Polar codes, recently invented by Arikan, are the first class of codes known to achieve the symmetric capacity for a large class of channels. The symmetric capacity is the highest achievable rate subject to using the binary input letters of the channel with equal probability. Polar code construction is based on a phenomenon called channel polarization. The encoding as well as the decoding operation of polar codes can be implemented with O(N logN) complexity, where N is the blocklength of the code. In this work, we study the factor graph representation of finite-length polar codes and their effect on the belief propagation (BP) decoding process over Binary Erasure Channel (BEC). Particularly, we study the parity-check-based (H-Based) as well as the generator based (G-based) factor graphs of polar codes. As these factor graphs are not unique for a code, we study and compare the performance of Belief Propagation (BP) decoders on number of well-known graphs. Error rates and complexities are reported for a number of cases. Comparisons are also made with the Successive Cancellation (SC) decoder. High errors are related to the so-called stopping sets of the underlying graphs. we discuss the pros and cons of BP decoder over SC decoder for various code lengths. / Thesis (Master, Electrical & Computer Engineering) -- Queen's University, 2012-01-31 17:10:59.955

Subjective video quality and video decoding complexity are jointly optimized in order to determine the video encoding parameters that will result in the best Quality of Experience (QoE) for an end user watching a video clip on a mobile device. Subjective video quality is estimated by an objective criteria, video quality metric (VQM), and a method for predicting the video quality of a test sequence from the available training sequences with similar content characteristics is presented. Standardized spatial index and temporal index metrics are utilized in order to measure content similarity. A statistical approach for modeling decoding complexity on a hardware platform using content features extracted from video clips is presented. The overall decoding complexity is modeled as the sum of component complexities that are associated with the computation intensive code blocks present in state-of-the-art hybrid video decoders. The content features and decoding complexities are modeled as random parameters and their joint probability density function is predicted as Gaussian Mixture Models (GMM). These GMMs are obtained off-line using a large training set comprised of video clips. Subsequently, decoding complexity of a new video clip is estimated by using the available GMM and the content features extracted in real time. A novel method to determine the video decoding capacity of mobile terminals by using a set of subjective decodability experiments that are performed once for each device is also proposed. Finally, the estimated video quality of a content and the decoding capacity of a device are combined in a utility-complexity framework that optimizes complexity-quality trade-off to determine video coding parameters that result in highest video quality without exceeding the hardware capabilities of a client device. The simulation results indicate that this approach is capable of predicting the user viewing satisfaction on a mobile device.

One of the most popular ways to exploit the advantages of a multiple-input multiple-output (MIMO) system is using space time block coding. A space time block code (STBC) is a finite set of complex matrices whose entries consist of the information symbols to be transmitted. A linear STBC is one in which the information symbols are linearly combined to form a two-dimensional code matrix. A well known method of maximum-likelihood (ML) decoding of such STBCs is using the sphere decoder (SD). In this thesis, new constructions of STBCs with low sphere decoding complexity are presented and various ways of characterizing and reducing the sphere decoding complexity of an STBC are addressed. The construction of low sphere decoding complexity STBCs is tackled using irreducible matrix representations of Clifford algebras, cyclic division algebras and crossed-product algebras. The complexity reduction algorithms for the STBCs constructed are explored using tree based search algorithms. Considering an STBC as a vector space over the set of weight matrices, the problem of characterizing the sphere decoding complexity is addressed using quadratic form representations. The main results are as follows. A sub-class of fast decodable STBCs known as Block Orthogonal STBCs (BOSTBCs) are explored. A set of sufficient conditions to obtain BOSTBCs are explained. How the block orthogonal structure of these codes can be exploited to reduce the SD complexity of the STBC is then explained using a depth first tree search algorithm. Bounds on the SD complexity reduction and its relationship with the block orthogonal structure are then addressed. A set of constructions to obtain BOSTBCs are presented next using Clifford unitary weight designs (CUWDs), Coordinate-interleaved orthogonal designs (CIODs), cyclic division algebras and crossed product algebras which show that a lot of codes existing in literature exhibit the block orthogonal property. Next, the dependency of the ordering of information symbols on the SD complexity is discussed following which a quadratic form representation known as the Hurwitz-Radon quadratic form (HRQF) of an STBC is presented which is solely dependent on the weight matrices of the STBC and their ordering. It is then shown that the SD complexity is only a function of the weight matrices defining the code and their ordering, and not of the channel realization (even though the equivalent channel when SD is used depends on the channel realization). It is also shown that the SD complexity is completely captured into a single matrix obtained from the HRQF. Also, for a given set of weight matrices, an algorithm to obtain a best ordering of them leading to the least SD complexity is presented using the HRQF matrix.

