Considering the communication systems with cyclic prefix (CP), such as orthogonal frequency-division multiplexing (OFDM) modulation and single-carrier cyclic prefixed (SCCP) modulation, when the length of CP is longer than the channel length, the use of cyclic prefix (CP) does not only eliminate the inter-block interference, but also convert linear convolution of the transmitted signal with the channel into circular convolution. Unfortunately, the use of CP significantly decreases the bandwidth utilization. Therefore, to reduce the length of CP is a critical issue. The thesis investigates that how to design a channel-shortening equalizer (CSE) at receiver which forces the length of the effective channel response as short as the CP length. The thesis describes the signal model as a matrix form. The effect channel response after CSE is investigated and then the coefficient of channel shortening filter is obtained using singular value decomposition method under various criterions. We further propose a novel CSE maximizing the shortening signal-to-interference ratio. In addition, it is demonstrated that the proposed CSE has the same performance as the conventional scheme but a lower computation complexity.

Wireless communication systems require advanced techniques at the transmitter and at the receiver that improve the performance in hostile radio environments. The received signal is significantly distorted due to the dynamic nature of the wireless channel caused by multipath fading and Doppler spread. In order to mitigate the negative impact of the channel to the received signal quality, techniques as equalization and diversity are usually employed in the system design. During the transmission, the phenomenon of inter-symbol interference (ISI) occurs at the receiver due to the time dispersion of the involved channels. Hence, several delayed replicas of previous symbols interfere with the current symbol. Equalization is usually employed in order to combat the effect of the ISI. Several implementations for equalization filters have been proposed, including linear and non-linear processing, providing complexity-performance trade-offs. It is known that the length of the equalization filter determines the complexity of the technique. Since the wireless channels are characterized by long and sparse impulse responses, the conventional equalizers require high computational complexity due to the large size of their filters. In this dissertation, we have further investigated the long standing problem of equalization in light of the recently derived theory of compressed sampling (CS) for sparse and redundant representations. The developed heuristic algorithms for equalization, can exploit either the sparsity of the channel impulse response (CIR), or the sparsity of the equalizer filters, in order to derive efficient implementation designs. To this end, building on basis pursuit and matching pursuit techniques new equalization schemes have been proposed that exhibit considerable computational savings, increased performance properties and short training sequence requirements. Our main contribution for this part is the Stochastic Gradient Pursuit algorithm for sparse adaptive equalization. An alternative approach to combat ISI is based on the orthogonal frequency division multiplexing (OFDM) system. In this system, the entire channel is divided into many narrow subchannels, so as the transmitted signals to be orthogonal to each other, despite their spectral overlap. However, in the case of doubly selective channels, the Doppler effect destroys the orthogonality between subcarriers. Thus, similarly to ISI, the effect of intercarrier interference (ICI) is introduced at the receiver, where symbols which belong to other subcarriers interfere with the current one. Considering this problem, we have developed iterative algorithms which recursively cancels the ICI at the receiver, providing performance-complexity trade-offs. For low or medium Doppler spreads, the typical approach is to approximate the frequency-domain channel matrix with a banded one. On this premise, we derived reduced-rank preconditioned conjugate gradient (PCG) algorithms in order to estimate the equalization matrix with a reduced number of iterations. Also developed an improved PCG algorithm with the same complexity order, using the Galerkin projections theory. However, in rapidly changing environments, a severe ICI is introduced and the banded approximation results in significant performance degradation. In order to recover this performance loss, we developed regularized estimation framework for ICI equalization, with linear complexity with respect the the number of the subcarriers. Moreover, we proposed a new equalization technique which has the potential to completely cancel the ICI. This approach works in a successive manner through a number of stages, conveying from the fully-connected ordered successive interference cancellation architecture (OSIC) in order to fully suppress the residual interference at each stage of the equalizer. On the other hand, diversity can improve the performance of the communication system by sending the information symbols through multiple signal paths, each of which fades independently. One approach to obtain diversity is through cooperative transmission, considering a group of nearby terminals (relays) as forming one virtual antenna array and applying a spatial beamforming technique so as to optimize the communication via them. Such beamforming techniques differ from their classical counterparts where the array elements are located in a common processing unit, due to the distribution of the relays in the space. In this setting, we developed new distributed algorithms which enable the relay cooperation for the computation of the beamforming weights leveraging the computational abilities of the relays. Each relay can estimate only the corresponding entry of the principal eigenvector, combining data from its network neighbours. The proposed algorithms are applied to two distributed beamforming schemes for relay networks. In the first scheme, the beamforming vector is computed through minimization of a total transmit power subject to the receiver quality-of-service (QoS) constraint. In the second scheme, the beamforming weights are obtained through maximization of the receiver SNR subject to a total transmit power constraint. Moreover, the proposed algorithms operate blindly, implying that no training data are required to be transmitted to the relays, and adaptively, exhibiting a quite short convergence period. / Τα συστήματα ασύρματων επικοινωνιών απαιτούν εξειδικευμένες τεχνικές στον πομπό και στον δέκτη, οι οποίες να βελτιώνουν την απόδοση του συστήματος σε εχθρικά περιβάλλοντα ασύρματης μετάδοσης. Λόγω της δυναμικής φύσης του ασύρματου καναλιού, που περιγράφεται από τα φαινόμενα της απόσβεσης, της πολυδιόδευσης και του Doppler, το λαμβανόμενο σήμα είναι παραμορφωμένο σε σημαντικό βαθμό. Για να αναιρέσουμε αυτήν την αρνητική επίδραση του καναλιού στην ποιότητα του λαμβανόμενου σήματος, κατά τον σχεδιασμό του συστήματος συνήθως υιοθετούνται τεχνικές όπως η ισοστάθμιση και η ποικιλομορφία. Ένα φαινόμενο που προκύπτει στο δέκτη ενός ασύρματου συστήματος επικοινωνίας, λόγω της χρονικής διασποράς που παρουσιάζουν τα κανάλια, είναι η διασυμβολική παρεμβολή, όπου χρονικά καθυστερημένα