En els sistemes de comunicacions on el transmissor disposa de certa informació sobre l'estat del canal (CSI), es possible dissenyar esquemes lineals de precodificació que assignin la potència de manera òptima induint guanys considerables, sigui en termes de capacitat, sigui en termes de la fiabilitat de l'enllaç de comunicacions. A la pràctica, aquest coneixement del canal mai és perfecte i, per tant, el senyal transmès es veurà degradat degut al desajust entre la informació que el transmissor disposi del canal i el seu estat real.En aquest context, aquesta tesi estudia dos problemes diferents però alhora estretament relacionats: el disseny d'un esquema pràctic de seguiment del canal en transmissió per canals variants en temps, i el disseny d'esquemes lineals de precodificació que siguin robustos a la incertesa del canal. La primera part de la tesi proposa el disseny d'un esquema de seguiment de canal que, mitjançant un enllaç de retorn de baixa capacitat, proporcioni al transmissor una informació acurada sobre el seu estat. Històricament, aquest tipus d'esquemes han rebut fortes crítiques degut a la gran quantitat d'informació que és necessari transmetre des del receptor cap el transmissor. Aquesta tesi, doncs, posa especial èmfasi en el disseny d'aquest canal de retorn. La solució que es proposa, basada en el filtre de Kalman, utilitza un esquema que recorda al transmissor DPCM. Les variacions del canal són tractades mitjançant dos predictors lineals idèntics situats en el transmissor i en el receptor, i un canal de retorn que assisteix el transmissor amb l'error de predicció. L'interès d'aquest esquema diferencial és que permet seguir les variacions del canal amb només dos o quatre bits per coeficient complex, fins i tot en canals ràpidament variants.La resta de la tesi cobreix el segon objectiu, l'estudi de diferents esquemes d'assignació de potències quan el coneixement del canal en transmissió no és perfecte. El problema es planteja per a un sistema MIMO OFDM com a formulació més general, incloent els casos d'una sola antena, de l'esquema beamforming i del canal multiplicatiu com a casos particulars.Primerament s'ha plantejat l'optimització dels criteris de mínim error quadràtic mig (MMSE) i mínima BER sense codificar. La innovació en el treball presentat a la tesi, respecte a altres treballs que segueixen els mateixos criteris de disseny, ha estat la formulació Bayesiana del problema per al disseny dels algoritmes robustos.La tesi continua amb el plantejament d'estratègies robustes d'assignació de potència destinades a minimitzar la BER codificada. Per aquesta tasca s'han utilitzat criteris de teoria de la informació. Possiblement una de les principals contribucions d'aquesta tesi ha estat el plantejament del cut-off rate com a paràmetre de disseny. Aquest criteri s'introdueix com alternativa a la capacitat de canal o a la informació mutual per al disseny del transmissor quan s'inclou codificació de canal. La ultima part de la tesi proposa un interleaver adaptatiu de baixa complexitat que, utilitzant el coneixement del canal disponible en el transmissor, assigna estratègicament els bits no només per combatre les ràfegues d'errors, sinó també per lluitar contra els esvaïments que puguin presentar les diferents portadores del canal per a una realització concreta. El disseny d'aquest interleaver, anomenat "interleaver RCPC" està basat en els codis Rate-Compatible Punctured Convolutional Codes. Com s'il·lustra a partir del resultats numèrics, l'ús d'aquest interleaver millora les prestacions dels algoritmes quan es comparen amb les que s'obtindrien si s'utilitzes un interleaver de bloc o un interleaver pseudo-aleatori. / When the transmitter of a communication system disposes of some Channel State Information (CSI), it is possible to design linear precoders that optimally allocate the power inducing high gains either in terms of capacity or in terms of reliable communications. In practical scenarios, this channel knowledge is not perfect and thus the transmitted signal suffers from the mismatch between the CSI at the transmitter and the real channel.In that context, this thesis deals with two different, but related, topics: the design of a feasible transmitter channel tracker for time varying channels, and