Ralph Jordan

Maximum slope convolutional codes

The slope is an important distance parameter for a convolutional code. It can be used to obtain a lower bound on the active burst distance and in this respect essentially determines the error-correcting capability of the code. An upper bound on the slope of rate R=b/c convolutional codes is derived. A new family of convolutional codes, called the maximum slope (MS) code family, is introduced. Tables for rate R=1/2 MS codes with memory 1/spl les/m/spl les/6 are presented. Additionally, some new rate R=(c-1)/c, 3/spl les/c/spl les/6, punctured convolutional codes with rate R=1/2 optimum free distance (OFD) and MS mother codes are presented. Simulation results for the bit error performance of serially concatenated turbo codes with MS component codes are presented.

Woven convolutional codes. II: decoding aspects

An iterative decoding scheme for woven convolutional codes is presented. It operates in a window sliding over the received sequence. This exploits the nature of convolutional codewords as infinite sequences and reflects the concept of considering convolutional encoding and decoding as a continuous process. The decoder is analyzed in terms of decoding delay and decoding complexity. Its basic building block is a symbol-by-symbol a posteriori probability (APP) decoder for convolutional codes, which is a windowed variant of the well-known Bahl-Cocke-Jelinek-Raviv (BCJR) algorithm. Additional interleaving for the woven constructions is introduced by employing convolutional scramblers. It is shown that row-wise random interleaving preserves the lower bound on the free distance of the original woven constructions. Based on the properties of the interleavers, new lower bounds on the free distance of woven constructions with both outer warp and inner warp are derived. Simulation results for woven convolutional codes with and without additional interleaving are presented.

Irregular turbo codes and unequal error protection

Turbo codes (TC) with unequal error protection (UEP) are introduced. We consider a serial cascade of an array of repetition codes, an interleaver, and a TC. Such a coding scheme was introduced by B.J. Frey and D.J.C. MacKay (1999) as an irregular turbo code (ITC). A lower bound on the protection level of the ITC is presented. Additionally, simulation results are shown.

Woven convolutional codes and unequal error protection

Woven convolutional codes with outer warp are used to construct a generator matrix with an effective free distance vector that is lower bounded by the free distances of the component codes. This enables the construction of convolutional codes with unequal error protection.

On permutations with second order separations between cascaded convolutional encoders

A new set of permutation parameters is introduced. It is called the set of second order separations and is a generalization of the well-known symbol separation. A construction method is presented that allows us to generate permutors with large second order separations. Additionally, we derive a lower bound on the minimum distance of serially concatenated codes with a permutation of sufficiently large second order separations.

On interleaver design for serially concatenated convolutional codes

Serially concatenated convolutional codes are considered. The free distance of this construction is shown to be lower-bounded by the product of the free distances of the outer and inner codes, if the precipices of the interleaver are sufficiently large. It is shown how to construct a convolutional scrambler with a given precipice.

On higher order permutors for serially concatenated convolutional codes

A new parameter set for designing permutors is introduced. It is called the set of higher order separations and can be considered as a generalization of the well-known symbol separation (spreading factor). The respective permutor is called a higher order permutor and we show how such a permutor can be constructed. For a second-order permutor in a serially concatenated convolutional encoding scheme we give a lower bound on the minimum distance of the resulting overall code. The integers that determine the sufficiently large separations, i.e., the smallest separations for which the distance properties can be guaranteed, are derived from the active distances of the convolutional component encoders. Additionally, a growth rate of the minimum distance like O((dfree o)lfloorrho/2rfloor+1) is proved for serially concatenated convolutional encoders with permutors having large separations of order rho

Decoding of woven convolutional codes and simulation results

An iterative decoding scheme for woven convolutional codes is presented. It is called pipeline decoding and operates in a window sliding over the received sequence. This exploits the nature of convolutional codes as sequences and suits the concept of convolutional encoding and decoding as a continuous process. The pipeline decoder is analyzed in terms of decoding delay and decoding complexity. Additional interleaving for woven convolutional constructions is introduced by employing a convolutional scrambler. It is shown that some types of interleaving preserve the lower bound on the free distance of the original woven construction. Simulation results for woven convolutional codes are presented.

On nested convolutional codes and their application to woven codes

Nested convolutional codes are a set of convolutional codes that is derived from a given generator matrix. The structural properties of nested convolutional codes and nested generator matrices are studied. A method to construct the set of all minimal (rational) generator matrices of a given convolutional code is presented. As an example, two different sets of nested convolutional codes are derived from two equivalent minimal generator matrices. The significant difference in their free-distance profiles emphasizes the importance of being careful when selecting the generator matrices that determine the nested convolutional codes. As an application of nested convolutional codes, woven codes with outer warp, and inner nested convolutional codes are considered. The free-distance profile of the inner generator matrix is shown to be an important design tool.

An upper bound on the slope of convolutional codes

The slope is an important distance parameter of a convolutional code. It essentially determines the error-correcting capability. Here we derive an upper bound on the slope.