Analysis and Design of Tuned Turbo Codes

It has been widely observed that there exists a fundamental trade-off between the minimum (Hamming) distance properties and the iterative decoding convergence behavior of turbo-like codes. While capacity achieving code ensembles typically are asymptotically bad in the sense that their minimum distance does not grow linearly with block length, and they therefore exhibit an error floor at moderate-to-high signal to noise ratios, asymptotically good codes usually converge further away from channel capacity. In this paper, we introduce the concept of tuned turbo codes, a family of asymptotically good hybrid concatenated code ensembles, where asymptotic minimum distance growth rates, convergence thresholds, and code rates can be traded-off using two tuning parameters, {\lambda} and {\mu}. By decreasing {\lambda}, the asymptotic minimum distance growth rate is reduced in exchange for improved iterative decoding convergence behavior, while increasing {\lambda} raises the asymptotic minimum distance growth rate at the expense of worse convergence behavior, and thus the code performance can be tuned to fit the desired application. By decreasing {\mu}, a similar tuning behavior can be achieved for higher rate code ensembles.Comment: Accepted for publication in IEEE Transactions on Information Theor

An Exit-Chart Aided Design Procedure for Near-Capacity N-Component Parallel Concatenated Codes

Shannon’s channel capacity specifies the upper bound on the amount of bits per channel use. In this paper, we explicitly demonstrate that twin-component turbo codes suffer from a capacity loss, when the component code rate is less than unity, which is shown by exploiting the so-called area properties of Extrinsic Information Transfer (EXIT) charts. This capacity loss is unavoidable for twin-component turbo codes, when the overall turbo coding rate is less than 1=2, while multiple-component turbo codes are capable of overcoming it by using unity-rate component codes. In order to demonstrate that multiplecomponent turbo codes are capable of exhibiting a better asymptotic performance, the minimum Signal Noise Ratio (SNR) required for the EXIT charts to have open convergence tunnels is used as our metric, which is referred to as ‘the open tunnel SNR threshold’. Furthermore, the employment of conventional two-dimensional EXIT charts is extended to facilitate the analysis of N-component turbo codes. Our results confirm that multiple-component turbo codes approach the Discrete-input Continuous-output Memoryless Channel’s (DCMC) capacity more closely and achieve a lower Bit Error Ratio (BER) than twin-component turbo codes at the same coding rate and the same complexity

Mathematical approach to channel codes with a diagonal matrix structure

Digital communications have now become a fundamental part of modern society. In communications, channel coding is an effective way to reduce the information rate down to channel capacity so that the information can be transmitted reliably through the channel. This thesis is devoted to studying the mathematical theory and analysis of channel codes that possess a useful diagonal structure in the parity-check and generator matrices. The first aspect of these codes that is studied is the ability to describe the parity-check matrix of a code with sliding diagonal structure using polynomials. Using this framework, an efficient new method is proposed to obtain a generator matrix G from certain types of parity-check matrices with a so-called defective cyclic block structure. By the nature of this method, G can also be completely described by a polynomial, which leads to efficient encoder design using shift registers. In addition, there is no need for the matrices to be in systematic form, thus avoiding the need for Gaussian elimination. Following this work, we proceed to explore some of the properties of diagonally structured lowdensity parity-check (LDPC) convolutional codes. LDPC convolutional codes have been shown to be capable of achieving the same capacity-approaching performance as LDPC block codes with iterative message-passing decoding. The first crucial property studied is the minimum free distance of LDPC convolutional code ensembles, an important parameter contributing to the error-correcting capability of the code. Here, asymptotic methods are used to form lower bounds on the ratio of the free distance to constraint length for several ensembles of asymptotically good, protograph-based LDPC convolutional codes. Further, it is shown that this ratio of free distance to constraint length for such LDPC convolutional codes exceeds the ratio of minimum distance to block length for corresponding LDPC block codes. Another interesting property of these codes is the way in which the structure affects the performance in the infamous error floor (which occurs at high signal to noise ratio) of the bit error rate curve. It has been suggested that “near-codewords” may be a significant factor affecting decoding failures of LDPC codes over an additive white Gaussian noise (AWGN) channel. A near-codeword is a sequence that satisfies almost all of the check equations. These nearcodewords can be associated with so-called ‘trapping sets’ that exist in the Tanner graph of a code. In the final major contribution of the thesis, trapping sets of protograph-based LDPC convolutional codes are analysed. Here, asymptotic methods are used to calculate a lower bound for the trapping set growth rates for several ensembles of asymptotically good protograph-based LDPC convolutional codes. This value can be used to predict where the error floor will occur for these codes under iterative message-passing decoding

Publications of the Jet Propulsion Laboratory, July 1969 - June 1970

JPL bibliography of technical reports released from July 1969 through June 197

Joint Permutor Analysis and Design for Multiple Turbo Codes

In this paper, we study the problem of joint permutor analysis and design for J-dimensional multiple turbo codes with J constituent encoders, J>2. The concept of summary distance is extended to multiple permutors of size N and used as the design metric. Using the sphere-packing concept, we prove that the minimum length-2 summary distance (spread) Dmin,2 is asymptoticly upper-bounded by O(N(J-1)/J). We also show that the asymptotic minimum length-2L summary distance Dmin,2L for the class of random permutors is lower-bounded by O(N(J-2)/J-ε), where ε>0 can be arbitrarily small. Then, using the technique of expurgating “bad” symbols, we show that the spread of random permutors can achieve the optimum growth rate, i.e., O(N(J-1)/J), and that the asymptotic growth rate of Dmin,2L can also be improved. The minimum length-2 and length- 4 summary distances are studied for an important practical class of permutors–linear permutors. We prove that there exist J-dimensional multiple linear permutors with optimal spread Dmin,2=O(N(J-1)/J). Finally, we present several joint permutor construction algorithms applicable to multiple turbo codes of short and medium lengths