the design of optimal linear precoders robust to imperfect channel estimates.The first part of the thesis proposes the design of a channel tracker that provides an accurate CSI at the transmitter by means of a low capacity feedback link. Historically, those schemes have been criticized because of the large amount of information to be transmitted from the receiver to the transmitter. This thesis focuses, thus, the attention in an accurate design of the return link. The proposed solution is based on the Kalman filter and follows a scheme that reminds the well known DPCM transmitter. The channel variability is processed by two identical linear predictors located at the transmitter and at the receiver, and a feedback link that assists the transmitter with the prediction error. The interest of this differential scheme is that allows to track the channel variations with only two or four bits per complex channel coefficient even in fast time-varying channels.The rest of the thesis covers the second topic, studying different robust power allocation algorithms when the CSI is not perfectly known at the transmitter. For the sake of generality, the problem is formulated for the general MIMO OFDM case, encompassing the single antenna transmission, the beamforming schemes and the frequency-flat fading channels as particular cases. First, the minimum MSE and the minimum uncoded BER parameters are chosen to be optimized, evaluating the performance of the algorithms in terms of uncoded BER. The basic novelty with respect to previous works that considers the same strategies of design is the proposal of a Bayesian approach for the design of the robust algorithms.Next the study is extended by proposing robust power allocation strategies focused on the minimization of the coded BER. For this purpose, information-theoretic criteria are used. Probably, one of the main contributions in the thesis is the proposal of the cut-off rate as a parameter of design whose maximization is directly related to the coded BER. This criterion is introduced as an alternative to the channel capacity and the mutual information for the design of optimal transceivers in the presence of any channel coding stage. The last part of the thesis proposes a low complexity adaptive interleaver that, making use of the CSI available at the transmitter, reallocates the bits not only to combat the bursty channel errors but also to combat the specific distribution of the faded subcarriers as a function of the channel response. The design of this interleaver, named as "RCPC interleaver", is based on the Rate-Compatible Punctured Convolutional Codes. As shown by numerical results, the use of this interleaver improves the performance of the algorithms when they are compared with the classical block interleavers and pseudo-random interleavers.

Τα τελευταία χρόνια, η ραγδαία αύξηση των χρηστών ασύρματης επικοινωνίας και η ολοένα και αυξανόμενη ζήτηση πιο αποδοτικών επικοινωνιών μεταξύ των χρηστών, έστρεψαν το ενδιαφέρον της επιστημονικής κοινότητας στην μελέτη πιο ευέλικτων ασύρματων δικτύων. Ιδιαίτερο ενδιαφέρον προς αυτή την κατεύθυνση παρουσιάζουν τα Γνωστικά Συστήματα Ραδιοεπικοινωνιών (Cognitive Radios). Τα Γνωστικά Συστήματα είναι ευφυή συστήματα τα οποία έχουν την ικανότητα να αντιλαμβάνονται τα χαρακτηριστικά του περιβάλλοντος στο οποίο βρίσκονται και να προσαρμόζουν κατάλληλα τις παραμέτρους της μετάδοσης τους με στόχο πιο αξιόπιστες και πιο ευέλικτες επικοινωνίες. Ιδιαίτερο χαρακτηριστικό αυτών των συστημάτων είναι το γεγονός ότι μπορούν να συνυπάρξουν ταυτόχρονα δύο είδη χρηστών, οι κύριοι και οι δευτερεύοντες. Κύριοι είναι οι χρήστες οι οποίοι έχουν νόμιμη άδεια χρήσης μιας ζώνης συχνοτήτων από κάποια αρμόδια αρχή, ενώ δευτερεύοντες ονομάζονται οι χρήστες του δικτύου που δεν έχουν άδεια χρήσης κάποιας ζώνης συχνοτήτων αλλά υπό κατάλληλες συνθήκες μπορούν να χρησιμοποιήσουν κάποια ζώνη που ανήκει στους κύριους χρήστες. Η μετάδοση των δευτερευόντων χρηστών γίνεται με τέτοιο τρόπο (συχνότητα και χρόνο μετάδοσης), ώστε να μην δημιουργείται παρεμβολή στους κύριους χρήστες. Πρόσφατα, το ερευνητικό ενδιαφέρον στράφηκε προς την χαρτογράφηση κάποιον φαινομένων σε τέτοιου είδους συστήματα. Πιο συγκεκριμένα, με τον όρο χαρτογράφηση εννοούμε την μελέτη ενός φαινομένου, όπως