αντίγραφα προηγούμενων συμβόλων παρεμβάλουν με το τρέχων σύμβολο. Ένας τρόπος για την αντιμετώπιση του φαινομένου αυτού, είναι μέσω της ισοστάθμισης στο δέκτη, όπου χρησιμοποιώντας γραμμικές και μη-γραμμικές τεχνικές επεξεργασίας, τα μεταδιδόμενα σύμβολα ανιχνεύονται από το ληφθέν σήμα. Ωστόσο, συνήθως τα ασύρματα κανάλια χαρακτηρίζονται από κρουστικές αποκρούσεις μεγάλου μήκους αλλά λίγων μη μηδενικών συντελεστών, και σε αυτήν την περίπτωση η υπολογιστική πολυπλοκότητα των συνήθων τεχνικών είναι πολύ υψηλή. Στα πλαίσια αυτής της διατριβής, αναπτύχθηκαν νέοι ευριστικοί αλγόριθμοι για το πρόβλημα της ισοστάθμισης, οι οποίοι εκμεταλλεύονται είτε την αραιότητα της κρουστικής απόκρισης είναι την αραιότητα του αντιστρόφου φίλτρου, προκειμένου να παραχθούν αποδοτικές υλοποιήσεις. Θεωρώντας τον μη γραμμικό ισοσταθμιστή ανατροφοδότησης-απόφασης, έχει δειχθεί ότι κάτω από συνήθεις υποθέσεις για τους συντελεστές της κρουστικής απόκρισης του καναλιού, το εμπρόσθιο φίλτρο και το φίλτρο ανατροφοδότησης μπορούν να αναπαρασταθούν από αραιά διανύσματα. Για τον σκοπό αυτό, τεχνικές Συμπιεσμένης Καταγραφής, οι οποίες έχουν χρησιμοποιηθεί κατα κόρον σε προβλήματα ταυτοποίησης συστήματος, μπορούν να βελτιώσουν σε μεγάλο βαθμό την απόδοση κλασσικών ισοσταθμιστών που δεν λαμβάνουν υπόψιν τους την αραιότητα των διανυσμάτων. Έχοντας ως βάση τις τεχνικές basis pursuit και matching pursuit, αναπτύχθηκαν νέα σχήματα ισοσταθμιστών τα οποία παρουσιάζουν αξιοσημείωτη μείωση στο υπολογιστικό κόστος. Επίσης, αντίθετα με τη συνήθη πρακτική ταυτοποίησης συστήματος, αναπτύχθηκε νέος ευριστικό αλγόριθμος για το πρόβλημα αραιής προσαρμοστικής ισοστάθμισης, με την ονομασία Stochastic Gradient Pursuit. Επιπλέον, ο αλγόριθμος αυτός επεκτάθηκε και για την περίπτωση όπου ο αριθμός των μη μηδενικών στοιχείων του ισοσταθμιστή είναι άγνωστος. Μία διαφορετική προσέγγιση για την αντιμετώπιση του φαινομένου της διασυμβολικής παρεμβολής είναι μέσω του συστήματος orthogonal frequency-division multiplexing (OFDM), όπου το συνολικό κανάλι χωρίζεται σε πολλά στενά υπο-κανάλια, με τέτοιον τρόπο ώστε τα μεταδιδόμενα σήματα να είναι ορθογώνια μεταξύ τους, παρότι παρουσιάζουν φασματική επικάλυψη. Ωστόσο, σε χρονικά και συχνοτικά επιλεκτικά κανάλια, το φαινόμενο Doppler καταστρέφει την ορθογωνιότητα των υπο-καναλιών. Σε αυτήν την περίπτωση, παρόμοια με το φαινόμενο της διασυμβολικής παρεμβολής, εμφανίζεται το φαινόμενο της διακαναλικής παρεμβολής, όπου τα σύμβολα που ανήκουν σε διαφορετικά υπο-κανάλια παρεμβάλουν στο τρέχον. Θεωρώντας αυτό το πρόβλημα, αναπτύχθηκαν νέα σχήματα ισοστάθμισης που ακυρώνουν διαδοχικά την παρεμβολή αυτή, παρέχοντας έναν συμβιβασμό μεταξύ της απόδοσης και της πολυπλοκότητας. Στις περιπτώσεις όπου το φαινόμενο Doppler δεν είναι τόσο ισχυρό, η συνήθης τακτική είναι η προσέγγιση του πίνακα του καναλιού με έναν πίνακα ζώνης. Με αυτό το σκεπτικό, αναπτύχθηκαν αλγόριθμοι μειωμένης τάξης που βασίζονται στην επαναληπτική μέθοδο preconditioned conjugate gradient (PCG), προκειμένου να εκτιμήσουμε τον πίνακα ισοστάθμισης με έναν μειωμένο αριθμό επαναλήψεων. Επίσης, αναπτύχθηκαν τεχνικές που βασίζονται σε προβολές Galerkin για την βελτίωση της απόδοσης των συστημάτων χωρίς να αυξάνουν σημαντικά την πολυπλοκότητα. Ωστόσο, για τις περιπτώσεις όπου το φαινόμενο Doppler έχει ισχυρή επίδραση στο δέκτη του τηλεπικοινωνιακού συστήματος, όπως στις περιπτώσεις πολύ δυναμικών καναλιών, τότε η προσέγγιση με τον πίνακα ζώνης μειώνει σημαντικά την απόδοση του συστήματος. Με στόχο να ανακτήσουμε την απώλεια αυτή, αναπτύχθηκαν τεχνικές κανονικοποιημένης εκτίμησης, με γραμμική πολυπλοκότητα σε σχέση με τον αριθμό των υπο-καναλιών. Επιπρόσθετα, αναπτύχθηκε ένα νέο σχήμα ισοστάθμισης που