The METU Gait Analysis System employs a computer program called Kiss-GAIT for the calculation of joint angles, moments and powers using force plate data and marker trajectories as input. Kiss-GAIT was developed using Delphi and is confined to calculations related to the standard gait protocol. Because the code lacks the flexibility required to carry out various test cases, the inverse dynamics formulation being used could not be verified and the extent of the error propagation problem could not be determined so far. The first aim of this study was to develop a code for the inverse dynamics model of the METU Gait Analysis System making use of the flexible programming environment provided by MATLAB. Verified and more reliable analysis results, obtained by reformulating the inverse dynamics algorithm in a new code, are presented. Secondly, data smoothing and differentiation techniques conventionally used in gait analysis were critically reviewed. A common tool used for filtering marker trajectories is the Butterworth digital filter. This thesis presents a modified, adaptive version of this classical tool that can handle non-stationary signals owing to its coefficients which are functions of local signal structure. The results of this thesis indicate the dominancy of ground reactions as compared to inertial effects in normal human gait. This implies that the accuracy needed in body segment inertial parameter estimation is not a critical factor. On the other hand, marker trajectories must be as accurate as possible for meaningful kinetic patterns. While any smoothing and differentiation routine that produces reasonable estimates is sufficient for joint moment calculation purposes, the estimation performance becomes a key requirement for the calculation of joint powers.

Smoothing is an important part of the pre-processing step in Signal Processing. A signal, which is purified from noise as much as possible, is necessary to achieve our aim. There are many smoothing algorithms which give good result on a stationary data, but these smoothing algorithms don&rsquo / t give expected result in a non-stationary data. Studying Acceleration data is an effective method to see whether the smoothing is successful or not. The small part of the noise that takes place in the Displacement data will affect our Acceleration data, which are obtained by taking the second derivative of the Displacement data, severely. In this thesis, some linear and non-linear smoothing algorithms will be analyzed in a non-stationary dataset.

Smoothing is an important part of the pre-processing step in Signal Processing. A signal, which is purified from noise as much as possible, is necessary to achieve our aim. There are many smoothing algorithms which give good result on a stationary data, but these smoothing algorithms don&rsquo / t give expected result in a non-stationary data. Studying Acceleration data is an effective method to see whether the smoothing is successful or not. The small part of the noise that takes place in the Displacement data will affect our Acceleration data, which are obtained by taking the second derivative of the Displacement data, severely. In this thesis, some linear and non-linear smoothing algorithms will be analyzed in a non-stationary data set.

Non-stationary signals, particularly frequency modulated (FM) signals which arecharacterized by their time-varying instantaneous frequencies (IFs), are fundamental to radar, sonar, radio astronomy, biomedical applications, image processing, speech processing, and wireless communications. Time-frequency (TF) analyses of such signals provide two-dimensional mapping of time-domain signals, and thus are regarded as the most preferred technique for detection, parameter estimation, analysis and utilization of such signals. In practice, these signals are often received with compressed measurements as a result of either missing samples, irregular samplings, or intentional under-sampling of the signals. These compressed measurements induce undesired noise-like artifacts in the TF representations (TFRs) of such signals. Compared to random missing data, burst missing samples present a more realistic, yet a more challenging, scenario for signal detection and parameter estimation through robust TFRs. In this dissertation, we investigated the effects of burst missing samples on different joint-variable domain representations in detail. Conventional TFRs are not designed to deal with such compressed observations. On the other hand, sparsity of such non-stationary signals in the TF domain facilitates utilization of sparse reconstruction-based methods. The limitations of conventional TF approaches and the sparsity of non-stationary signals in TF domain motivated us to develop effective TF analysis techniques that enable improved IF estimation of such signals with high resolution, mitigate undesired effects of cross terms and artifacts and achieve highly concentrated robust TFRs, which is the goal of this dissertation. In this dissertation, we developed several TF analysis techniques that achieved the aforementioned objectives. The developed methods are mainly classified into two three broad categories: iterative missing data recovery, adaptive local filtering based TF approach, and signal stationarization-based approaches. In the first category, we recovered the missing data in the instantaneous auto-correlation function (IAF) domain in conjunction with signal-adaptive TF kernels that are adopted to mitigate undesired cross-terms and preserve desired auto-terms. In these approaches, we took advantage of the fact that such non-stationary signals become stationary in the