για παράδειγμα της παρεμβολής ή του κέρδους του καναλιού όχι μόνο ως προς το χρόνο αλλά και ως προς το χώρο. Για παράδειγμα, στα γνωστικά συστήματα η χαρτογράφηση του καναλιού δίνει τη δυνατότητα στους δευτερεύοντες χρήστες ανά πάσα στιγμή να γνωρίζουν το πιθανό κανάλι επικοινωνίας προς όλα τα σημεία του χώρου που μελετούμε. Κάτι τέτοιο, λόγω της ιδιαιτερότητας αυτού του συστήματος, μπορεί να δώσει πολύτιμες πληροφορίες για τον τρόπο με τον οποίο μπορούν να εκπέμψουν οι δευτερεύοντες χρήστες έτσι ώστε να μην παρεμβάλλονται στην μετάδοση των κύριων χρηστών. Οι εργασίες που σχετίζονται με τα γνωστικά συστήματα μέχρι τώρα είχαν ως στόχο την μελέτη είτε της επισκίασης είτε της παρεμβολής που μετριέται σε ένα σημείο του χώρου. Στα πλαίσια της παρούσας διπλωματικής εργασίας, στόχος μας ήταν η μελέτη της μακροπρόθεσμης πρόβλεψης (long range prediction) ενός καναλιού σε ένα δίκτυο κινούμενων γνωστικών χρηστών, κάτι το οποίο δεν έχει μελετηθεί έως τώρα. Με τον όρο μακροπρόθεσμη πρόβλεψη καναλιού αναφερόμαστε στην εκτίμηση του καναλιού σε κάποια επόμενη χρονική στιγμή. Δεδομένου ότι το κανάλι επηρεάζεται τόσο από φαινόμενα μικρής όσο και από ευρείας κλίμακας στρέψαμε το ενδιαφέρον μας σε ένα ενοποιημένο μοντέλο καναλιού που περιλαμβάνει και τα δύο είδη εξασθένησης. Αυτή η μακροπρόθεσμη πρόβλεψη του χρονικά μεταβαλλόμενου καναλιού αποσκοπεί στην χαρτογράφηση των διαθέσιμων καναλιών στο χώρο αλλά και στο χρόνο, πληροφορία που μπορεί να φανεί πολύ χρήσιμη σε διάφορες εφαρμογές όπως αυτή της δρομολόγησης πακέτων πληροφορίας σε ένα δίκτυο, ή της κατανομής ενέργειας στα γνωστικά και σε άλλα ασύρματα συστήματα. Πιο συγκεκριμένα, στο κεφάλαιο 1, θα αναφερθούμε στις νέες τάσεις που ακολουθούνται στις ασύρματες επικοινωνίες. Αρχικά θα αναφερθούμε στις συνεργατικές επικοινωνίες και τις λεγόμενες συντονισμένες μεταδόσεις πολλαπλών σημείων ενώ στην συνέχεια θα επικεντρωθούμε στα γνωστικά δίκτυα και τις λειτουργίες που επιτελούν. Στο κεφάλαιο 2 θα περιγράψουμε με λεπτομέρεια τους μηχανισμούς διάδοσης των ηλεκρομαγνητικών κυμάτων και τα δύο βασικά είδη εξασθένησης που παραμορφώνουν το λαμβανόμενο σήμα στον δέκτη. Στη συνέχεια του κεφαλαίου αυτού θα περιγράψουμε διάφορα μοντέλα που περιγράφουν τα δύο είδη εξασθένησης και τέλος, θα παρουσιάσουμε ένα μοντέλο που τα συνδυάζει. Στο κεφάλαιο 3, θα παρουσιάσουμε ένα ενοποιημένο μοντέλο το οποίο συμπεριλαμβάνει τόσο τα μικρής όσο και τα μεγάλης κλίμακας φαινόμενα το οποίο θα υιοθετήσουμε στην συνέχεια στην πειρματική μας διαδικασία. Επιπλέον, θα παρουσιάσουμε τους λόγους για τους οποίους στρέψαμε την προσοχή μας στο εν λόγω θέμα. Στο κεφάλαιο 4 θα αναφερθούμε στην έννοια της χαρτογράφησης ενός φαινομένου σε ένα γνωστικό σύστημα και στην τρέχουσα βιβλιογραφία. Πιο συγκεκριμένα, θα κάνουμε μία ιστορική αναδρομή της χρήσης της χαρτογράφησης στα γνωστικά συστήματα. Ξεκινώντας από την χαρτογράφηση της παρεμβολής σε ένα δίκτυο και την χαρτογράφηση του φάσματος, θα επικεντρωθούμε στην χαρτογράφηση των φαινομένων ευρείας κλίμακας και ειδικά στην χαρτογράφηση της επισκίασης. Στο κεφάλαιο 5 θα μελετήσουμε την έννοια της εκτίμησης και παρακολούθησης καναλιού και θα επικεντρωθούμε σε παραμετρικά κανάλια. Στο τέλος του κεφαλαίου, θα διευρύνουμε την παρακολούθηση ενός καναλιού με την διαδικασία της μακροπρόθεσμης πρόβλεψης του κέρδους του καναλιού. Αυτή η επιπλέον γνώση της μακροπρόθεσμης πρόβλεψης σε συνδυασμό με διαδικασίες χωρικής παρεμβολής σε ένα γνωστικό σύστημα μπορεί να βελτιώσει πολύ τις συνθήκες μετάδοσης. Λόγω της ιδιαιτερότητας που παρουσιάζουν τα εν λόγω συστήματα κάθε επιπλέον μακροπρόθεσμη πληροφορία της κατάστασης του δικτύου μπορεί να βοηθήσει σε σημαντικές αποφάσεις κατανομής ισχύος ή μετάδοσης ώστε να ελαχιστοποιηθεί η παρεμβολή προς τους κύριους χρήστες. Στο κεφάλαιο 6 θα περιγράψουμε με λεπτομέρεια το πρόβλημα μακροπρόθεσμης πρόβλεψης με το οποίο ασχοληθήκαμε και θα παρουσιάσουμε μία σειρά από πειραματικά αποτελέσματα που σχετίζονται με την ποιότητα της πρόβλεψης του καναλιού κάτω από διαφορετικές συνθήκες. Τέλος, στο κεφάλαιο 7 θα αναφερθούμε στα συμπεράσματα που προέκυψαν από την συγκεκριμένη διπλωματική εργασία καθώς και σε κάποιες μελλοντικές κατευθύνσεις που παρουσιάζουν ιδιαίτερο ενδιαφέρον. Οι κατευθύνσεις αυτές σχετίζονται τόσο για το συγκεκριμένο αλγόριθμο που