έχει την δυνατότητα να ακυρώσει πλήρως την διακαναλική παρεμβολή. Το συγκεκριμένο σχήμα λειτουργεί βασιζόμενο σε έναν αριθμό διαδοχικών σταδίων, ακολουθώντας την φιλοσοφία της αρχιτεκτονικής fully-connected ordered successive interference cancellation (OSIC), με στόχο να μειώσει την εναπομείναντα παρεμβολή σε κάθε στάδιο του ισοσταθμιστή Η απόδοση ενός τηλεπικοινωνιακού συστήματος μπορεί επίσης να βελτιωθεί με την χρήση τεχνικών ποικιλομορφίας, δηλαδή με την μετάδοση των συμβόλων μέσω πολλών ανεξάρτητων μονοπατιών. Μία τεχνική ποικιλομορφίας είναι η συνεργατική μετάδοση, όπου μία ομάδα κοντινών τερματικών (relays) σχηματίζουν μία εικονική συστοιχία κεραιών και τεχνικές διαμόρφωσης λοβού μετάδοσης χρησιμοποιούνται προκειμένου να βελτιστοποιηθεί η επικοινωνία μέσω των τερματικών. Οι συγκεκριμένες τεχνικές διαμόρφωσης λοβού μετάδοσης, διαφέρουν από τις κλασσικές όπου η συστοιχία κεραιών βρίσκεται τοποθετημένη σε έναν κόμβο, καθώς τα τερματικά κατανέμονται στον χώρο. Υπό αυτές τις συνθήκες, αναπτύχθηκαν κατανεμημένοι αλγόριθμοι οι οποίοι εκμεταλλεύονται την επικοινωνία και τις υπολογιστικές δυνατότητες των τερματικών για τον υπολογισμό των συνιστωσών του διανύσματος διαμόρφωσης λοβού μετάδοσης. Κάθε τερματικό εκτιμά μόνο την αντίστοιχη συνιστώσα από το κύριο ιδιοδιάνυσμα, συνδιάζοντας δεδομένα από τα γειτονικά τερματικά. Οι προτεινόμενοι αλγόριθμοι εφαρμόστηκαν σε δύο σχήματα κατανεμημένης μετάδοσης μέσω ενδιάμεσων κόμβων. Στο πρώτο σχήμα, τα βάρη του διανύσματος διαμόρφωσης λοβού μετάδοσης υπολογίστηκαν με βάση την ελαχιστοποίηση της συνολικής ισχύος μετάδοσης υπό τον περιορισμό συγκεκριμένου κατωφλίου για την ποιότητα του λαμβανόμενου σήματος. Στο δεύτερο σχήμα, υπολογίστηκαν μεγιστοποιώντας την ποιότητα του λαμβανόμενου σήματος υπό τον περιορισμό ενός κατωφλίου για την συνολική ισχύ μετάδοσης. Επιπλέον, οι αλγόριθμοι που αναπτύχθηκαν λειτουργούν τυφλά, δηλαδή χωρίς φάση εκπαίδευσης, και προσαρμοστικά με μικρό διάστημα σύγκλισης.

Along with the rapidly increasing demand for wireless data while more and more crowded radio frequency (RF) spectrum, optical wireless communications (OWC) become a promising candidate to complement conventional RF communications, especially for indoor short and medium range data transmissions. Orthogonal frequency division multiplexing (OFDM) is considered for OWC due to its ability to boost data rates. However, the average emitted optical power and dynamic range of driving signals of LEDs are two major constraints in OWC. OFDM waveforms exhibits high upper and lower peak-to-average power ratios (PAPRs), which make OFDM signals optical power inefficient and easy to violate the dynamic range of LEDs, resulting clipping and nonlinear distortions. In this dissertation, we analyze and design optical power and dynamic range constrained OWC systems, for which OFDM is our major subject. We first derive distributions of upper PAPR and lower PAPR of OWC-OFDM signals. Then we analyze the clipped OFDM signals in term of error vector magnitude (EVM), signal-to-distortion ratio (SDR), and achievable data rates under both optical power and dynamic range constraints. The next part of this dissertation is the OFDM system design for visible light communications (VLC) considering illumination requirement. We investigate the illumination-to-communication efficiency (ICE) in VLC-OFDM, and design the brightness control and flickering mitigation schemes for VLC-OFDM. In the end, to reduce the complexity of driving circuits of LEDs , we propose using delta-sigma modulators in VLC-OFDM systems to convert continuous magnitude OFDM symbols into two-level LED driver signals without loss of the communication theory advantages of OFDM.

