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...