[Resumen]
Since Shannon demonstrated in 1948 the feasibility of achieving an arbitrarily low error probability in a communications system provided that the transmission rate was kept below a certain limit, one of the greatest challenges in the realm of digital communications and, more specifically, in the channel coding field, has been finding codes that are able to approach this limit as much as possible with a reasonable encoding and decoding complexity, However, it was not until 1993, when Berrou et al. presented the turbo codes, that a coding scheme capable of performing at less than 1dB from Shannon's limit with an extremely low error probability was found. The idea on which these codes are based is the iterative decoding of concatenated components that exchange information about the transmitted bits, which is known as the "turbo principle".
The generalization of this idea led in 1995 to the rediscovery of LDPC (Low Density Parity Check) codes, proposed for the first time by Gallager in the 60s. LDPC codes are linear block codes with a sparse parity check matrix that are able to surpass the performance of turbo codes with a smaller decoding complexity. However, due to the fact that the generator matrix of general LDPC codes is not sparse, their encoding complexity can be excessively high. LDGM (Low Density Generator Matrix) codes, a particular case of LDPC codes, are codes with a sparse generator matrix, thanks to which they present a lower encoding complexity. However, except for the case of very high rate codes, LDGM codes are "bad", i.e., they have a non-zero error probability that is independent of the code block length. More recently, IRA (Irregular Repeat-Accumulated) codes, consisting of the serial concatenation of a LDGM code and an accumulator, have been proposed. IRA codes are able to get close to the performance of LDPC codes with an encoding complexity similar to that of LDGM codes.
In this thesis we explore an alternative to IRA codes consisting in the serial concatenation of two LDGM codes, a scheme that we will denote SCLDGM (Serially-Concatenated Low-Density Generator Matrix). The basic premise of SCLDGM codes is that an inner code of rate close to the desired transmission rate fixes most of the errors, and an external code of rate close to one corrects the few errors that result from decoding the inner code.
For any of these schemes to perform as close as possible to the capacity limit it is necessary to determine the code parameters that best fit the channel over which the transmission will be done. The two techniques most commonly used in the literature to optimize LDPC codes are Density Evolution (DE) and EXtrinsic Information Transfer (EXIT) charts, which have been employed to obtain optimized codes that perform at a few tenths of a decibel of the AWGN channel capacity. However, no optimization techniques have been presented for SCLDGM codes, which so far have been designed heuristically and therefore their performance is far from the performance achieved by IRA and LDPC codes.
Other of the most important advances that have occurred in recent years is the utilization of multiple antennas at the trasmitter and the receiver, which is known as MIMO (Multiple-Input Multiple-Output) systems. Telatar showed that the channel capacity in these kind of systems scales linearly with the minimum number of transmit and receive antennas, which enables us to achieve spectral efficiencies far greater than with systems with a single transmit and receive antenna (or Single Input Single Output (SISO) systems). This important advantage has attracted a lot of attention from the research community, and has caused that many of the new standards, such as WiMax 802.16e or WiFi 802.11n, as well as future 4G systems are based on MIMO systems.
The main problem of MIMO systems is the high complexity of optimum detection, which grows exponentially with the number of transmit antennas and the number of modulation levels. Several suboptimum algorithms have been proposed to reduce this complexity, most notably the SIC-MMSE (Soft-Interference Cancellation Minimum Mean Square Error) and spherical detectors. Another major issue is the high complexity of the channel estimation, due to the large number of coefficients which determine it. There are techniques, such as Maximum-Likelihood-Expectation-Maximization (ML-EM), that have been successfully applied to estimate MIMO channels but, as in the case of detection, they suffer from the problem of a very high complexity when the number of transmit antennas or the size of the constellation increase.
The main objective of this work is the study and optimization of SCLDGM codes in SISO and MIMO channels. To this end, we propose an optimization method for SCLDGM codes based on EXIT charts that allow these codes to exceed the performance of IRA codes existing in the literature and get close to the performance of LDPC codes, with the advantage over the latter of a lower encoding complexity.
We also propose optimized SCLDGM codes for both spherical and SIC-MMSE suboptimal MIMO detectors, constituting a system that is capable of approaching the capacity limits of MIMO channels with a low complexity encoding, detection and decoding. We analyze the BICM (Bit-Interleaved Coded Modulation) scheme and the concatenation of SCLDGM codes with Space-Time Codes (STC) in ergodic and quasi-static MIMO channels.
Furthermore, we explore the combination of these codes with different channel estimation algorithms that will take advantage of the low complexity of the suboptimum detectors to reduce the complexity of the estimation process while keeping a low distance to the capacity limit.
Finally, we propose coding schemes for low rates involving the serial concatenation of several LDGM codes, reducing the complexity of recently proposed schemes based on Hadamard codes