IAF domain at each time instant. In the second category, we developed a novel adaptive local filtering-based TF approach that involves local peak detection and filtering of TFRs within a window of a specified length at each time instant. The threshold for each local TF segment is adapted based on the local maximum values of the signal within that segment. This approach offers low-complexity, and is particularly useful for multi-component signals with distinct amplitude levels. Finally, we developed knowledge-based TFRs based on signal stationarization and demonstrated the effectiveness of the proposed TF techniques in high-resolution Doppler analysis of multipath over-the-horizon radar (OTHR) signals. This is an effective technique that enables improved target parameter estimation in OTHR operations. However, due to high proximity of these Doppler signatures in TF domain, their separation poses a challenging problem. By utilizing signal self-stationarization and ensuring IF continuity, the developed approaches show excellent performance to handle multiple signal components with variations in their amplitude levels. / Electrical and Computer Engineering

Les ondes acoustiques subissent peu de dispersion dans le milieu marin, comparé au milieu aérien. Certaines espèces de cétacés communiquent ainsi à grande distance, d'autres utilisent leurs émissions sonores pour s'orienter. La bioacoustique consiste à étudier ces espèces à partir de l'analyse de leurs sons, c'est-à-dire à les détecter, classer, localiser. Cela peut se faire via un réseau d'hydrophones au déploiement fastidieux. Afin de contribuer au passage à l'échelle de la bioacoustique, cette thèse propose des modèles originaux mono-hydrophone pour l'analyse de ces signaux stationnaires ou transitoires. Premièrement, nous dérivons un nouveau modèle d'estimation de la distance entre une source impulsive (ex. biosonar) et un hydrophone. Notre modèle théorique, l'Intra Spectral ATténuation(ISAT), dérive des lois acoustiques de déformation spectrale du signal transitoire induite par l'atténuation durant sa propagation. Ce modèle relie les rapports énergétiques des bandes de fréquences pondérés par un modèle de perte par atténuation fréquentielle (Thorp ou Leroy) à la distance de propagation. Nous approximons aussi ISAT par un modèle neuromimétique. Ces deux modèles sont validés sur le sonar du cachalot (Physeter macrocephalus) enregistré avec notre bouée acoustique autonome BOMBYX et notre système d'acquisition DECAV en collaboration avec le Parc National de Port-Cros et le sanctuaire Pelagos pour la protection des mammifères marins en Méditerranée. Les mesures d'erreur (RMSE) d'environ 500 mètres sur nos références du centre d'essai OTAN aux Bahamas présentent un intérêt opérationnel. Deuxièmement, nous proposons une analyse originale de l'évolution des voisements de cétacé par codage parcimonieux. Notre encodage des cepstres par apprentissage non supervisé d'un dictionnaire met en évidence l'évolution temporelle des bigrammes des chants que les baleines à bosse mâles émettent durant la période de reproduction. Nous validons ce modèle sur nos enregistrements du canal de Sainte-Marie à Madagascar entre 2008 et 2014, via notre réseau d'hydrophones BAOBAB qui constitue une première dans l'Océan Indien. Nos modèles s'inscrivent dans le projet Scaled Bioacoustics (SABIOD, MI CNRS) et ouvrent de nouvelles perspectives pour les passages à l'échelle temporelle et spatiale de la bioacoustique. / Acoustic waves show low dispersion due to the underwater propagation, compared to the propagation in the air. Some species of cetaceans communicate at long distance, others use their sound production for orientation. The goal of the scientic area called bioacoustics is to study animal species based on the analysis of their emitted sound. Their sounds can be used to detect, to classify and to locate the cetaceans. Recordings can be done with an passive acoustic array of multiple hydrophones, but this method is expensive and difficult to deploy. Thus, in order to scale this approach, we propose in this Phd thesis several original single hydrophone models to analyze these stationary or transient signals.Firstly, we provide a new theoretical model to estimate the distance between the impulsive source (ex. biosonar of the cetacean) and the hydrophone. Our model, the Intra Spectral ATtenuation (ISAT), is based on the spectral signal alteration due to the underwater acoustic propagation, especially the differences in different frequency bands. We also approximated ISAT by an artificial neural network. Both models are validated on clicks emitted by sperm whales (Physeter macrocephalus) recorded by our sonobuoy BOMBYX and our data-acquisition system DECAV developed incollaboration with the National Park of Port-Cros (France) and the Pelagos sanctuary for the protection of marine mammals in the Mediterranean sea (France). The error (RMSE) measures on the recordings of the NATO test center in the Bahamas are about500 meters, promising further real applications. Secondly, we worked on the variations of the cetacean vocalizations using the sparse coding method. The encoding of thecepstrums by unsupervised learning of a dictionary shows bigrammic time changes of the songs of humpback whales (Megaptera novaeangliae). We validate this model on signals recorded in the Ste Marie Channel (Madagascar) between 2008 and 2014, through our network of hydrophones BAOBAB which is the first passive acoustic array deployed in the Indian Ocean.Our models are part of the Saled Bioacoustics project (SABIOD, MI CNRS) and open perspectives for temporal and spatial scaling of bioacoustics.