The central objective of this thesis is to suggest and develop one simple, unified method for communication over optical fiber networks, valid for all values of dispersion and nonlinearity parameters, and for a single-user channel or a multiple-user network. The method is based on the nonlinear Fourier transform (NFT), a powerful tool in soliton theory and exactly solvable models for solving integrable partial differential equations governing wave propagation in certain nonlinear media. The NFT decorrelates signal degrees of freedom in such models, in much the same way that the Fourier transform does for linear systems. In this thesis, this observation is exploited for data transmission over integrable channels such as optical fibers, where pulse propagation is governed by the nonlinear Schr\"odinger (NLS) equation. In this transmission scheme, which can be viewed as a nonlinear analogue of orthogonal frequency-division multiplexing commonly used in linear channels, information is encoded in the nonlinear spectrum of the signal. Just as the (ordinary) Fourier transform converts a linear convolutional channel into a number of parallel scalar channels, the nonlinear Fourier transform converts a nonlinear dispersive channel described by a \emph{Lax convolution} into a number of parallel scalar channels. Since, in the spectral coordinates the NLS equation is multiplicative, users of a network can operate in independent nonlinear frequency bands with no deterministic inter-channel interference. Unlike most other fiber-optic transmission schemes, this technique deals with both dispersion and nonlinearity directly and unconditionally without the need for dispersion or nonlinearity compensation methods. This thesis lays the foundations of such a nonlinear frequency-division multiplexing system.

The central objective of this thesis is to suggest and develop one simple, unified method for communication over optical fiber networks, valid for all values of dispersion and nonlinearity parameters, and for a single-user channel or a multiple-user network. The method is based on the nonlinear Fourier transform (NFT), a powerful tool in soliton theory and exactly solvable models for solving integrable partial differential equations governing wave propagation in certain nonlinear media. The NFT decorrelates signal degrees of freedom in such models, in much the same way that the Fourier transform does for linear systems. In this thesis, this observation is exploited for data transmission over integrable channels such as optical fibers, where pulse propagation is governed by the nonlinear Schr\"odinger (NLS) equation. In this transmission scheme, which can be viewed as a nonlinear analogue of orthogonal frequency-division multiplexing commonly used in linear channels, information is encoded in the nonlinear spectrum of the signal. Just as the (ordinary) Fourier transform converts a linear convolutional channel into a number of parallel scalar channels, the nonlinear Fourier transform converts a nonlinear dispersive channel described by a \emph{Lax convolution} into a number of parallel scalar channels. Since, in the spectral coordinates the NLS equation is multiplicative, users of a network can operate in independent nonlinear frequency bands with no deterministic inter-channel interference. Unlike most other fiber-optic transmission schemes, this technique deals with both dispersion and nonlinearity directly and unconditionally without the need for dispersion or nonlinearity compensation methods. This thesis lays the foundations of such a nonlinear frequency-division multiplexing system.

This work concentrates on the application of diversity techniques and space time block coding for future high speed mobile wireless communications on multicarrier systems. At first, alternative multicarrier kernels robust for high speed doubly-selective fading channel are sought. They include the comparisons of discrete Fourier transform (DFT), fractional Fourier transform (FrFT) and wavelet transform (WT) multicarrier kernels. Different wavelet types, including the raised-cosine spectrum wavelets are implemented, evaluated and compared. From different wavelet families, orthogonal wavelets are isolated from detailed evaluations and comparisons as suitable for multicarrier applications. The three transforms are compared over a doubly-selective channel with the WT significantly outperforming all for high speed conditions up to 300 km/hr. Then, a new wavelet is constructed from an ideal filter approximation using established wavelet design algorithms to match any signal of interest; in this case under bandlimited criteria. The new wavelet showed better performance than other traditional orthogonal wavelets. To achieve MIMO communication, orthogonal space-time block coding, OSTBC, is evaluated next. First, the OSTBC is extended to assess the performance of the scheme over extended receiver diversity order. Again, with the extended diversity conditions, the OSTBC is implemented for a multicarrier system over a doubly-selective fading channel. The MIMO-OFDM systems (implemented using DFT and WT kernels) are evaluated for different operating frequencies, typical of LTE standard, with Doppler effects. It was found that, during high mobile speed, it is better to transmit OFDM signals using lower operating frequencies. The information theory for the 2-transmit antenna OSTBC does not support higher order implementation of multi-antenna systems, which is required for the future generation wireless communications systems. Instead of the OSTBC, the QO-STBC is usually deployed to support the design of higher order multi-antenna systems other than the 2-transmit antenna scheme. The performances of traditional QO-STBC methods are diminished by some off-diagonal (interference) terms such that the resulting system does not attain full diversity. Some methods for eliminating the interference terms have earlier been discussed. This work follows the construction of cyclic matrices with Hadamard matrix to derive QO-STBC codes construction which are N-times better than interference free QO-STBC, where N is the number of transmit antenna branches.