χρησιμοποιήσαμε όσο και με πιθανές εφαρμογές του. / In recent years, the use of wireless communication systems has rapidly increased along with the demand for more resource hungry applications. To support such demands, the research community has been focusing into new more efficient and more flexible communications techniques and systems. To this end, the so-called cognitive radio systems are of particular interest. In a cognitive system, the participating communication nodes are able of understanding the special characteristics of the surrounding environment and adjust, accordingly, their transmission parameters in order to achieve a reliable level of communication. An identifying element of such systems is the co-existence of two kinds of users, i.e. the primary and the secondary ones. The first are licensed users of a specific transmission band as opposed to the second, who are not licensed to use such a band, however under certain constraints and conditions they are allowed to. Their transmission is performed in an appropriate manner so as not to interfere with the primary users. Currently, there is an active research interest in the so-called cartography of certain phenomena that occur in the wireless medium used by systems like the cognitive ones. Specifically, the term cartography is used for the study of a phenomenon such as the gain of channel through time and space for any pair of points in a given region. This information could, for example, support the decision making procedure of a secondary user related to the selection of a transmission band to use for transmission, reducing in this way the produced interference level. The research efforts, so far, has been focusing on the cartography of either the shadowing phenomenon or the interference that is present at a particular point at space. In this master thesis, the main target is the study of long-range channel prediction algorithms in a cognitive network of mobile users that take into account both small and large scale fading. The phrase “long-range channel prediction” refers to the process of predict the value of the channel at a future time instant. In order to capture the twofold nature to fading, i.e. small and large scale one, a unifying channel model is adopted. This operation and the predicted information aim at providing the necessary tools in order to map the available channels into space as well as time. This information can be valuable in numerous applications such as routing of data packets and resource allocation. In more detail, chapter 1 is an introduction to wireless communications focusing on their history and the new directions that look to the future. Specifically, a description will be provided for the evolution of cellular and ad hoc wireless networks along with three new tendencies, i.e. cooperative communications, coordinated multipoint or transmissions and cognitive networks. In chapter 2, an introduction of the wireless medium for transmitting communications signals is presented. Specifically, at first, the propagation mechanisms of electromagnetic waves along with their two main sources of fading that distort signals will be provided. Then, appropriate models for the fading sources will be described which are useful for developing and evaluating communications algorithms. A unified model that incorporates both small and large-scale fading is presented in chapter 3. This model will be used later in this thesis during the experimental analysis. Moreover, a motivation of looking into this direction will be provided. In chapter 4, a bibliographic presentation of cartography will be presented. The chapter will start with a historical review of chartography. First, the focus will be on the mapping of interference and spectrum usage. Then, the focus will shift towards the mapping of large scale phenonena, especially, the mapping of shadowing. The estimation/tracking and prediction of wireless channels will be the main focus of chapter 5. Parametric channels and associated algorithms will be presented. Moreover, a long rage prediction algorithm, that is studied in this thesis, will be described in detail. This additional information combined with spatial interpolation techniques can, in general, be used to improve the transmission