the thread that is embedded throughout in the algorithms, examples, applications, and case studies. It is an embedded model-based theme, together with the developed hierarchy of physics-based models, that contributes to its uniqueness coupled with SVD methods that even enable potential real-time methods to become a reality. This text has evolved from five previous texts and has been broadened by a wealth of practical applications to real-world problems [27, 34, 36–53]. The introduction of robust SVD methods for model-building and signal/system decomposition that have been available in the literature for quite a while, but require more of a systems theoretical background to comprehend. After introducing these fundamental concepts in the first chapter, random signals and systems are briefly discussed leading to the concept of spectral estimation. Once developed, the parametric estimators lead directly into the concept of subspaces and the decomposition of the data space into the signal and noise subspaces evolving to the well-known multiple signal classification (MUSIC) spectral estimator [28, 36].
The third chapter is focused on representing a signal in a nonparametric as well as parametric constructs. Nonparametrically, the usual time-series representation of a signal can take the structure of a set of coefficients or weights with specific properties (e.g. orthogonality) or simply a polynomial-in-time, while its frequency domain representation offers a variety of structures ranging from the usual Fourier decomposition to the statistical approach of component theory. Random signals can be converted or transformed to a deterministic form using correlation and power spectra replacing time and frequency domain characterizations. Parametric representations offer an even larger variety of constructs ranging from the fundamental transfer function along with its statistical characterization to capture noise and uncertainties with the autoregressive family along with related exponential/sinusoidal models leading to the decomposed modal model. Parametric physics-based models evolve that are captured by state-space models that are easily transformed to any of the previous representations as well as enabling both single to multichannel characterizations. Signal decomposition approaches, the heart of this text, are developed in the fourth chapter. Here, starting with the so-called “eigen-signals” that are based on statistical projection theory and shown to easily be extracted from both deterministic and random data where the basics are developed. The usual Fourier modal and principal component decompositions are also briefly discussed to offer some viable alternatives to the fundamental approach completing the chapter. Chapter 5 provides the heart of signal/system decomposition methods starting with data-based methods followed by spectral-based techniques and concluded with model-based decomposition.
Chapter 6 develops the “subspace” approach for generic state-space systems with the underlying system theoretical properties that enable the application of many algorithms to capture physics-based data [36]. It is here that both the nonparametric and parametric techniques are married to show equivalence and compare performance. Modal examples are discussed along with actual application data.
The sixth chapter provides the technical breath of this text in the time domain, briefly developing the time domain and frequency domain approaches to processing data. It is here that signal decomposition techniques are carefully developed and applied to both simulation and application data sets. Starting with the usual preprocessing of data based on a priori knowledge, both noise and disturbances are mitigated to improve the overall signal-to-noise ratio (SNR). Two of the most popular and robust techniques, N4SID and MOESP, are carefully developed based on projection theory and shown how the SVD plays a significant role in their implementation. The well-known prediction error method (PEM) is also briefly discussed to illustrate the usual approach to decomposing the signal as well.
Next, in chapter seven not as well known, frequency domain techniques are developed starting with the simplest basic frequency domain (BFD) method of peak-picking the estimated frequency response function (FRF) data and evolving to the more sophisticated least squares-based methods completing the chapter. Both the least-squares frequency domain (LSFD) method as well as the polyreference (P-LSFD) approaches to resonant frequency estimation are developed leading to the sophisticated iterative maximum likelihood method (ML-LSFD). Here, the complex mode indicator function (CMIF) is developed providing a viable technique to extract modal resonant frequencies from measured spectra–again based on the SVD methodology to decompose the FRF matrix.
Chapter eight focuses on the performance analysis of the signal decomposition techniques based on statistical methods and many of the techniques inherited from modal analysis methodologies. This performance analysis metric coupled with the well-known modal assurance criterion (MAC) enables a complete extraction of a decoupled signal and underlying system.
Finally, the text is completed with a set of real-world applications and simulations briefly developed to reinforce the signal decomposition approach developed throughout the text. The applications include: the response evolving from rocket flight dynamics, a simple vibrating cylinder (pipe) experiment, followed by material property extraction using resonant ultrasound spectroscopy, to the analysis of an 8-story (mass) structure and its subsystem decomposition, concluding with the time-reversal decomposition of a spatiotemporal signal for flaw detection in nondestructive evaluation.
