Les codes espace-temps sont des codes correcteurs dédiés aux transmissions MIMO. Mathématiquement, un code espace-temps est un ensemble fini de matrices complexes. Ses performances dépendent de plusieurs critères, dont la distance minimale en métrique rang. Les codes de Gabidulin sont des codes dans cette métrique, connus pour leur optimalité et pour l'existence d'algorithmes de décodage efficaces. C'est pourquoi ils sont utilisés pour concevoir des codes espace-temps. La principale difficulté est alors de construire des matrices complexes à partir de matrices binaires. Les travaux présentés dans ce documents consistent à généraliser les codes de Gabidulin à des corps de nombres, en particulier des extensions cyclique. Nous verrons qu'ils ont les mêmes propriétés que leurs analogues sur les corps finis. Nous étudierons plusieurs modèles d'erreurs et d'effacements et présenterons un algorithme qui permettra de retrouver l'information transmise avec une complexité quadratique. En calculant dans des corps infinis, nous serons confrontés au problème de la taille des éléments, qui augmente exponentiellement au gré des calculs. Pour éviter ce désagrément, nous verrons qu'il est possible de réduire le code afin de calculer dans un corps fini. Enfin, nous proposerons une famille de codes espace-temps dont la construction est basée sur les codes de Gabidulin généralisés. Nous verrons que leurs performances sont similaires à celles des codes existants, et qu'ils disposent d'une structure supplémentaire. / Space-time codes are error correcting codes dedicated to MIMO transmissions. Mathematically, a space-time code is a finite family of complex matrices. Its preformances rely on several parameters, including its minimal rank distance. Gabidulin codes are codes in this metric, famous for their optimality and thanks to efficient decoding algorithms. That's why they are used to design space-time codes. The main difficulty is to design complex matrices from binary matrices. The aim of the works collected here is to generalize Gabidulin codes to number fields, especially cyclique extesnions. We see that they have the same properties than Gabidulin codes over finite fields. We study several errors and erasures models and introduce a quadratic algorithm to recover transmitted information. When computing in finite fields, we are faced with the growing size problem. Indeed, the size of the coefficients grows exponentielly along the algorithm. To avoid this problem, it is possible to reduce the code, in order to compute in a finite field. Finally, we design a family of space-time codes, based on generalised Gabidulin codes. We see that our codes have performances similar to those of existing codes, and that they have additional structure.

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Communication over a wireless channel is a challenging task because of the inherent fading effects. Any wireless communication system employs some form of diversity improving techniques in order to improve the reliability of the channel. This thesis deals with efficient code design for two different spatial diversity techniques, viz, diversity by employing multiple antennas at the transmitter and/or the receiver, and diversity through cooperative commu- nication between users. In other words, this thesis deals with efficient code design for (1) multiple-input multiple-output (MIMO) channels, and (2) wireless relay channels. Codes for the MIMO channel are termed space-time (ST) codes and those for the relay channels are called distributed ST codes. The first part of the thesis focuses on ST code construction for MIMO fading channel with perfect channel state information (CSI) at the receiver, and no CSI at the transmitter. As a measure of performance we use the rate-diversity tradeoff and the Diversity-Multiplexing Gain (D-MG) Tradeoff, which are two different tradeoffs characterizing the tradeoff between the rate and the reliability achievable by any ST code. We provide two types of code constructions that are optimal with respect to the rate-diversity tradeoff; one is based on the rank-distance codes which are traditionally applied as codes for storage devices, and the second construction is based on a matrix representation of a cayley algebra. The second contribution in ST code constructions is related to codes with a certain nonvanishing determinant (NVD) property. Motivation for these constructions is a recent result on the necessary and sufficient conditions for an ST code to achieve the D-MG tradeoff. Explicit code constructions satisfying these conditions are provided for certain number of transmit antennas. The second part of the thesis focuses on distributed ST code construction for wireless relay channel. The transmission protocol follows a two-hop model wherein the source broadcasts a vector in the first hop and in the second hop the relays transmit a vector that is a transformation of the received vector by a relay-specific unitary transformation. While the source and relays do not have CSI, at the destination we assume two different scenarios (a) destina- tion with complete CSI (b) destination with only the relay-destination CSI. For both these scenarios, we derive a Chernoff bound on the pair-wise error probability and propose code design criteria. For the first case, we provide explicit construction of distributed ST codes with lower decoding complexity compared to codes based on some earlier system models. For the latter case, we propose a novel differential encoding and differential decoding technique and also provide explicit code constructions. At the heart of all these constructions is the cyclic division algebra (CDA) and its matrix representations. We translate the problem of code construction in each of the above scenarios to the problem of constructing CDAs satisfying certain properties. Explicit examples are provided to illustrate each of these constructions.