conditions in a cognitive system. This information, in such systems, can assist in decision making procedures related to power allocations and transmissions, so as to mitigate the interference among the users. The aforementioned synthesized channel model along with the long rage prediction problem that is studied in this thesis will be evaluated in chapter 6. In this chapter, exhaustive simulations have been conducted targeting the prediction performance under different propagation conditions. Finally, in chapter 7, the main conclusions drawn in this thesis along with some future research directions will be provided. The directions are related both with the specific algorithm that was studied and some possible applications.

In this thesis, various pilot symbol aided channel estimation and tracking methods are investigated and their performances are compared for an OFDM system with packet based communication on HF channel. For the HF channel, Watterson HF channel model is used. The compared methods are least squares (LS) channel estimation, linear minimum mean square error (LMMSE) channel estimation, least mean squares (LMS) channel tracking, recursive least squares (RLS) channel tracking, constant position model based Kalman filter channel tracking, and constant velocity model based Kalman filter channel tracking. For LMS and RLS methods some adaptive approaches are also investigated.

One of the fundamental limitations of mobile radio communications is their time-varying fading channel. This thesis addresses the efficient use of channel state information to improve the communication systems, with a particular emphasis on practical issues such as compatibility with the existing wireless systems and low complexity implementation. The closed-loop transmit diversity technique is used to improve the performance of the downlink channel in MIMO communication systems. For example, the WCDMA standard endorsed by 3GPP adopts a mode of downlink closed-loop scheme based on partial channel state information known as mode 1 of 3GPP. Channel state information is fed back from the mobile unit to the base station through a low-rate uncoded feedback bit stream. In these closed-loop systems, feedback error and feedback delay, as well as the sub-optimum reconstruction of the quantized feedback data, are the usual sources of deficiency. In this thesis, we address the efficient reconstruction of the beamforming weights in the presence of the feedback imperfections, by exploiting the residual redundancies in the feedback stream. We propose a number of algorithms for reconstruction of beamforming weights at the base-station, with the constraint of a constant transmit power. The issue of the decoding at the receiver is also addressed. In one of the proposed algorithms, channel fading prediction is utilized to combat the feedback delay. We introduce the concept of Blind Antenna Verification which can substitute the conventional Antenna Weight Verification process without the need for any training data. The closed-loop mode 1 of 3GPP is used as a benchmark, and the performance is examined within a WCDMA simulation framework. It is demonstrated that the proposed algorithms have substantial gain over the conventional method at all mobile speeds, and are suitable for the implementation in practice. The proposed approach is applicable to other closed-loop schemes as well. The problem of (long-range) prediction of the fading channel is also considered, which is a key element for many fading-compensation techniques. A linear approach, usually used to model the time evolution of the fading process, does not perform well for long-range prediction applications. We propose an adaptive algorithm using a state-space approach for the fading process based on the sum-sinusoidal model. Also to enhance the widely-used linear approach, we propose a tracking method for a multi-step linear predictor. Comparing the two methods in our simulations shows that the proposed algorithm significantly outperforms the linear method, for both stationary and non-stationary fading processes, especially for long-range predictions. The robust structure, as well as the reasonable computational complexity, makes the proposed algorithm appealing for practical applications.