Appendices are included for critical review as well as problem sets and notes for the MATLAB software used in the signal processing/controls/identification areas at the end of each chapter.
12 August 2022Danville, CA
James Vincent Candy
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Acknowledgments
My wife, Patricia, is the major motivational element needed to undertake this endeavor through her support and encouragement. My family, extended family, and friends having endured the many regrets, but still offer encouragement in spite of all of my excuses. Of course, the constant support of my great colleagues and friends, especially Mr. B. Beauchamp, Dr. K. Fisher, and Dr. S. Lehman who carefully reviewed the manuscript and suggested many improvements, cannot go without a acknowledgment.
Glossary
AIC
Akaike information criterion
AR
autoregressive (model)
ARMA
autoregressive moving average (model)
ARMAX
autoregressive moving average exogenous input (model)
ARX
autoregressive exogenous input (model)
BFD
basic frequency domain (method)
BSP
Bayesian signal processing
BW
bandwidth
CC-T
correlation coefficient statistical test
CD
central difference
CDF
cumulative distribution function
CDM
common denominator model
CM
conditional mean
CMIF
complex mode indicator function
CRLB
Cramér–Rao lower bound
C-Sq
Chi-squared (distribution or test)
CT
continuous-time
DPSS
discrete prolate spheroid sequences
DExp
damped exponential decomposition
DFT
discrete Fourier transform
DOF
degrees of freedom
DtFT
discrete-time Fourier transform
EMA
experimental modal analysis
FDD
frequency domain decomposition (method)
FFT
fast Fourier transform
FIR
finite impulse response
FPE
final prediction error
FRF
frequency response function
GLRT
generalized likelihood-ratio test
G-M
Gaussian mixture
GM
Gauss Markov
G–S
Gram–Schmidt (orthogonal decomposition)
HD
Hellinger distance
ICA
independent component analysis
IDtFT
inverse discrete-time Fourier transform
IFFT
inverse fast Fourier transform
i.i.d.
independent-identically distributed (samples)
IIR
infinite impulse response
IRF
impulse response function
KD
Kullback divergence
KL
Kullback Leibler
KLD
Kullback–Leibler divergence
LD
lower-diagonal (matrix) decomposition
LE
Lyapunov equation
LKF
linear Kalman filter
LMFD
left matrix fraction description (system)
LMS
least mean square
LR
lower residual (terms)
LS
least-squares
LSCD
least-squares common denominator (method)
LSFD
least-squares frequency domain (method)
LSCF
least-squares complex frequency (method)
LTI
linear time-invariant (system)
MA
moving average (model)
MAC
modal assurance criterion
MAD
median absolute deviation (statistic)
MAICE
minimum Akaike information criterion
MAP
maximum a posteriori
MATLAB®
mathematical software package
MBID
model-based identification
MBP
model-based processor
MBSP
model-based signal processing
MC
Monte Carlo
MCK
mass damper spring (system)
MDL
minimum description length
MDOF
multiple degree of freedom (system)
MIMO
multiple input/multiple output (system)
MinE
minimum probability-of-error
MNORM
minimum norm (spectral estimator)
ML
maximum likelihood
MOC
modal observability criterion
ML-LSCF
maximum likelihood polyreference least-squares complex frequency (method)
MOESP
multivariable output error state-space algorithm
MMSE
minimum mean-squared error
MSE
mean-squared error
MSV
modal singular value criterion
MTM
multiple taper method (spectral estimator)
MV
minimum variance
MVDR
minimum variance distortionless response (spectral estimator)
MUSIC
multiple signal classification (spectral estimator)
N4SID
numerical algorithm for subspace state-space system identification
NMSE
normalized mean-squared error
OMA
operational modal analysis
PCA
principal component analysis
PHD
Pisarenko harmonic decomposition (spectral estimator)
pLSCF
polyreference least-squares complex frequency (method)
probability density function (continuous)
PEM
prediction error method
PID
proportional integral derivative (control)
PI-MOESP
past-input multivariable output error state-space algorithm
PMF
probability mass function (discrete)
PO-MOESP
past-output multivariable output error state-space algorithm
PSD
power spectral density
RC
resistor capacitor (circuit)
RLC
resistor–inductor–capacitor (circuit)
RLS
recursive least-squares
RMFD
right matrix fraction description (system)
RMS
root mean squared
RMSE
