The Improvement And Implementation Of Fountain Codes And Polar Codes

In modern society, with the rapid growth of Internet technology and wirelesscommunication technology, it becomes quite convenient, real-time and complete to transmitvarious data to all over the world through digital communication systems. In order toguarantee the reliability of digital communication systems during transmitting process, highattention has been paid to the improvement and research of error correcting codes.In recent ten years, as a main type of error correcting codes, the novel linear block codesare proposed to lead the revolution of error correcting codes. Their continuous developmentand improvement certainly will play an crucial role in the future digital communication.Novel linear block codes mainly include fountain codes and polar codes. Their appearance isan important achievement of error correcting codes field in the21st century. For fountaincodes, source singals are encoded by the degree distribution function for selecting restrictionand then are broadcasted forwardly. When the decoder receives a specified number of thecoded words, the source singals can be recovered successfully. This process is favoredbecause of the flexible codes rate and good performance. Based on channel polarizationtheory, polar codes mainly adopt the iteration of Bhattachayya parameters or the quantizationof transmitting channels to construct the codewords. The key points of construction are shownas: the reliability of virtual channels is simplified by Bhattachayya parameter iterations orquantization of transmitting channels. The values of reliability are obtained while theinformation bits can be screened. The key issue of optimal construction is how to calculate thereliability of virtual channels rapidly and accurately. The deviation between proposedencoding algorithm and the actual value is less, the accuracy of channel reliability is higher.Then, polar codes can be constructed with better performance. This paper first focuses on howto improve decoding efficiency for fountain codes. Moreover, the properties of Bhattachayyaparameters are analyzed in polar codes while the new construction of polar codes is proposedin other binary-input momoryless symmetric channels. Finally, the schemes based on polarcodes are considered in relay channels and wireless optical communication systems.Main contributions include:1. Some new linear block codes are introduced, in terms of fountain codes and polarcodes. Their technology backgrounds and development situations are presented with detailedanalysis, summary and conclusion. Some features and properties of encoder/decoder are obtained because of special construction and different degree distribution functions infountain codes. In addition, some properties and characteristics of polar codes are procured byanalyzing the numberical characteristics of combining, splitting and the judging standards ofsuccessive cancellation decoding algorithm for polarization channels. Thereby, some existingrelations between the new types of linear block codes are summarized. Some of the currenttechnical issues and suggestions which are waiting to be solved urgently are pointed out.2. The optimal partial decoding algorithm are proposed for fountain codes. For shortlength LT codes, the traditional belief propagation (BP) and Gaussian elimination (GE)decoding algorithms are given also with their advantages and disadvantages. Fast beliefpropagation (FBP) algorithm which combines some advantages of traditional algorithms isapplied to the decoder of fountain codes. Besides, it has been proved that FBP algorithm isone of the optimal decoding algorithm for the short length LT codes. The FBP algorithm notonly improves successful decoding probability of LT codes, but also reduces the decodingdelay and demand of data storage due to its unique arrangement. Simulation results indicatethat the successful probability of FBP algorithm is48.09%higher than BP algorithm in binarysymmetric channel (BSC). For delay of transmission system, FBP algorithm is less than GEalgorithm but a little higher than BP algorithm. Moreover, there is a time platform threshold inFBP algorithm. Then FBP algorithm is superior to the traditional algorithms in the shortlength and low rate LT codes.3. The upper/lower bounds of error exponent function and Bhattachayya parameter areproved in polar codes. First, the error exponent function should be defined. The relationbetween the channel capacity, which is the variable domain, and error exponent function isconsidered in binary memoryless symmetric channel (BMSC). The error exponent functionand its auxiliary function have extreme properties in binary error channel (BEC) and BSC. So,the binary memoryless extremal theorem is proposed for any BMSC. For the auxiliaryfunction, the extremal theorem can be extended to Bhattachayya parameter in polar codes. Ifthe channel capacity is to be the viriable, it will be proved that the Bhattachayya parameters ofvirtual channels exist a similar relationship of the upper and lower bounds in polar codes.When polar codes are constructed with estimating exact reliability of virtual channels, theBhattachayya parameter values of BEC and BSC should be concerned. Theoretically, byanalyzing both the transmitting reliability of two channels, the optimal scheme should bereconstructed for polar codes in the other BMSC.4. A series of encoding constructing schemes are proposed by the lower bound in polarcodes. Based on the Bhattachayya parameter extremal theorem, the properties about Bhattachayya parameters are analyzed before and after polar codes. The more accurate andconvergent Bhattachayya parameters for unreliable channel are obtained in polar codes. Thelower bound formula is used to estimate the reliability of virtual channels, which improvesand optimizes the polarization structure of unreliable channels. Then, the lower boundconstruction scheme is proposed for BSC which has more accurate estimation and lessimprovement of performance compared with the traditional iterations. For additive whiteGaussian noise channels, the linear construting scheme is proposed and the selecting methodof parameter is obtained by the simulation.5. Polar codes are applied to the time-division half-duplex relay channel. Due to thecharacteristics of channels combining and splitting, polar codes are considered to transmit inthe relay channels. For the half-duplex relay systems, the characteristics of the model andeach node are analyzed. Based on the forwarding protocols and the orthogonal vectors oftransmission, a suitable polarized transmission structure and a transmitting strategy arepresented. Moreover, polar codes are proved to achieve the Shannon limit in the half-duplexrelay systems. The optimizations of time-division and information-division parameters areproposed, while the random selection strategy is optimal in the half-duplex relay systems,.Finally, the simulation results and conclusions are presented.6. In optical turbulence channels, error probability performances of free-space optical(FSO) communication are analyzed for irradiance modulation and direct detection (IM/DD)systems with subcarrier intensity modulation using binary phase-shift keying (SIM-BPSK)and polar codes. The analysis is carried out by different climates in the Gamma-Gammaturbulence channels. For bit-by-bit interleaved channels, the pairwise error probability (PEP)makes a high accurate series presentation about the virtual channels which deduces anasymptotic PEP. Each block has the independent fading in quasi-static fading channels. Thenwe study the upper and lower bound of the frame error ratio performance with Bhattachayyaparameter and density evolution estimations in polar coding FSO systems. Density evolutionestimation is better than Bhattachayya parameter. The numerical results are given to presentthe improvement and performance in polar coding FSO systems.In conclusion, software simulations, tests and comparisons with traditional algorithmsare considered to verify their effectiveness and progressiveness of the proposed algorithms.For all the proposed theorems, their preciseness and scope is proved by the analysis ofmathematical derivation in this paper.