Window decoding schedules are very attractive for message passing decoding of spatially coupled LDPC codes. They take advantage of the inherent convolutional code structure and allow continuous transmission with low decoding latency and complexity. In this paper we show that the decoding complexity can be further reduced if suitable message passing schedules are applied within the decoding window. An improvement based schedule is presented that easily adapts to different ensemble structures, window sizes, and channel parameters. Its combination with a serial (on-demand) schedule is also considered. Results from a computer search based schedule are shown for comparison.

It is well known that communication using multiple antennas provides high data rate and reliability. Coding across space and time is necessary to fully exploit the gains offered by multiple input multiple output (MIMO) systems. One such popular method of coding for MIMO systems is space-time block coding. In applications where the terminals do not have enough physical space to mount multiple antennas, relaying or cooperation between multiple single antenna terminals can help achieve spatial diversity in such scenarios as well. Relaying techniques can also help improve the range and reliability of communication. Recently it has been shown that certain space-time block codes (STBCs) can be employed in a distributed fashion in single antenna relay networks to extract the same benefits as in point to point MIMO systems. Such STBCs are called distributed STBCs. However an important practical issue with STBCs and DSTBCs is its associated high maximum likelihood (ML) decoding complexity. The central theme of this thesis is to systematically construct STBCs and DSTBCs applicable for various scenarios such that are amenable for low decoding complexity. The first part of this thesis provides constructions of high rate STBCs from crossed product algebras that are minimum mean squared error (MMSE) optimal, i.e., achieves the least symbol error rate under MMSE reception. Moreover several previous constructions of MMSE optimal STBCs are found to be special cases of the constructions in this thesis. It is well known that STBCs from orthogonal designs offer single symbol ML decoding along with full diversity but the rate of orthogonal designs fall exponentially with the number of transmit antennas. Thus it is evident that there exists a tradeoff between rate and ML decoding complexity of full diversity STBCs. In the second part of the thesis, a definition of rate of a STBC is proposed and the problem of optimal tradeoff between rate and ML decoding complexity is posed. An algebraic framework based on extended Clifford algebras is introduced to study the optimal tradeoff for a class of multi-symbol ML decodable STBCs called ‘Clifford unitary weight (CUW) STBCs’ which include orthogonal designs as a special case. Code constructions optimally meeting this tradeoff are also obtained using extended Clifford algebras. All CUW-STBCs achieve full diversity as well. The third part of this thesis focusses on constructing DSTBCs with low ML decoding complexity for two hop, amplify and forward based relay networks under various scenarios. The symbol synchronous, coherent case is first considered and conditions for a DSTBC to be multi-group ML decodable are first obtained. Then three new classes of four-group ML decodable full diversity DSTBCs are systematically constructed for arbitrary number of relays. Next the symbol synchronous non-coherent case is considered and full diversity, four group decodable distributed differential STBCs (DDSTBCs) are constructed for power of two number of relays. These DDSTBCs have the best error performance compared to all previous works along with low ML decoding complexity. For the symbol asynchronous, coherent case, a transmission scheme based on orthogonal frequency division multiplexing (OFDM) is proposed to mitigate the effects of timing errors at the relay nodes and sufficient conditions for a DSTBC to be applicable in this new transmission scheme are given. Many of the existing DSTBCs including the ones in this thesis are found to satisfy these sufficient conditions. As a further extension, differential encoding is combined with the proposed transmission scheme to arrive at a new transmission scheme that can achieve full diversity in symbol asynchronous, non-coherent relay networks with no knowledge of the timing errors at the relay nodes. The DDSTBCs in this thesis are proposed for application in the proposed transmission scheme for symbol asynchronous, non-coherent relay networks. As a parallel to the non-coherent schemes based on differential encoding, we also propose non-coherent schemes for symbol synchronous and symbol asynchronous relay networks that are based on training. This training based transmission scheme leverages existing coherent DSTBCs for non-coherent communication in relay networks. Simulations show that this training scheme when used along with the coherent DSTBCs in this thesis outperform the best known DDSTBCs in the literature. Finally, in the last part of the thesis, connections between multi-group ML decodable unitary weight (UW) STBCs and groups with real elements are established for the first time. Using this connection, we translate the necessary and sufficient conditions for multi-group ML decoding of UW-STBCs entirely in group theoretic terms. We discuss various examples of multi-group decodable UW-STBCs together with their associated groups and list the real elements involved. These examples include orthogonal designs, quasi-orthogonal designs among many others.

Window decoding schedules are very attractive for message passing decoding of spatially coupled LDPC codes. They take advantage of the inherent convolutional code structure and allow continuous transmission with low decoding latency and complexity. In this paper we show that the decoding complexity can be further reduced if suitable message passing schedules are applied within the decoding window. An improvement based schedule is presented that easily adapts to different ensemble structures, window sizes, and channel parameters. Its combination with a serial (on-demand) schedule is also considered. Results from a computer search based schedule are shown for comparison.