Non-stationary signals, that is signals with time-varying (TV) statistical properties, are commonly encountered in engineering practice. The vibration responses of structures, such as traffic-excited bridges, robotic devices, rotating machinery, and so on, constitute typical examples of non-stationary signals. Structures characterized by properties that vary with time are generally referred as TV structures and their vibration-based identification under normal operating conditions is a significant and challenging problem. An important class of parametric methods for the solution of this problem is based on Functional Series Time-dependent AutoRegressive Moving Average (FS-TARMA) models. These models have parameters that explicitly depend on time, with the dependence described by deterministic functions belonging to specific functional sub-spaces. The focus of the present thesis is on the development of complete and advanced FS-TARMA methods that will offer important improvements in overcoming drawbacks of existent methods and will further foster practical use and application of FS-TARMA models in non-stationary vibration analysis. The specific objectives of the thesis are: a) The introduction of a novel class of Adaptable FS-TARMA (AFS-TARMA) models and the development of a method for their effective identification. AFS-TARMA models are adaptable in the sense that they are not based on basis functions of a fixed form, but instead, they use basis functions with a-priori unknown properties that may adapt to the specific random signal characteristics. b) The postulation of a vector FS-TARMA method for output-only structural identification and the development of effective tools for both model parameter estimation and model structure selection. c) The introduction of a statistical method for vibration-based fault diagnosis in TV structures. d) The presentation of a thorough review on FS-TARMA models covering both theoretical and practical aspects of the model parameter estimation and structure selection problems with special emphasis being placed on promising recent methods. The methods that are developed in each chapter of this thesis are validated through their application in both numerical and experimental case studies and comparisons with currently available non-stationary signal identification methods. The results of the study demonstrate the new methods' applicability, effectiveness, and high potential for parsimonious and accurate identification and dynamic analysis of TV structures. / Μη-στάσιμα σήματα, δηλαδή σήματα με χρονικά μεταβαλλόμενες (ΧΜ) στατιστικές ιδιότητες, απαντώνται συχνά στην επιστήμη του μηχανικού. Τυπικά παραδείγματα αποτελούν οι ταλαντωτικές αποκρίσεις κατασκευών, όπως γέφυρες με κινούμενα οχήματα, ρομποτικές διατάξεις, περιστρεφόμενες μηχανές και άλλες. Κατασκευές που χαρακτηρίζονται από ιδιότητες οι οποίες μεταβάλλονται με τον χρόνο αναφέρονται ως ΧΜ κατασκευές και η δυναμική αναγνώριση και ανάλυση τους επί τη βάση ταλαντωτικών σημάτων απόκρισης αποτελεί σημαντικό και ταυτόχρονα δύσκολο πρόβλημα. Μια σημαντική τάξη παραμετρικών μεθόδων για την επίλυση αυτού του προβλήματος βασίζεται στα συναρτησιακά χρονικά μεταβαλλόμενα μοντέλα αυτοπαλινδρόμησης κινητού μέσου όρου (FS-TARMA, Functional Series Time-Dependent Auto-Regressive Moving Average). Τα μοντέλα αυτά χαρακτηρίζονται απο ΧΜ παραμέτρους οι οποίες ακολουθούν καθοριστικό πρότυπο και κατά συνέπεια μπορούν να προβληθούν σε κατάλληλα επιλεγμένους συναρτησιακούς υποχώρους. Ως βασικός στόχος της παρούσας διατριβής ορίζεται η ανάπτυξη εξελιγμένων μεθόδων μοντελοποίησης FS-TARMA οι οποίες θα προσφέρουν σημαντικές βελτιώσεις στις υπάρχουσες προσεγγίσεις και θα βοηθήσουν στην αντιμετώπιση πρακτικών προβλημάτων που σχετίζονται τόσο με την αναγνώριση των δυναμικών χαρακτηριστικών όσο και την διάγνωση βλαβών σε ΧΜ κατασκευές. Οι συγκεκριμένοι στόχοι της διατριβής μπορούν να περιγραφούν ως ακολούθως: α) Εισαγωγή καινοτόμων προσαρμόσιμων μοντέλων FS-TARMA και ανάπτυξη κατάλληλης μεθόδου για