<p>The family of Low Density Parity Check (LDPC) codes is a strong candidate to be used as Forward Error Correction (FEC) in future communication systems due to its strong error correction capability. Most LDPC decoders use the Message Passing algorithm for decoding, which is an iterative algorithm that passes messages between its variable nodes and check nodes. It is not until recently that computation power has become strong enough to make Message Passing on LDPC codes feasible. Although locally simple, the LDPC codes are usually large, which increases the required computation power. Earlier work on LDPC codes has been concentrated on the binary Galois Field, GF(2), but it has been shown that codes from higher order fields have better error correction capability. However, the most efficient LDPC decoder, the Belief Propagation Decoder, has a squared complexity increase when moving to higher order Galois Fields. Transmission over a channel with M-PSK signalling is a common technique to increase spectral efficiency. The information is transmitted as the phase angle of the signal.</p><p>The focus in this Master’s Thesis is on simplifying the Message Passing decoding when having inputs from M-PSK signals transmitted over an AWGN channel. Symbols from higher order Galois Fields were mapped to M-PSK signals, since M-PSK is very bandwidth efficient and the information can be found in the angle of the signal. Several simplifications of the Belief Propagation has been developed and tested. The most promising is the Table Vector Decoder, which is a Message Passing Decoder that uses a table lookup technique for check node operations and vector summation as variable node operations. The table lookup is used to approximate the check node operation in a Belief Propagation decoder. Vector summation is used as an equivalent operation to the variable node operation. Monte Carlo simulations have shown that the Table Vector Decoder can achieve a performance close to the Belief Propagation. The capability of the Table Vector Decoder depends on the number of reconstruction points and the placement of them. The main advantage of the Table Vector Decoder is that its complexity is unaffected by the Galois Field used. Instead, there will be a memory space requirement which depends on the desired number of reconstruction points.</p>

The family of Low Density Parity Check (LDPC) codes is a strong candidate to be used as Forward Error Correction (FEC) in future communication systems due to its strong error correction capability. Most LDPC decoders use the Message Passing algorithm for decoding, which is an iterative algorithm that passes messages between its variable nodes and check nodes. It is not until recently that computation power has become strong enough to make Message Passing on LDPC codes feasible. Although locally simple, the LDPC codes are usually large, which increases the required computation power. Earlier work on LDPC codes has been concentrated on the binary Galois Field, GF(2), but it has been shown that codes from higher order fields have better error correction capability. However, the most efficient LDPC decoder, the Belief Propagation Decoder, has a squared complexity increase when moving to higher order Galois Fields. Transmission over a channel with M-PSK signalling is a common technique to increase spectral efficiency. The information is transmitted as the phase angle of the signal. The focus in this Master’s Thesis is on simplifying the Message Passing decoding when having inputs from M-PSK signals transmitted over an AWGN channel. Symbols from higher order Galois Fields were mapped to M-PSK signals, since M-PSK is very bandwidth efficient and the information can be found in the angle of the signal. Several simplifications of the Belief Propagation has been developed and tested. The most promising is the Table Vector Decoder, which is a Message Passing Decoder that uses a table lookup technique for check node operations and vector summation as variable node operations. The table lookup is used to approximate the check node operation in a Belief Propagation decoder. Vector summation is used as an equivalent operation to the variable node operation. Monte Carlo simulations have shown that the Table Vector Decoder can achieve a performance close to the Belief Propagation. The capability of the Table Vector Decoder depends on the number of reconstruction points and the placement of them. The main advantage of the Table Vector Decoder is that its complexity is unaffected by the Galois Field used. Instead, there will be a memory space requirement which depends on the desired number of reconstruction points.

The increasing demand for high data rate transmission over broadband radio channels has imposed significant challenges in wireless communications. Accurate channel estimation has a major impact on the whole system performance. Specifically, reliable estimate of the channel state information (CSI) is more challenging for orthogonal frequency division multiplexing (OFDM) systems in doubly selective fading channels than for the slower fading channels over which OFDM has been deployed traditionally. With the help of a basis expansion model (BEM), a novel multivariate autoregressive (AR) process is developed to model the time evolution of the fast fading channel. Relying on pilot symbol aided modulation (PSAM), a novel Kalman smoothing algorithm based on a second-order dynamic model is exploited, where the mean square error (MSE) of the channel estimator is near to that of the optimal Wiener filter. To further improve the performance of channel estimation, a novel low-complexity iterative joint channel estimation and symbol detection procedure is developed for fast fading channels with a small number of pilots and low pilot power to achieve the bit error rate (BER) performance close to when the CSI is known perfectly. The new channel estimation symbol detection technique is robust to variations of the radio channel from the design values and applicable to multiple modulation and coding types. By use of the extrinsic information transfer (EXIT) chart, we investigate the convergence behavior of the new algorithm and analyze the modulation, pilot density, and error correction code selection for good system performance for a given power level. The algorithms developed in this thesis improve the performance of the whole system requiring only low ratios of pilot to data for excellent performance in fast fading channels. / Graduate