Approved for public release, distribution is unlimited / This thesis investigates the simulation performance of multiple-input multiple-output (MIMO) systems utilizing Alamoutibased space-time block coding (STBC) technique. The MIMO communication systems using STBC technique employing both single- carrier modulation and orthogonal frequency division multiplexing (OFDM) are simulated in Matlab. The physical layer part of the IEEE 802.16a standard is used in constructing the simulated OFDM schemes. Stanford University Interim (SUI) channel models are selected for the wireless channel in the simulation process. The performance results of the simulated MIMO systems are compared to those of conventional single antenna systems. / Lieutenant Junior Grade, Turkish Navy

Approved for public release; distribution in unlimited. / OFDM is a modulation technique that achieves high data rates, increased bandwidth efficiency and robustness in multipath environments. However, OFDM has some disadvantages, such as sensitivity to channel fading, large peak to average ratio and sensitivity to frequency offset. The latter causes intercarrier interference (ICI) and a reduction in the amplitude of the desired subcarrier which results in loss of orthogonality. In this thesis, the effects of frequency offset are studied in terms of loss of orthogonality. A number of techniques for frequency offset estimation are presented and tested in computer simulations. / Lieutenant Junior Grade, Turkish Navy

This thesis consists of two parts addressing two different problems in fading channels. The first part deals with asynchronous cooperative relay communication. The assumption of nodes in a cooperative communication relay network operating in synchronous fashion is often unrealistic. In this work we consider two different models of asynchronous operation in cooperative-diversity networks experiencing slow fading and examine the corresponding Diversity-Multiplexing Tradeoffs (DMT). For both models, we propose protocols and distributed space-time codes that asymptotically achieve the transmit diversity bound on DMT for all multiplexing gains and for number of relays N ≥ 2. The distributed space-time codes for all the protocols considered are based on Cyclic Division Algebras (CDA). The second part of the work addresses the DMT analysis of diversity embedded codes for MIMO channels. Diversity embedded codes are high rate codes that are designed so that they have a high diversity code embedded within them. This allows a form of opportunistic communication depending on the channel conditions. The high diversity code ensures that at least a part of the information is received reliably, whereas the embedded high rate code allows additional information to be transferred if the channel is good. This can be thought of coding the data into two streams: high priority and low priority streams so that the high priority stream gets a better reliability than the lower priority stream. We show that superposition based diversity embedded codes in conjunction with naive single stream decoding is sub-optimal in terms of the DM tradeoff. We then construct explicit diversity embedded codes by the superposition of approximately universal space-time codes from CDAs. The relationship between broadcast channels and the diversity embedded setting is then utilized to provide some achievable Diversity Gain Region (DGR) for MIMO broadcast Channels.

The aim of the dissertation was to design a realistic, low-complexity non-binary (NB) low density parity check (LDPC) coded space-time-frequency (STF) coded multiple-input multiple-output (MIMO) orthogonal frequency division multiplexing (OFDM) system with an iterative joint decoder and detector structure at the receiver. The goal of the first part of the dissertation was to compare the performance of different design procedures for NB-LDPC codes on an additive white Gaussian noise (AWGN) channel, taking into account the constraint on the code length. The effect of quantisation on the performance of the code was also analysed. Different methods for choosing the NB elements in the parity check matrix were compared. For the STF coding, a class of universal STF codes was used. These codes use linear pre-coding and a layering approach based on Diophantine numbers to achieve full diversity and a transmission rate (in symbols per channel use per frequency) equal to the number of transmitter antennas. The study of the system considers a comparative performance analysis of di erent ST, SF and STF codes. The simulations of the system were performed on a triply selective block fading channel. Thus, there was selectivity in the fading over time, space and frequency. The effect of quantisation at the receiver on the achievable diversity of linearly pre-coded systems (such