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Abstract
This dissertation tries to unify the three classes of capacity-approaching codes: time-varying convolutional codes, turbo codes, and low-density parity-check codes. The structural properties of periodically time-varying convolutional codes are studied first. It is shown that every periodically time-varying convolutional encoder is equivalent to a time-invariant convolutional encoder. A method to find the time-invariant convolutional encoder with feedback equivalent to a periodically time-varying convolutional encoder with feedback is found. Based on the equivalence between periodically time-varying and time-invariant convolutional encoders, a new catastrophic condition for periodically time-varying convolutional encoders is derived. A new technique to convert a catastrophic periodically time-varying convolutional encoder into a noncatastrophic encoder is also presented. The average transfer functions of periodically time-varying convolutional codes and their relationship to the spectrum-thinning conjecture of periodically time-varying convolutional codes are discussed.
Time-varying convolutional codes are found to be compatible with the turbo iterative decoding algorithm and thus can be used to build good turbo codes. A time-varying turbo code, i.e., turbo code with periodically time-varying convolutional component codes, that outperforms some of the well-known turbo codes is constructed. This turbo code is found to have thinner distance spectrum than the other turbo codes.
To understand the relationship between turbo and low-density parity-check codes better, a general form of the generator and parity-check matrices of an arbitrary turbo code is derived. Turbo codes are then decoded by the sum-product algorithm based on their parity-check matrices. Guided by the general expression of the parity-check matrix of an arbitrary turbo code, turbo encoders with large memory and selected generator polynomials are used to generate turbo codes with low-density parity-check matrices. These codes can be decoded by the sum-product algorithm, whereas the turbo decoding algorithm is impractical due to the large encoder memory. The bit-error-rate performance of these turbo codes is still worse than that of the conventional low-density parity-check codes. This loss in performance may be justified by the largely decreased encoder complexity.
