Error control coding in some form is present in virtually every communication system today. Recently, Low Density Parity Check (LDPC) codes have been proposed along with a simple iterative decoding algorithm. These codes have been demonstrated to perform very close to the Shannon Limit. The simplicity of LDPC codes have also led to many interesting asymptotic and finite-length properties of these codes. The techniques for designing good LDPC codes over a wide variety of channels have been studied. LDPC codes are being used in a wide variety of applications, such as fading channels, Orthogonal Frequency Division Multiplexing (OFDM) systems, source compression etc. This proposal investigates the use of LDPC codes in wiretap channel systems such as quantum key distribution and for congestion localization in large networks. Quantum Key Distribution (QKD) is secure key exchange method where the two legitimate parties first transmit information over a quantum channel, which can be eavesdropped on by the eavesdropper. The QKD system can be modeled as a special case of an wiretap channel system. An wiretap chanel system is a broadcast system, where the sender has to send a message to a legitimate party over a main channel. The wiretapper also receives the message through another channel called the wiretap channel. The sender has to code the transmitted message in such a way so that the legitimate party is able to recover the message without errors, whereas the wiretapper essentially has no information about the message. As we will see, the encoder for such a system is stochastic as opposed to a deterministic encoder in error correction coding. In this research, we propose a coding scheme using LDPC codes for such systems. Congestion in a network occurs when some nodes receive more traffic than they can process. It leads to dropping packets and thus lowering the throughput. On the contrary, if other nodes in the network are aware of the congested nodes, new packets can be dynamically routed through less congested routes. We developed a congestion detection mechanism wherein a few high priority probe packets are routed through the network. A central entity collects the contents of all the probe packets and estimates the state (congested or not) of every node in the network. One important parameter of congeston localization schemes is scalability, i.e. how the number of measurements scales with the size of network as the size of the network grows. We have shown that it is possible to do congestion detection using our scheme for a properly designed network with the number of measurements required growing linearly with the size of the network.

The new developments in coding theory research have revolutionized the application of coding to practical systems. Low-Density Parity Check (LDPC) codes form a class of Shannon limit approaching codes opted for digital communication systems that require high reliability. This thesis investigates the underlying relationship between the spectral properties of the parity check matrix and LDPC decoding convergence. The bit error rate of an LDPC code is plotted for the parity check matrix that has different Second Smallest Eigenvalue Modulus (SSEM) of its corresponding Laplacian matrix. It is found that for a given (n,k) LDPC code, large SSEM has better error floor performance than low SSEM. The value of SSEM decreases as the sparseness in a parity-check matrix is increased. It was also found from the simulation that long LDPC codes have better error floor performance than short codes. This thesis outlines an approach to analyze LDPC decoding based on the eigenvalue analysis of the corresponding parity check matrix.