One of the fundamental limitations of mobile radio communications is their time-varying fading channel. This thesis addresses the efficient use of channel state information to improve the communication systems, with a particular emphasis on practical issues such as compatibility with the existing wireless systems and low complexity implementation. The closed-loop transmit diversity technique is used to improve the performance of the downlink channel in MIMO communication systems. For example, the WCDMA standard endorsed by 3GPP adopts a mode of downlink closed-loop scheme based on partial channel state information known as mode 1 of 3GPP. Channel state information is fed back from the mobile unit to the base station through a low-rate uncoded feedback bit stream. In these closed-loop systems, feedback error and feedback delay, as well as the sub-optimum reconstruction of the quantized feedback data, are the usual sources of deficiency. In this thesis, we address the efficient reconstruction of the beamforming weights in the presence of the feedback imperfections, by exploiting the residual redundancies in the feedback stream. We propose a number of algorithms for reconstruction of beamforming weights at the base-station, with the constraint of a constant transmit power. The issue of the decoding at the receiver is also addressed. In one of the proposed algorithms, channel fading prediction is utilized to combat the feedback delay. We introduce the concept of Blind Antenna Verification which can substitute the conventional Antenna Weight Verification process without the need for any training data. The closed-loop mode 1 of 3GPP is used as a benchmark, and the performance is examined within a WCDMA simulation framework. It is demonstrated that the proposed algorithms have substantial gain over the conventional method at all mobile speeds, and are suitable for the implementation in practice. The proposed approach is applicable to other closed-loop schemes as well. The problem of (long-range) prediction of the fading channel is also considered, which is a key element for many fading-compensation techniques. A linear approach, usually used to model the time evolution of the fading process, does not perform well for long-range prediction applications. We propose an adaptive algorithm using a state-space approach for the fading process based on the sum-sinusoidal model. Also to enhance the widely-used linear approach, we propose a tracking method for a multi-step linear predictor. Comparing the two methods in our simulations shows that the proposed algorithm significantly outperforms the linear method, for both stationary and non-stationary fading processes, especially for long-range predictions. The robust structure, as well as the reasonable computational complexity, makes the proposed algorithm appealing for practical applications.

This thesis aims to contribute to the developments of wireless communication systems. The work generally consists of three parts: the first part is a discussion on general digital communication systems, the second part focuses on wireless channel modelling and fading mitigation techniques, and in the third part we discuss the possible application of advanced digital signal processing, especially time-frequency representation and blind source separation, to wireless communication systems. The first part considers general digital communication systems which will be incorporated in later parts. Today's wireless communication system is a subbranch of a general digital communication system that employs various techniques of A/D (Analog to Digital) conversion, source coding, error correction, coding, modulation, and synchronization, signal detection in noise, channel estimation, and equalization. We study and develop the digital communication algorithms to enhance the performance of wireless communication systems. In the Second Part we focus on wireless channel modelling and fading mitigation techniques. A modified Jakes' method is developed for Rayleigh fading channels. We investigate the level-crossing rate (LCR), the average duration of fades (ADF), the probability density function (PDF), the cumulative distribution function (CDF) and the autocorrelation functions (ACF) of this model. The simulated results are verified against the analytical Clarke's channel model. We also construct frequency-selective geometrical-based hyperbolically distributed scatterers (GBHDS) for a macro-cell mobile environment with the proper statistical characteristics. The modified Clarke's model and the GBHDS model may be readily expanded to a MIMO channel model thus we study the MIMO fading channel, specifically we model the MIMO channel in the angular domain. A detailed analysis of Gauss-Markov approximation of the fading channel is also given. Two fading mitigation techniques are investigated: Orthogonal Frequency Division Multiplexing (OFDM) and spatial diversity. In the Third Part, we devote ourselves to the exciting fields of Time-Frequency Analysis and Blind Source Separation and investigate the application of these powerful Digital Signal Processing (DSP) tools to improve the performance of wireless communication systems.

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.