Digital communication systems which use Orthogonal Frequency Division Multiplexing (OFDM) are now widely used and have many advantages. The main disadvantage is the requirement for highly linear analogue electronics including the high power amplifier (HPA). This requirement cannot be met in all circumstances because of the occurrence of symbols with high peak to average power ratio (PAPR). Such symbols may be non-linearly distorted by limiting. Approaches to solve this problem have been either to reduce the PAPR at the transmitter or to try to mitigate the effect of the non-linearity at the receiver. Soft-limiting, i.e. applying limiting in software prior to the HPA is a simple way to reduce the PAPR. It produces non-linear distortion which will cause an increase in the bit-error-rate (BER) at the receiver. This thesis surveys existing alternatives ways of reducing the effect of non-linearity and proposes some new ones. Two iterative receiver techniques, based on statistical analysis of the nature of the non-linearity, have been implemented and investigated. These are the ‘Bussgang Noise Cancellation’ (BNC) technique and the ‘Decision Aided Reconstruction’ (DAR) techniques. As these techniques are valid for any memory-less nonlinearity, an alternative form of limiting, named as Inverted-Wraparound (IWRAP) has been included in the BNC investigation. A new method is proposed which is capable of correcting the received time-domain samples that are clipped, once they have been identified. This is named the ‘Equation-Method’ and it works by identifying constellation symbols that are likely to be correct at the receiver. If there are a sufficient number of these and they are correctly identified, the FFT may be partitioned to produce a set of equations that may be solved for the clipped time-domain samples. The thesis proposes four enhancements to this new method which improve its effectiveness. It is shown that the best form of this method outperforms conventional techniques especially for severe clipping levels. The performance of these four enhancements is evaluated over channels with additive white Gaussian noise (AWGN) in addition to clipping distortion. A technique based on a ‘margin factor’ is designed to make these methods work more effectively in the presence of AWGN noise. A new combining algorithm referred as ‘HARQ for Clipping’ is presented where soft bit decisions are combined from multiple transmissions. ‘HARQ for Clipping’ has been combined with the best version of the Equation-Method, and the performance of this approach is evaluated in terms of the BER with different levels of AWGN. It has been compared to other approaches from the literature and was found to out-perform the BNC iterative receiver by 3dB at signal to noise ratios around 10dB. Without HARQ, the best version of the Equation-Method performs better than the BNC receiver, at signal-to-nose ratios above about 17dB.

This paper presents a new hybrid PAPR reduction technique for the OFDM signal, which combines a multiple symbol representations method with a signal clipping method. The clipping method is a nonlinear PAPR reduction scheme, where the amplitude of the signal is limited to a given threshold. Considering the fact that the signal must be interpolated before A/D conversion, a variety of clipping methods has been proposed. Some methods suggest the clipping before interpolation, having the disadvantage of the peaks re-growth. Other methods contributed that the clipping after interpolation, having the disadvantage of out-of-band power production. In order to overcome this problem different filtering techniques have been proposed. Filtering can also cause peak re-growth, but less than the clipping before interpolation. Another clipping technique supposes that only subcarriers having the highest phase difference between the original signal and its clipped variant will be changed. This is the case of the partial clipping method. To further reduce the PAPR, the dynamic of the clipped signal can be compressed. Linear methods like partial transmit sequence or selective mapping has been proposed for the reduction of PAPR as well. Another PAPR reduction method is the tone reservation. It uses tones on which no data is sent to reduce the transmitted signal peaks. Derivatives of this method with lower computation complexity and improved performance have been proposed: One-Tone One-Peak and one by-one iteration. A similar PAPR reduction method is the multiple symbol representations, where alternative signalling points are used to represent one symbol. The simulation results highlight the advantages of the proposed PAPR reduction method. / The proposed technique namely Adaptive Active Constellation Extension (Adaptive ACE) Algorithm reduced the high Peak-to-Average Power Ratio (PAPR) of the Orthogonal Frequency Division Multiplexing (OFDM) systems. The Peak-to-Average Power Ratio (PAPR) is equal to 6.8 dB for the target clipping ratios