Η διπλωματική εργασία επικεντρώνεται στη μελέτη της κωδικοποίησης για κώδικες LDPC. Στα πλαίσιά της, θα μελετηθούν τα προβλήματα και η πολυπλοκότητα κωδικοποίησης συναρτήσει του μήκους της κωδικής λέξης. Έμφαση θα δοθεί σε εφαρμογές με μεγάλο μήκος κωδικής λέξης όπως εκείνες που χρησιμοποιούνται σε νέες τηλεπικοινωνιακές εφαρμογές, όπως δορυφορικό Digital Video Broadcast (DVB) DVB-S2, IEEE 802.3an (10GBASE-T) και IEEE 802.16(WiMAX). Σε τέτοιες εφαρμογές όπου η κωδική λέξη μπορεί να έχει μήκος αρκετά μεγαλύτερο των 1000 bits, η πολυπλοκότητα κωδικοποίησης είναι σημαντική. Αυτό συμβαίνει διότι απαιτούνται μεγάλες σε μέγεθος μνήμες για την αποθήκευση του Πίνακα Έλεγχου Ισοτιμίας (Parity-check Matrix H), πολύ μεγάλη χρονική επεξεργαστική πολυπλοκότητα O(n^2) αλλά και πολλά επεξεργαστικά στοιχεία τάξης Ο(n^2). Ο σκοπός λοιπόν είναι να μελετηθούν οι αλγόριθμοι κωδικοποίησης και να μελετηθεί πώς μπορεί να αξιοποιηθεί η αραιότητα του Πίνακα Έλεγχου Ισοτιμίας έτσι ώστε να επιτευχθεί κατά το δυνατόν γραμμική πολυπλοκότητα O(n) κωδικοποίησης. Στη συνέχεια, αφού αναπτυχθεί η κατάλληλη μέθοδος κωδικοποίησης, θα ακολουθήσει η μελέτη και ο σχεδιασμός μίας βέλτιστης VLSI αρχιτεκτονικής για την υλοποίηση σε υλικό του LDPC κωδικοποιητή, ώστε να ικανοποιεί και άλλα πρακτικά κριτήρια, με έμφαση στη μείωση της καθυστέρησης και της απαιτούμενης επιφάνειας. Θα αναπτυχθεί επίσης μια κατάλληλη αρχιτεκτονική για διάφορους βαθμούς παραλληλίας του κωδικοποιητή. / An LDPC code is a linear block code specified by a very sparse parity check matrix (PCM). LDPC codes are usually represented by a bi-partite graph in which a variable node corresponds to a ’coded bit’ or a PCM column, and a check node corresponds to a parity check equation or a PCM row. There is an edge between each pair of nodes if there is a ’one’ in the corresponding PCM entry. In a general analysis an (n, k) LDPC code has k information bits and n coded bits with code rate r = k/n. An important issue in the implementation of LDPC-code based forward error correction systems is the encoding of LDPC codes. Generally, LDPC codes cannot have the simple encoding structures based on of shift registers as in the case of convolutional, turbo codes, or cyclic block codes. However, general LDPC codes do not fall in this category. Except QC-cyclic LDPC codes, most efficient LDPC codes, especially irregular LDPC codes are hard to encode with the idea of shift registers. A straightforward way is to derive a systematic generator matrix from a PCM, and then to encode LDPC code systematically with the generator matrix. This can work for every LDPC code in theory, but practically it is a very bad idea because it has high complexity, as the generator matrix derived from parity-check matrix is not sparse contrasted to the PCM. Generator matrix can be very dense matrix. The objective is to utilize the sparseness to achieve LDPC encoding in linear time. This Master’s thesis presents a flexible encoder architecture using QC-cyclic LDPC codes and efficient two-step encoding algorithm in order to achieve linear time encoding. The particular approach considers several VLSI design issues of LDPC encoder. In particular efficient approaches are presented for reducing memory requirements, for reducing hardware complexity, and increasing the speed and throughput of LDPC encoding. The proposed structure is also generic and scalable, supporting multiple configurations, in terms of bits per symbol and code rate. A generic scalable processing unit is also presented. It supports LDPC codes that define parity check matrix as a combination of identity matrix, shifted identity matrix and all-zero matrix (QC-cyclic LDPC codes). The particular LDPC encoder architecture is synthesized and experimental results are reported.

Since the rediscovery, LDPC codes attract a large amount of research efforts. In 1998, nonbinary LDPC codes were firstly investigated and the results shown that they are better than their binary counterparts in performance. Recently, there is always a requirement from the industry to design applied nonbinary LDPC codes. In this dissertation, we firstly propose a novel class of quasi-cyclic (QC) LDPC codes. This class of QC-LDPC codes embraces both linear encoding complexity and excellent compatibility in various degree distributions and nonbinary expansions. We show by simulation results that our proposed QC-LDPC codes perform as well as their comparable counterparts. However, this proposed code structure is more flexible in designing. This feature may show its power when we change the code length and rate adaptively. Further more, we present two algorithms to generate codes with short girth and better girth distribution. The two algorithms are designed based on progressive edge growth (PEG) algorithm and they are specifically designed for quasi-cyclic structure. The simulation results show the improvement they achieved. In this thesis, we also investigate the believe propagation based iterative algorithms for decoding of nonbinary LDPC codes. The algorithms include sum-product (SP) algorithm, SP algorithm using fast Fourier transform, min-sum (MS) algorithm and complexity reduced extended min-sum (EMS) algorithm. In particular, we present the proposed modified min-sum algorithm with threshold filtering which further reduces the computation complexity.