την αποτελεσματική εκτίμηση τους. Τα νέα μοντέλα είναι προσαρμόσιμα υπό την έννοια ότι δεν βασίζονται σε προκαθορισμένες συναρτήσεις βάσης, αλλά αντιθέτως χρησιμοποιούν συναρτήσεις βάσης με εκ των προτέρων άγνωστες ιδιότητες οι οποίες μπορούν να προσαρμοστούν στα χαρακτηριστικά συγκεκριμένου σήματος. β) Ανάπτυξη διανυσματικής μεθόδου εκτίμησης μοντέλων FS-TARMA για την αναγνώριση κατασκευών μέσα από διανυσματικά σήματα ταλαντωτικής απόκρισης. Ανάπτυξη αποδοτικών εργαλείων τόσο για το πρόβλημα εκτίμησης των παραμέτρων όσο και της επιλογής της δομής του μοντέλου. γ) Εισαγωγή στατιστικής μεθόδου για την διάγνωση βλαβών σε ΧΜ κατασκευές μέσω μοντέλων FS-TAR. δ) Παρουσίαση μιας διεξοδικής επισκόπησης των μοντέλων FS-TARMA η οποία καλύπτει τόσο θεωρητικά όσο και πρακτικά ζητήματα των προβλημάτων εκτίμησης των παραμέτρων και επιλογής της δομής των μοντέλων. Η αποτελεσματικότητα των μοντέλων και των μεθόδων που αναπτύσσονται σε κάθε κεφάλαιο αυτής της διατριβής διερευνάται µέσω της εφαρµογής τους τόσο σε αριθµητικές όσο και πειραµατικές µελέτες και συγκρίσεις µε υπάρχουσες µη-στάσιµες µεθόδους αναγνώρισης σηµάτων. Τα αποτελέσματα της εργασίας αυτής επιδεικνύουν την ικανότητα των νέων μοντέλων να παρέχουν εξαιρετικά ακριβείς αναπαραστάσεις ΧΜ κατασκευών κατάλληλων τόσο για την δυναμική ανάλυση όσο και για την διάγνωση βλαβών σε αυτές.

Cette thèse propose trois méthodes de traitement du signal orientées vers la surveillance d’état et le diagnostic. Les techniques proposées sont surtout adaptées pour la surveillance d’état, effectuée à la base de vibrations, des machines tournantes qui fonctionnent dans des conditions d’opération non-stationnaires comme par exemple les éoliennes mais elles ne sont pas limitées à un tel usage. Toutes les méthodes proposées sont des algorithmes automatiques et gérés par les données.La première technique proposée permet de sélectionner la partie la plus stationnaire d’un signal en cadrant la représentation temps-fréquence d’un signal.La deuxième méthode est un algorithme pour l’association des dispositions spectrales, des séries harmoniques et des séries à bandes latérales avec des fréquences caractéristiques provennant du cinématique d'un système analysé. Cette méthode propose une approche unique dédiée à l’élément roulant du roulement qui permet de surmonter les difficultés causées par le phénomène de glissement.La troisième technique est un algorithme de démodulation de bande latérale entière. Elle fonctionne à la base d’un filtre multiple et propose des indicateurs de santé pour faciliter une évaluation d'état du système sous l’analyse.Dans cette thèse, les méthodes proposées sont validées sur les signaux simulés et réels. Les résultats présentés montrent une bonne performance de toutes les méthodes. / This thesis proposes a three signal-processing methods oriented towards the condition monitoring and diagnosis. In particular the proposed techniques are suited for vibration-based condition monitoring of rotating machinery which works under highly non-stationary operational condition as wind turbines, but it is not limited to such a usage. All the proposed methods are automatic and data-driven algorithms.The first proposed technique enables a selection of the most stationary part of signal by cropping time-frequency representation of the signal.The second method is an algorithm for association of spectral patterns, harmonics and sidebands series, with characteristic frequencies arising from kinematic of a system under inspection. This method features in a unique approach dedicated for rolling-element bearing which enables to overcome difficulties caused by a slippage phenomenon.The third technique is an all-sideband demodulation algorithm. It features in a multi-rate filter and proposes health indicators to facilitate an evaluation of the condition of the investigated system.In this thesis the proposed methods are validated on both, simulated and real-world signals. The presented results show good performance of all the methods.