root minimum mean-squared error
RPE
recursive prediction error
RPEM
recursive prediction error method
RUS
resonant ultrasound spectroscopy
SDIAG
stabilization diagram
SDOF
single degree of freedom (system)
SID
subspace identification
SISO
single input/single output (system)
SNR
signal-to-noise ratio
SSA
singular spectral analysis
SSP
state-space processor decomposition
SSQE
sum-squared error
SVD
singular value (matrix) decomposition
UD
upper-diagonal matrix decomposition
UR
upper residual (terms)
WPM
Welch periodogram method
WSSR
weighted sum-squared residual statistical test
W-Test
whiteness statistical test
Z
Z
-transform
Z-M
zero-mean statistical test
About the Companion Website
This book is accompanied by a companion website:
www.wiley.com/go/CandySignProc
The website includes:
Instructor Solutions Manual
1Introduction
1.1 Background
When we are confronted by a problem, simple or complex, we tend to decompose it in order to achieve a solution. For instance, if the lights go out in our room, we first ask ourselves, if the switch is still on; if on, we next search for a fuse box and if unsuccessful, we proceed to question whether the actual power source is available. In more complex scenario, if our car stops unexpectedly while driving, we first check the fuel gauge, then the battery, and then search various components of the engine based on telltale sounds that may have occurred. These are problems that occur in everyday life and exemplify the concept of decomposition. Engineers and scientists “decompose” complex signal/system problems into separate components in order to comprehend their superposition or overall performance. Thus, the understanding of signal components or subsystems is necessary in order to analyze or modify complex signals or systems. In this text, we develop techniques that enable us to perform this decomposition in order to simplify the signal/system into more comprehensible components. For example, in chemistry, when presented with an unknown substance to be analyzed, a chemist will perform the analysis by applying mass spectrometry to decompose it into constituent components. In criminal forensics, investigators decompose the crime scene to determine the cause, weapons, to piece together a scenario that will lead to the identification of the perpetrators. In energy science, scientists decompose nuclear reactions into constituent components to achieve higher energy yields. Control/structural engineers intensely study second-order system performance that can be used to evaluate complex systems like motors or bridges or building responses based on decomposing them into these simpler subsystem responses. Given noisy, uncertain measurement data, signal processors perform routine spectral analysis to reveal constituent components in order to separate signals from the noise. Applications like extracting the pregnant mother’s heart beats from those of the fetus are achieved by decomposition techniques. Complex detection problems can also be solved by techniques such as decentralized detector design by decomposing the problem into local detectors reporting to a central processor. All-in-all, the decomposition approach to problem-solving from the simple to the complex is routine, often going unnoticed, in everyday life as well as in sophisticated scientific engineering applications. This text presents “decomposition” from the signal/system perspective.
We must first decide on the essence of the signal itself. Is it deterministic or random? A deterministic signal is characterized by its repeatability, that is, it is free from variations and repeatable from measurement to measurement, while if it varies extraneously, then it is no longer repeatable and it is defined simply as a random signal. Here we are concerned with the development of processing techniques to extract the pertinent information from random signals using any a priori information available. These techniques are called signal estimation with enhancement achieved by applying a particular algorithm called a signal estimator or just estimator that is annotated by a “.” We start with a simple example.
Example 1.1 Suppose we are given sinusoidal data, , sampled at approximately 0.1 sec depicted in Figure 1.1a, and wish to decompose it by separating the signal, , from any disturbances, , and noise, .