of 4 dB, 2 dB and 0 dB by using Adaptive Active Constellation Extension (Adaptive ACE) Algorithm. Thus, the minimum PAPR can be achieved for low target clipping ratios. The Signal-to-Noise Ratio (SNR) of the Orthogonal Frequency Division Multiplexing (OFDM) signal obtained by the Adaptive Active Constellation Extension (Adaptive ACE) algorithm is equal to 1.2 dB at a Bit Error Rate (BER) of 10-0..4 for different constellation orders like 4-Quadrature Amplitude Modulation (4-QAM), 16-Quadrature Amplitude Modulation (16-QAM) and 64-Quadrature Amplitude Modulation (16-QAM). Here, the Bit Error Rate of 10-0.4 or 0.398, that means a total of 398-bits are in error when 1000-bits are transmitted via a communication channel or approximately 4-bits are in error when 10-bits are transmitted via a communication channel, which is high when compared to that of the original Orthogonal Frequency Division Multiplexing (OFDM) signal. The other problems faced by the Adaptive Active Constellation Extension (Adaptive ACE) algorithm are Out-of-Band Interference (OBI) and peak regrowth. Here, the Out-of-Band Interference (OBI) is a form of noise or an unwanted signal, which is caused when the original Orthogonal Frequency Division Multiplexing (OFDM) signal is clipped for reducing the peak signals which are outside of the predetermined area and the peak regrowth is obtained after filtering the clipped signal. The peak regrowth results to, increase in the computational time and computational complexity. In this paper, we have proposed a PAPR reduction scheme to improve the bit error rate performance by applying companding transform technique. Hence, 1-1.5 dB reduction in PAPR with this Non-companding technique is achieved. In Future, We can accept to implement the same on Rician and Rayleigh channels. / Clement Ranjith Anthikkad (E-mail: clement.ranjith@gmail.com / clan11@bth.se) & Irfan Ahmed Baig (E-mail: baig.irfanahmed@gmail.com / ir-a11@bth.se )

Fusion frames and block sparse recovery are of interest in signal processing and communication applications. In these applications it is required that the fusion frame have some desirable properties. One such requirement is that the fusion frame be tight and its subspaces form an optimal packing in a Grassmannian manifold. Such fusion frames are called Grassmannian fusion frames. Grassmannian frames are known to be optimal dictionaries for sparse recovery as they have minimum coherence. By analogy Grassmannian fusion frames are potential candidates as optimal dictionaries in block sparse processing. The present work intends to study fusion frames in finite dimensional vector spaces assuming a specific structure useful in block sparse signal processing. The main focus of our work is the design of Grassmannian fusion frames and their implication in block sparse recovery. We will consider burst error correction as an application of block sparsity and fusion frame concepts. We propose two new algebraic methods for designing Grassmannian fusion frames. The first method involves use of Fourier matrix and difference sets to obtain a partial Fourier matrix which forms a Grassmannian fusion frame. This fusion frame has a specific structure and the parameters of the fusion frame are determined by the type of difference set used. The second method involves constructing Grassmannian fusion frames from Grassmannian frames which meet the Welch bound. This method uses existing constructions of optimal Grassmannian frames. The method, while fairly general, requires that the dimension of the vector space be divisible by the dimension of the subspaces. A lower bound which is an analog of the Welch bound is derived for the block coherence of dictionaries along with conditions to be satisfied to meet the bound. From these results we conclude that the matrices constructed by us are optimal for block sparse recovery from block coherence viewpoint. There is a strong relation between sparse signal processing and error control coding. It is known that burst errors are block sparse in nature. So, here we attempt to solve the burst error correction problem using block sparse signal recovery methods. The use of Grassmannian fusion frames which we constructed as optimal dictionary allows correction of maximum possible number of errors, when used in conjunction with reconstruction algorithms which exploit block sparsity. We also suggest a modification to improve the applicability of the technique and point out relationship with a method which appeared previously in literature. As an application example, we consider the use of the burst error correction technique for impulse noise cancelation in OFDM system. Impulse noise is bursty in nature and severely degrades OFDM performance. The Grassmannian fusion frames constructed with Fourier matrix and difference sets is ideal for use in this application as it can be easily incorporated into the OFDM system.

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