For time-varying signals such as speech and audio, short-time analysis becomes necessary to compute specific signal attributes and to keep track of their evolution. The standard technique is the short-time Fourier transform (STFT), using which one decomposes a signal in terms of windowed Fourier bases. An advancement over STFT is the wavelet analysis in which a function is represented in terms of shifted and dilated versions of a localized function called the wavelet. A specific modeling approach particularly in the context of speech is based on short-time linear prediction or short-time Wiener filtering of noisy speech. In most nonstationary signal processing formalisms, the key idea is to analyze the properties of the signal locally, either by first truncating the signal and then performing a basis expansion (as in the case of STFT), or by choosing compactly-supported basis functions (as in the case of wavelets). We retain the same motivation as these approaches, but use polynomials to model the signal on a short-time basis (“short-time polynomial representation”). To emphasize the local nature of the modeling aspect, we refer to it as “local polynomial modeling (LPM).” We pursue two main threads of research in this thesis: (i) Short-time approaches for speech enhancement; and (ii) LPM for enhancing smooth signals, with applications to ECG, noisy nonuniformly-sampled signals, and voiced/unvoiced segmentation in noisy speech. Improved iterative Wiener filtering for speech enhancement A constrained iterative Wiener filter solution for speech enhancement was proposed by Hansen and Clements. Sreenivas and Kirnapure improved the performance of the technique by imposing codebook-based constraints in the process of parameter estimation. The key advantage is that the optimal parameter search space is confined to the codebook. The Nonstationary signal enhancement solutions assume stationary noise. However, in practical applications, noise is not stationary and hence updating the noise statistics becomes necessary. We present a new approach to perform reliable noise estimation based on spectral subtraction. We first estimate the signal spectrum and perform signal subtraction to estimate the noise power spectral density. We further smooth the estimated noise spectrum to ensure reliability. The key contributions are: (i) Adaptation of the technique for non-stationary noises; (ii) A new initialization procedure for faster convergence and higher accuracy; (iii) Experimental determination of the optimal LP-parameter space; and (iv) Objective criteria and speech recognition tests for performance comparison. Optimal local polynomial modeling and applications We next address the problem of fitting a piecewise-polynomial model to a smooth signal corrupted by additive noise. Since the signal is smooth, it can be represented using low-order polynomial functions provided that they are locally adapted to the signal. We choose the mean-square error as the criterion of optimality. Since the model is local, it preserves the temporal structure of the signal and can also handle nonstationary noise. We show that there is a trade-off between the adaptability of the model to local signal variations and robustness to noise (bias-variance trade-off), which we solve using a stochastic optimization technique known as the intersection of confidence intervals (ICI) technique. The key trade-off parameter is the duration of the window over which the optimum LPM is computed. Within the LPM framework, we address three problems: (i) Signal reconstruction from noisy uniform samples; (ii) Signal reconstruction from noisy nonuniform samples; and (iii) Classification of speech signals into voiced and unvoiced segments. The generic signal model is x(tn)=s(tn)+d(tn),0 ≤ n ≤ N - 1. In problems (i) and (iii) above, tn=nT(uniform sampling); in (ii) the samples are taken at nonuniform instants. The signal s(t)is assumed to be smooth; i.e., it should admit a local polynomial representation. The problem in (i) and (ii) is to estimate s(t)from x(tn); i.e., we are interested in optimal signal reconstruction on a continuous domain starting from uniform or nonuniform samples. We show that, in both cases, the bias and variance take the general form: The mean square error (MSE) is given by where L is the length of the window over which the polynomial fitting is performed, f is a function of s(t), which typically comprises the higher-order derivatives of s(t), the order itself dependent on the order of the polynomial, and g is a function of the noise variance. It is clear that the bias and variance have complementary characteristics with respect to L. Directly optimizing for the MSE would give a value of L, which involves the functions f and g. The function g may be estimated, but f is not known since s(t)is unknown. Hence, it is not practical to compute the minimum MSE (MMSE) solution. Therefore, we obtain an approximate result by solving the bias-variance trade-off in a probabilistic sense using the ICI technique. We also propose a new approach to optimally select the ICI technique parameters, based on a new cost function that is the sum of the probability of false alarm and the area covered over the confidence interval. In addition, we address issues related to optimal model-order selection, search space for window lengths, accuracy of noise estimation, etc. The next issue addressed is that of voiced/unvoiced segmentation of speech signal. Speech segments show different spectral and temporal characteristics based on whether the segment is voiced or unvoiced. Most speech processing techniques process the two segments differently. The challenge lies in making detection techniques offer robust performance in the presence of noise. We propose a new technique for voiced/unvoiced clas-sification by taking into account the fact that voiced segments have a certain degree of regularity, and that the unvoiced segments do not possess any smoothness. In order to capture the regularity in voiced regions, we employ the LPM. The key idea is that regions where the LPM is inaccurate are more likely to be unvoiced than voiced. Within this frame-work, we formulate a hypothesis testing problem based on the accuracy of the LPM fit and devise a test statistic for performing V/UV classification. Since the technique is based on LPM, it is capable of adapting to nonstationary noises. We present Monte Carlo results to demonstrate the accuracy of the proposed technique.