That is, we would like to analyze its components that represent the desired signal to be extracted along with undesirable disturbances to be removed along with inherent measurement (numerical) noise. The first step a signal processor would pursue is to perform an analysis of the frequency spectrum as shown in Figure 1.1b. The signal is assumed to be deterministic (nonrandom) ignoring any uncertainties. Performing a discrete Fourier transform (DFT) on the data reveals spectral peaks corresponding to the sinusoids,
It is clear from the spectrum that resonances exist at three temporal frequencies of 0.1225, 0.3228, and 0.496 Hz. If we were analytically aware (a priori) that the signal was composed of three sinusoids
the sinusoidal transform relation assuming zero phases () is given by
Figure 1.1Deterministic sinusoidal data: (a) composite signal consisting of three sinusoids at frequencies of , , and Hz, zero phase at a large SNR ( dB). (b) Composite Fourier spectrum illustrating three sinusoidal peaks consisting of two disturbances and the actual signal (thick line) along with numerical noise.
and we expect to observe perfect spectral lines (impulses) located at these specified frequencies. However, since there is leakage in the discrete Fourier spectral estimation created by truncating the data abruptly, a small DC-component appears creating a smearing of the peaks that result in a spectral width yet still providing correct peak frequencies. So we see ideally that the Fourier transform decomposes the composite signal of superposed sinusoids into the three spectral peaks enabling us to analyze its composition and consider separating them into its constituent components.
Next, the signal processor may know (a priori) from the underlying physics of the problem that generated this data that the actual signal is located between 0.15 and 0.4 Hz, while a pair of disturbances, caused by local signals lie outside that band, that is,
Example 1.2 Consider the same sinusoidal data of Example 1.1 contaminated by instrumentation noise and disturbances at a 0 dB signal-to-noise ratio (SNR). The noisy data and corresponding spectrum are shown in Figure 1.2a,b. Clearly from the figure, the corresponding spectral (Fourier) data still indicate the presence of three sinusoidal peaks as before; however, the measurement instrumentation noise has distorted their locations somewhat at frequencies of 0.1219, 0.3450, and 0.4960 Hz instead of the true frequencies.
1.2 Spectral Decomposition
Simply treating these data as stochastic requires an improved spectral estimator to extract the peaks leading to the basis of power spectral density (PSD) estimation rather than simple Fourier (FFT) spectra. Perhaps the simplest spectral estimator follows by treating the data as a random signal. Techniques similar to deterministic theory hold when the random signal is transformed to its correlation sequence and its Fourier spectrum is transformed to its power spectrum. We know that the correlation sequence and power spectrum form a DFT pair, analogous to a deterministic signal and its corresponding spectrum, that is, we have that
and as in the deterministic case, we can analyze the spectral content of a random signal by investigating its power spectrum. The PSD function for a discrete random process, is defined as:
Figure 1.2Random sinusoidal data with disturbances and noise: (a) Composite signal consisting of three sinusoids at frequencies of , , and Hz. (b) Composite Fourier spectrum illustrating three noisy sinusoidal peaks at frequencies of , , and Hz consisting of two disturbances and the actual signal (thick line) along with numerical noise.
where is the complex conjugate. The expected value operation can be thought of as “mitigating” the randomness. Similarly, the correlation (at lag ) is
Recall that the DFT pair is defined by
for .
Therefore, for , a random signal, we have that the correlation-power spectrum pair is (Wiener–Khintchine theorem[1])
With the a priori information available about the signal and disturbance spectra, it is possible for the signal processor to perform a simple extraction by applying bandpass filters illustrating how the uncertain composite signal can be decomposed into its constituent components as previously performed by the Fourier decomposition (DFT) of the deterministic sinusoidal data of Figure 1.1. We illustrate this application in the following example.
Example 1.3 Consider the noisy composite data of Example 1.2 with measurements acquired at a 0 dB SNR shown in Figure 1.2a. Bandpass filters are designed by the processor to extract the desired signal at 0.345 Hz rejecting disturbances at 0.1225 and 0.4960 Hz. The 256-weight finite impulse response (FIR) filters are designed to operate within the respective frequency bands of (0.025–0.15) Hz, (0.2–0.4) Hz, and (0.4–0.5) Hz. The results are shown in Figure 1.3 for each band.
This example illustrates the fundamental concepts of decomposing a composite noisy measurement into its constituent components using any a priori information to perform a simple decomposition and extraction of the desired signal. Next, we consider more sophisticated methods of achieving such a decomposition.
1.3 Data Decomposition
Data are acquired in a variety of domains depending on the particular application and hardware available. In this text, we will consider the usual time domain data (time series), frequency