In recent years, LDPC codes are gaining a lot of attention among researchers. Its near- Shannon performance combined with its highly parallel architecture and lesser complexity compared to Turbo-codes has made LDPC codes one of the most popular forward error correction (FEC) codes in most of the recently ratied wireless communication standards. This thesis focuses on one of these standards, namely the DVB-S2 standard that was ratied in 2005.<p> In this thesis, the design and architecture of a FPGA implementation of an LDPC decoder for the DVB-S2 standard are presented. The decoder architecture is an improvement over others that are published in the current literature. Novel algorithms are devised to use a memory mapping scheme that allows for 360 functional units (FUs) used in decoding to be implemented using the Sum-Product Algorithm (SPA). The functional units (FU) are optimized for reduced hardware resource utilization on a FPGA with a large number of congurable logic blocks (CLBs) and memory blocks. A novel design of a parity-check module (PCM) is presented that veries the parity-check equations of the LDPC codes. Furthermore, a special characteristic of ve of the codes dened in the DVB-S2 standard and their in uence on the decoder design is discussed. Three versions of the LDPC decoder are implemented, namely the 360-FU decoder, the 180-FU decoder and the hybrid 360/180-FU decoder. The decoders are synthesized for two FPGAs. A Xilinx Virtex-II Pro family FPGA is used for comparison purposes and a Xilinx Virtex-6 family FPGA is used to demonstrate the portability of the design. The synthesis results show that the hardware resource utilization and minimum throughput of the decoders presented are competitive with a DVB-S2 LDPC decoder found in the current literature that also uses FPGA technology.

In recent years, LDPC codes are gaining a lot of attention among researchers. Its near-Shannon performance combined with its highly parallel architecture and lesser complexity compared to Turbo-codes has made LDPC codes one of the most popular forward error correction (FEC) codes in most of the recently ratified wireless communication standards. This thesis focuses on one of these standards, namely the DVB-S2 standard that was ratified in 2005. In this thesis, the design and architecture of a FPGA implementation of an LDPC decoder for the DVB-S2 standard are presented. The decoder architecture is an improvement over others that are published in the current literature. Novel algorithms are devised to use a memory mapping scheme that allows for 360 functional units (FUs) used in decoding to be implemented using the Sum-Product Algorithm (SPA). The functional units (FU) are optimized for reduced hardware resource utilization on a FPGA with a large number of configurable logic blocks (CLBs) and memory blocks. A novel design of a parity-check module (PCM) is presented that verifies the parity-check equations of the LDPC codes. Furthermore, a special characteristic of five of the codes defined in the DVB-S2 standard and their influence on the decoder design is discussed. Three versions of the LDPC decoder are implemented, namely the 360-FU decoder, the 180-FU decoder and the hybrid 360/180-FU decoder. The decoders are synthesized for two FPGAs. A Xilinx Virtex-II Pro family FPGA is used for comparison purposes and a Xilinx Virtex-6 family FPGA is used to demonstrate the portability of the design. The synthesis results show that the hardware resource utilization and minimum throughput of the decoders presented are competitive with a DVB-S2 LDPC decoder found in the current literature that also uses FPGA technology.

In recent years, LDPC codes are gaining a lot of attention among researchers. Its near- Shannon performance combined with its highly parallel architecture and lesser complexity compared to Turbo-codes has made LDPC codes one of the most popular forward error correction (FEC) codes in most of the recently ratied wireless communication standards. This thesis focuses on one of these standards, namely the DVB-S2 standard that was ratied in 2005.<p> In this thesis, the design and architecture of a FPGA implementation of an LDPC decoder for the DVB-S2 standard are presented. The decoder architecture is an improvement over others that are published in the current literature. Novel algorithms are devised to use a memory mapping scheme that allows for 360 functional units (FUs) used in decoding to be implemented using the Sum-Product Algorithm (SPA). The functional units (FU) are optimized for reduced hardware resource utilization on a FPGA with a large number of congurable logic blocks (CLBs) and memory blocks. A novel design of a parity-check module (PCM) is presented that veries the parity-check equations of the LDPC codes. Furthermore, a special characteristic of ve of the codes dened in the DVB-S2 standard and their in uence on the decoder design is discussed. Three versions of the LDPC decoder are implemented, namely the 360-FU decoder, the 180-FU decoder and the hybrid 360/180-FU decoder. The decoders are synthesized for two FPGAs. A Xilinx Virtex-II Pro family FPGA is used for comparison purposes and a Xilinx Virtex-6 family FPGA is used to demonstrate the portability of the design. The synthesis results show that the hardware resource utilization and minimum throughput of the decoders presented are competitive with a DVB-S2 LDPC decoder found in the current literature that also uses FPGA technology.