Compressed sensing has been proposed as a computationally efficient method to estimate the finite-dimensional signals. The idea is to develop an undersampling operator that can sample the large but finite-dimensional sparse signals with a rate much below the required Nyquist rate. In other words, considering the sparsity level of the signal, the compressed sensing samples the signal with a rate proportional to the amount of information hidden in the signal. In this dissertation, first, we employ compressed sensing for physical layer signal processing of directional millimeter-wave communication. Second, we go through the theoretical aspect of compressed sensing by running a comprehensive theoretical analysis of compressed sensing to address two main unsolved problems, (1) continuous-extension compressed sensing in locally convex space and (2) computing the optimum subspace and its dimension using the idea of equivalent topologies using Köthe sequence. In the first part of this thesis, we employ compressed sensing to address various problems in directional millimeter-wave communication. In particular, we are focusing on stochastic characteristics of the underlying channel to characterize, detect, estimate, and track angular parameters of doubly directional millimeter-wave communication. For this purpose, we employ compressed sensing in combination with other stochastic methods such as Correlation Matrix Distance (CMD), spectral overlap, autoregressive process, and Fuzzy entropy to (1) study the (non) stationary behavior of the channel and (2) estimate and track channel parameters. This class of applications is finite-dimensional signals. Compressed sensing demonstrates great capability in sampling finite-dimensional signals. Nevertheless, it does not show the same performance sampling the semi-infinite and infinite-dimensional signals. The second part of the thesis is more theoretical works on compressed sensing toward application. In chapter 4, we leverage the group Fourier theory and the stochastical nature of the directional communication to introduce families of the linear and quadratic family of displacement operators that track the join-distribution signals by mapping the old coordinates to the predicted new coordinates. We have shown that the continuous linear time-variant millimeter-wave channel can be represented as the product of channel Wigner distribution and doubly directional channel. We notice that the localization operators in the given model are non-associative structures. The structure of the linear and quadratic localization operator considering group and quasi-group are studied thoroughly. In the last two chapters, we propose continuous compressed sensing to address infinite-dimensional signals and apply the developed methods to a variety of applications. In chapter 5, we extend Hilbert-Schmidt integral operator to the Compressed Sensing Hilbert-Schmidt integral operator through the Kolmogorov conditional extension theorem. Two solutions for the Compressed Sensing Hilbert Schmidt integral operator have been proposed, (1) through Mercer's theorem and (2) through Green's theorem. We call the solution space the Compressed Sensing Karhunen-Loéve Expansion (CS-KLE) because of its deep relation to the conventional Karhunen-Loéve Expansion (KLE). The closed relation between CS-KLE and KLE is studied in the Hilbert space, with some additional structures inherited from the Banach space. We examine CS-KLE through a variety of finite-dimensional and infinite-dimensional compressible vector spaces. Chapter 6 proposes a theoretical framework to study the uniform convergence of a compressible vector space by formulating the compressed sensing in locally convex Hausdorff space, also known as Fréchet space. We examine the existence of an optimum subspace comprehensively and propose a method to compute the optimum subspace of both finite-dimensional and infinite-dimensional compressible topological vector spaces. To the author's best knowledge, we are the first group that proposes continuous compressed sensing that does not require any information about the local infinite-dimensional fluctuations of the signal.