Finite-state Markov modeling of fading channels - a survey of principles and applications <s> PROPOSED TECHNIQUES <s> The finite-state Markov channel (FSMC) is a discrete time-varying channel whose variation is determined by a finite-state Markov process. These channels have memory due to the Markov channel variation. We obtain the FSMC capacity as a function of the conditional channel state probability. We also show that for i.i.d. channel inputs, this conditional probability converges weakly, and the channel's mutual information is then a closed-form continuous function of the input distribution. We next consider coding for FSMCs. In general, the complexity of maximum-likelihood decoding grows exponentially with the channel memory length. Therefore, in practice, interleaving and memoryless channel codes are used. This technique results in some performance loss relative to the inherent capacity of channels with memory. We propose a maximum-likelihood decision-feedback decoder with complexity that is independent of the channel memory. We calculate the capacity and cutoff rate of our technique, and show that it preserves the capacity of certain FSMCs. We also compare the performance of the decision-feedback decoder with that of interleaving and memoryless channel coding on a fading channel with 4PSK modulation. <s> BIB001 </s> Finite-state Markov modeling of fading channels - a survey of principles and applications <s> PROPOSED TECHNIQUES <s> We propose a decision-feedback decoder for coded signals transmitted over finite-state Markov channels. The decoder achieves maximum-likelihood sequence detection (in the absence of feedback errors) with very low complexity by exploiting previous bit decisions and the Markov structure of the channel. We also propose a similar decoder, the output-feedback decoder, that does not use previous bit decisions and therefore does not suffer from error propagation. The decoder performance is determined using a new sliding window analysis technique as well as by simulation. Both decoders exhibit excellent bit error rate performance with a relatively low complexity that is independent of the channel decorrelation time. <s> BIB002 </s> Finite-state Markov modeling of fading channels - a survey of principles and applications <s> PROPOSED TECHNIQUES <s> We describe parallel concatenated codes for communication over finite-state binary Markov channels. We present encoder design techniques and decoder processing modifications that utilize the a priori statistics of the channel and show that the resulting codes allow reliable communication at rates which are above the capacity of a memoryless channel with the same stationary bit error probability as the Markov channel. These codes outperform systems based on the traditional approach of using a channel interleaver to create a channel which is assumed to be memoryless. In addition, we introduce a joint estimation/decoding method that allows the estimation of the parameters of the hidden Markov model when they are not known a priori. <s> BIB003 </s> Finite-state Markov modeling of fading channels - a survey of principles and applications <s> PROPOSED TECHNIQUES <s> We propose a modified algorithm for decoding of low-density parity-check codes over finite-state binary Markov channels. The proposed approach clearly outperforms systems in which the channel statistics are not exploited in the decoding, even when the channel parameters are not known a priori at the decoder. <s> BIB004 </s> Finite-state Markov modeling of fading channels - a survey of principles and applications <s> PROPOSED TECHNIQUES <s> Density evolution analysis of low-density parity-check (LDPC) codes in memoryless channels is extended to the Gilbert-Elliott (GE) channel, which is a special case of a large class of channels with hidden Markov memory. In a procedure referred to as estimation decoding, the sum-product algorithm (SPA) is used to perform LDPC decoding jointly with channel-state detection. Density evolution results show (and simulation results confirm) that such decoders provide a significantly enlarged region of successful decoding within the GE parameter space, compared with decoders that do not exploit the channel memory. By considering a variety of ways in which a GE channel may be degraded, it is shown how knowledge of the decoding behavior at a single point of the GE parameter space may be extended to a larger region within the space, thereby mitigating the large complexity needed in using density evolution to explore the parameter space point-by-point. Using the GE channel as a straightforward example, we conclude that analysis of estimation decoding for LDPC codes is feasible in channels with memory, and that such analysis shows large potential gains. <s> BIB005 </s> Finite-state Markov modeling of fading channels - a survey of principles and applications <s> PROPOSED TECHNIQUES <s> This paper shows the existence of the optimal training, in terms of achievable mutual information rate, for an output feedback implicit estimator for finite-state Markov communication channels. A proper quantification of source redundancy information, implicitly used for channel estimation, is performed. This enables an optimal training rate to be determined as a tradeoff between input signal entropy rate reduction (source redundancy) and channel process entropy rate reduction (channel estimation). The maximal mutual information rate, assuming the optimal implicit training and the presence of channel noise, is shown to be strictly below the ergodic channel information capacity. It is also shown that this capacity penalty, caused by noisy time-varying channel process estimation, vanishes only if the channel process is known or memoryless (channel estimation cannot improve system performance). <s> BIB006 </s> Finite-state Markov modeling of fading channels - a survey of principles and applications <s> PROPOSED TECHNIQUES <s> This paper proposes a novel successive decoding paradigm for fading channels. K independent codewords of different rates and lengths are interleaved at the transmitter according to a pattern, and successively decoded at the receiver. This enables progressively better channel estimates for the receiver, at a minimal cost of pilot symbols. At each decoding stage, channel estimation and decoding are performed separately, or iteratively if the residual channel correlation within a codeword can be exploited. The flexible interleaver pattern, after some level of optimization, achieves near-capacity system performance with only a small number of codewords K, as demonstrated numerically by a finite state Markov modelled flat fading channel. <s> BIB007
The first technique for approaching information rates in the GEC higher than that of the memoryless case C NM was proposed in  , which was later generalized to FSMC models in BIB001 . The authors proposed a decision-feedback decoder, which is comprised of a channel-state estimator and an ML decoder. Since the channel-state estimator uses previous ML estimates of input symbols, this method is prone to error propagation BIB001 . Assume that the output of the ML estimated symbol is in error. Since this decision is fed back to the channel state estimator, the channel state estimator will be in error too. The erroneous channel state estimate will be used in the next ML symbol estimation round and hence, the error propagates. To avoid error propagation, BIB002 has proposed an alternative strategy, called the output-feedback decoder, which does not use previous symbols decisions and only relies on previous received symbols. In BIB006 , the authors studied achievable information rates in the output-feedback decoder proposed in BIB002 . It was shown that the state estimation is only possible at the price of decreasing information rate and reducing the input symbol entropy rate. In particular, for the class of uniformly-symmetric and variable-noise FSMC models defined in BIB001 , channel state estimation in the output-feedback decoder is not possible with the capacity-achieving i.u.d. input distribution. By choosing an input distribution which is non-i.u.d., one is effectively introducing redundancy or structure into the channel input signal, which enables (implicit or output-feedback) channel state process estimation, but reduces information rate below the channel information capacity. In another approach, existing capacity approaching codes for AWGN channels, such as turbo codes and LDPC codes, have been modified and extended for decoding through FSMC models. The authors in BIB003 proposed turbo-decoding for the GEC that avoided channel interleaving. The decoding technique builds a supertrellis to jointly describe the finite state of the channel and the finite state of the turbo encoder and modifies decoding equations. Code design is shown to have a considerable effect on the error rate performance. The optimum number of constituent turbo encoders is chosen to balance between providing diversity and weakening each decoder due to puncturing. The numerical error rates show that information rates higher than the memoryless capacity C NM are achievable. Incorporating the GEC and FSMC model into the message passing in LDPC decoding has been studied in  and BIB004 , respectively and superior error rate performance compared to decoding techniques that do not take channel statistics into account is shown. In BIB005 , density evolution analysis is extended from memoryless channels to the GEC, which can be used to determine the region of GEC parameters over which LDPC decoding is successful. Recently, a successive decoding technique for FSMC models was proposed in BIB007 , which is optimal with near-capacity performance, but at the expense of large codewords and decoding stages and hence, complexity and delay.