as the STF codes used) was mathematically derived and verified with simulations. A sphere decoder (SD) was used as a MIMO detector. The standard method used to create a soft-input soft output (SISO) SD uses a hard-to-soft process and the max-log-map approximation. A new approach was developed which combines a Hopfield network with the SD. This SD-Hopfield detector was connected with the fast Fourier transform belief propagation (FFT-BP) algorithm in an iterative structure. This iterative system was able to achieve the same bit error rate (BER) performance as the original SISO-SD at a reduced complexity. The use of the iterative Hopfield-SD and FFT-BP decoder system also allows performance to be traded off for complexity by varying the number of decoding iterations. The complete system employs a NB-LDPC code concatenated with an STF code at the transmitter with a SISO-SD and FFT-BP decoder connected in an iterative structure at the receiver. The system was analysed in varying channel conditions taking into account the effect of correlation and quantisation. The performance of different SF and STF codes were compared and analysed in the system. An analysis comparing different numbers of FFT-BP and outer iterations was also done. AFRIKAANS : Die doel van die verhandeling was om ’n realistiese, lae-kompleksiteit nie-binˆere (NB) LDPC gekodeerde ruimte-tyd-frekwensie-gekodeerde MIMO-OFDM-sisteem met iteratiewe gesamentlike dekodeerder- en detektorstrukture by die ontvanger te ontwerp. Die eerstem deel van die verhandeling was om die werkverrigting van verskillende ontwerpprosedures vir NB-LDPC kodes op ’n gesommeerde wit Gausruiskanaal te vergelyk met inagneming van die beperking op die lengte van die kode. Verskillende metodes om die nie-bineêre elemente in die pariteitstoetsmatriks te kies, is gebruik. Vir die ruimte-tyd-frekwensiekodering is ’n klas universele ruimte-tyd-frekwensiekodes gebruik. Hierdie kodes gebruik lineêre pre-kodering en ’n laagbenadering gebaseer op Diofantiese syfers om volle diversiteit te bereik en ’n oordragtempo (in simbole per kanaalgebruik per frekwensie) gelyk aan die aantal senderantennes. Die studie van die sisteem oorweeg ’n vergelykende werkverrigtinganalisie van verskillende ruimte-tyd-, ruimte-freksensie- en ruimte-tyd-frekwensiekodes. Die simulasies van die sisteem is gedoen op ’n drievoudig selektiewe blokwegsterwingskanaal. Daar was dus selektiwiteit in die wegsterwing oor tyd, ruimte en frekwensie. Die effek van kwantisering by die ontvanger op die bereikbare diversiteit van lineêr pre-gekodeerde sisteme (soos die ruimte-tyd-frekwensiekodes wat gebruik is) is matematies afgelei en bevestig deur simulasies. ’n Sfeerdekodeerder (SD) is gebruik as ’n MIMO-detektor. Die standaardmetode wat gebuik is om ’n sagte-inset-sagte-uitset (SISO) SD te skep, gebruik ’n harde-na-sagte proses en die maksimum logaritmiese afbeelding-benadering. ’n Nuwe benadering wat ’n Hopfield-netwerk met die SD kombineer, is ontwikkel. Hierdie SD-Hopfield-detektor is verbind met die FFT-BP-algoritme in iteratiewe strukture. Hierdie iteratiewe sisteem was in staat om dieselfde bisfouttempo te bereik as die oorspronklike SISO-SD, met laer kompleksiteit. Die gebruik van die iteratiewe Hopfield-SD en FFT-BP-dekodeerdersisteem maak ook daarvoor voorsiening dat werkverrigting opgeweeg kan word teen kompleksiteit deur die aantal dekodering-iterasies te varieer. Die volledige sisteem maak gebruik van ’n QC-NB-LDPC-kode wat met ’n ruimte-tyd-frekwensiekode by die sender aaneengeskakel is met ’n SISO-SD en FFT-BP-dekodeerder wat in ’n iteratiewe struktuur by die ontvanger gekoppel is. Die sisteem is onder ’n verskeidenheid kanaalkondisies ge-analiseer met inagneming van die effek van korrelasie en kwantisering. Die werkverrigting van verskillende ruimte-frekwensie- en ruimte-tyd-frekwensiekodes is vergelyk en in die sisteem ge-analiseer. ’n Analise om ’n wisselende aantal FFT-BP en buite-iterasies te vergelyk, is ook gedoen. Copyright / Dissertation (MEng)--University of Pretoria, 2010. / Electrical, Electronic and Computer Engineering / unrestricted