In recent years, LDPC codes are gaining a lot of attention among researchers. Its near-Shannon performance combined with its highly parallel architecture and lesser complexity compared to Turbo-codes has made LDPC codes one of the most popular forward error correction (FEC) codes in most of the recently ratified wireless communication standards. This thesis focuses on one of these standards, namely the DVB-S2 standard that was ratified in 2005. In this thesis, the design and architecture of a FPGA implementation of an LDPC decoder for the DVB-S2 standard are presented. The decoder architecture is an improvement over others that are published in the current literature. Novel algorithms are devised to use a memory mapping scheme that allows for 360 functional units (FUs) used in decoding to be implemented using the Sum-Product Algorithm (SPA). The functional units (FU) are optimized for reduced hardware resource utilization on a FPGA with a large number of configurable logic blocks (CLBs) and memory blocks. A novel design of a parity-check module (PCM) is presented that verifies the parity-check equations of the LDPC codes. Furthermore, a special characteristic of five of the codes defined in the DVB-S2 standard and their influence on the decoder design is discussed. Three versions of the LDPC decoder are implemented, namely the 360-FU decoder, the 180-FU decoder and the hybrid 360/180-FU decoder. The decoders are synthesized for two FPGAs. A Xilinx Virtex-II Pro family FPGA is used for comparison purposes and a Xilinx Virtex-6 family FPGA is used to demonstrate the portability of the design. The synthesis results show that the hardware resource utilization and minimum throughput of the decoders presented are competitive with a DVB-S2 LDPC decoder found in the current literature that also uses FPGA technology.

The low complexity of layered processing makes the layered structure a promising candidate for MIMO systems with a large number of transmit antennas and higher order modulation. For broadband systems, orthogonal frequency division multiplexing (OFDM) appears promising for its immunity against delay spread. In addition, OFDM is especially suitable for frequency selective MIMO systems since the introduction of orthogonal subcarriers makes system design and implementation as simple as those for flat fading channels. Therefore, the combination of layered structure with OFDM is a promising technique for high-speed wireless data transmission. The proposed research is focused on the layered structure for MIMO-OFDM systems, where several techniques are proposed for performance enhancement, namely, channel estimation based on subspace tracking, parallel detection of group-wise space-time codes by predictive soft interference cancellation, quasi-block diagonal low-density parity-check codes (LDPC) coding and statistical data rate allocation for layered systems. For MIMO-OFDM systems, rank reduction by some linear transform matrix is necessary for channel estimation. In the proposed research, we propose a channel estimation algorithm for MIMO-OFDM systems, which uses the optimum low-rank channel approximation obtained by tracking the frequency autocorrelation matrix of the channel response. Then parallel detection algorithm is proposed for a modified layered system with group-wise space-time coding, where the structure of particular component space-time code trellises is exploited using partial information from the Viterbi decoder of the simultaneously decoded interfering component codes. Next we incorporate the layered structure with LDPC to develop a quasi-block diagonal LDPC space-time structure. The lower triangular structure of the parity check matrix introduces correlation between layers. Each layer, as a part of the whole codeword, can be decoded while taking information from other undetected layers to improve the decoding performance. In the end, a modified layered structure is proposed where the layer detection order is fixed and the data rate for each layer is allocated based on the detection order and channel statistics. With Gaussian approximation of layer capacities, we derive the optimum data rate allocation.

This thesis presents a low-power LDPC decoder design based on speculative scheduling of energy necessary to decode dynamically varying data frame in both block-fading channels and general AWGN channels. A model of a memory-efficient low-power high-throughput multi-rate array LDPC decoder as well as its FPGA implementa- tion results is first presented. Then, I propose a decoding scheme that provides the feature of constant-time decoding and thus facilitates real-time applications where guaranteed data rate is required. It pre-analyzes each received data frame to estimate the maximum number of necessary iterations for frame convergence. The results are then used to dynamically adjust decoder frequency and switch between multiple-voltage levels; thereby energy use is minimized. This is in contrast to the conventional fixed-iteration decoding schemes that operate at a fixed voltage level regardless of the quality of data received. Analysis shows that the proposed decoding scheme is widely applicable for both two-phase message-passing (TPMP) decoding algorithm and turbo decoding message passing (TDMP) decoding algorithm in block fading channels, and it is independent of the specific LDPC decoder architecture. A decoder architecture utilizing our recently published multi-rate decoding architecture for general AWGN channels is also presented. The result of this thesis is a decoder design scheme that provides a judicious trade-off between power consumption and coding gain.