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Abstract Multiple-input multiple-output transmitted signal design for the partially coherent Rayleigh fading channels with discrete inputs under a given average transmit power constraint is consider in this thesis. The objective is to design the space-time constellations and linear precoders to adapt to the degradation caused by the imperfect channel estimation at the receiver and the transmit-receive antenna correlation. The system is partially coherent so that the multiple-input multiple-output channel coefficients are estimated at the receiver and its error covariance matrix is fed back to the transmitter. Two constellation design criteria, one for the single and another for the multiple transmit antennae are proposed. An upper bound on the average bit error probability for the single transmit antenna and cutoff rate, i.e., a lower bound on the mutual information, for multiple transmit antennae are derived. Both criteria are functions of channel estimation error covariance matrix. The designed constellations are called as partially coherent constellation. Additionally, to use the resulting constellations together with forward error control codes requires efficient bit mapping schemes. Because these constellations lack geometrical symmetry in general, the Gray mapping is not always possible in the majority of the constellations obtained. Moreover, different mapping schemes may lead to highly different bit error rate performances. Thus, an efficient bit mapping algorithm called the modified binary switching algorithm is proposed. It minimizes an upper bound on the average bit error probability. It is shown through computer simulations that the designed partially coherent constellation and their optimized bit mapping algorithm together with turbo codes outperform the conventional constellations. Linear precoder design was also considered as a simpler, suboptimal alternative. The cutoff rate expression is again used as a criterion to design the linear precoder. A linear precoder is obtained by numerically maximizing the cutoff rate with respect to the precoder matrix with a given average transmit power constraint. Furthermore, the precoder matrix is decomposed using singular-value-decomposition into the input shaping, power loading, and beamforming matrices. The beamforming matrix is found to coincide with the eigenvectors of the transmit correlation matrix. The power loading and input shaping matrices are solved numerically using the difference of convex functions programming algorithm and optimization under the unitary constraint, respectively. Computer simulations show that the performance gains of the designed precoders are significant compared to the cutoff rate optimized partially coherent constellations without precoding. / Tiivistelmä Väitöskirjassa tarkastellaan lähetyssignaalien suunnittelua osittain koherenteissa Rayleigh-häipyvissä kanavissa toimiviin monitulo-monilähtöjärjestelmiin (MIMO). Lähettimen keskimääräinen lähetysteho oletetaan rajoitetuksi ja lähetyssignaali diskreetiksi. Tavoitteena on suunnitella tila-aikakonstellaatioita ja lineaarisia esikoodereita jotka mukautuvat epätäydellisen kanavaestimoinnin aiheuttamaan suorituskyvyn heikkenemiseen sekä lähetin- ja vastaanotinantennien väliseen korrelaatioon. Tarkasteltavien järjestelmien osittainen koherenttisuus tarkoittaa sitä, että MIMO-kanavan kanavakertoimet estimoidaan vastaanottimessa, josta niiden virhekovarianssimatriisi lähetetään lähettimelle. Työssä esitetään kaksi konstellaatiosuunnittelukriteeriä, toinen yhdelle lähetinantennille ja toinen moniantennilähettimelle. Molemmat kriteerit ovat kanavan estimaatiovirheen kovarianssimatriisin funktioita. Työssä johdetaan yläraja keskimääräiselle bittivirhetodennäköisyydelle yhden lähetinantennin tapauksessa sekä rajanopeus (cutoff rate), joka on alaraja keskinäisinformaatiolle, usean lähetinantennin tapauksessa. Konstellaatioiden käyttö yhdessä virheenkorjauskoodien kanssa edellyttää tehokaita menetelmiä, joilla bitit kuvataan konstellaatiopisteisiin. Koska tarvittavat konstellaatiot eivät ole tyypillisesti geometrisesti symmetrisiä, Gray-kuvaus ei ole yleensä mahdollinen.Lisäksi erilaiset kuvausmenetelmät voivat johtaa täysin erilaisiin bittivirhesuhteisiin. Tästä johtuen työssä esitetään uusi kuvausalgoritmi (modified bit switching algorithm), joka minimoi keskimääräisen bittivirhetodennäköisyyden ylärajan. Simulointitulokset osoittavat, että työssä kehitetyt konstellaatiot antavat paremman suorituskyvyn turbokoodatuissa järjestelmissä kuin perinteiset konstellaatiot. Työssä tarkastellaan myös lineaarista esikoodausta yksinkertaisena, alioptimaalisena vaihtoehtona uusille konstellaatioille. Esikoodauksen suunnittelussa käytetään samaa kriteeriä kuin konstellaatioiden kehityksessä eli rajanopeutta. Lineaarinen esikooderi löydetään numeerisesti maksimoimalla rajanopeus kun rajoitusehtona on lähetysteho. Esikoodausmatriisi hajotetaan singulaariarvohajotelmaa käyttäen esisuodatus, tehoallokaatio ja keilanmuodostusmatriiseiksi, jonka havaitaan vastaavan lähetyskorrelaatiomatriisin ominaisvektoreita. Tehoallokaatiomatriisi ratkaistaan numeerisesti käyttäen difference of convex functions -optimointia ja esisuodatusmatriisi optimoinnilla unitaarista rajoitusehtoa käyttäen. Simulaatiotulokset osoittavat uusien esikoodereiden tarjoavan merkittävän suorituskykyedun sellaisiin rajanopeusoptimoituihin osittain koherentteihin konstellaatioihin nähden, jotka eivät käytä esikoodausta.