One of the main challenges in wireless communication is the fluctuation of signal amplitude caused by fading. Many efforts have been put to mitigate this adverse effect.
Trellis coded modulation (TCM), originally proposed by Ungerboeck for bandwidth- efficient communication over the additive white Gaussian noise (AWGN), has shown some drawbacks when transmitting over fading environment. In the design of TCM, modulation and coding is combined as an entity to improve the performance. The design goal is to maximize the minimum free Euclidean distance, and therefore it is often optimized over AWGN channel. However, when transmitting over fading channels, its performance is significantly degraded since the diversity order is usually low. To combat the adverse effect of fading channel, symbol interleaver is added and parallel transitions in the trellis should be avoided. However, since the minimum number of distinct symbols between two codewords limits the diversity order, the constraint length should be increased. The increased constraint length further results in exponentially increased decoding complexity which is unacceptable.
In [2], Zehavi proposed an alternative approach called bit-interleaved coded modulation (BICM) to increase the diversity order to the minimum Hamming distance of the code. By placing a bit-wise interleaver at the encoder output, this allows large diversity order with moderate system complexity. In [4], Li and Ritcey showed that the performance of BICM can be further improved by iterative decoding between the demapper and the decoder, a scheme called bit-interleaved coded modulation with iterative decoding (BICM-ID). It has been shown in the literature that the design of the demapper is crucial to achieve a high coding gain over iteration. In [6], EXIT chart was proposed to describe the iterative decoding behavior through a decoding trajectory between the transfer curve of the demapper and decoder.
On the other hand, error control is also a main issue for data communication.
Combined with the advantage of automatic-repeat-request (ARQ) mechanisms and forward-error-correction (FEC) schemes, HARQ is often adopted to achieve high reliability and high system throughput. In HARQ schemes, additional redundant parity bits are appended to the original message for both error correction and detection. When the presence of errors is detected, the receiver first tries to correct the erroneous bits. If the number of errors is beyond the designed error-correcting capability of the code, a retransmission request is send to the transmitter. The retransmitted packets can be exactly the same as the initial one or contain some extra redundant bits. When a new packet is received, the newly received packet can be decode alone or jointly decode with the previous ones.
In this thesis we are concerned about the case that retransmission carry identical bits and all the received packets are combined together for decoding. Furthermore, BICM and BICM-ID in conjunction with HARQ is considered.
It is known that the performance will be significantly improved by introducing packet combing. Chase combining [12] is the well known ML combing technique. It combines arbitrary number of coded packets into a single coded packet with lower code rate, thus improves the error-correcting capability of the code. However, Wengerter [15] showed that different bit to symbol mapping for retransmission can further improves the system performance. By simply swapping or taking logic inversion on the modulation bits to average out the unequal bit reliabilities, a method called constellation rearrangement, significant improvement has been observed.
However, no optimality can be claimed on this method. In [16], an optimization criterion base on the BER upper bound has been proposed. The main deficiency of the mapping found by the minimization of the BER upper bound is that the upper bound is only tight at high SNR, the performance at low SNR can not be guaranteed. Murthy
[17] further suggested to change the criterion to maximize the sum of the magnitudes of the LLR of the bits forming the M-QAM symbols in different retransmissions.
However, the maximization is made on the sum not on the individual bits LLR, no optimality can be guaranteed. Another criterion based on the augmented signal space after retransmission is to maximize the minimal accumulated (over transmission) squared Euclidean distance [18]. Gidlund [19] also aimed at increasing the Euclidean distance between signal points, thus applying the idea of set-partition in TCM to spread the signal points well in the augmented signal space. These designs ignore the relationship of the number of bit differences between nearest symbols; however, the number of bit differences is a crucial parameter for the design of BICM mappings.
Hence is also not optimized for BICM systems.
Those mapping designs described above are all independent of SNR. However, an analysis based on the BICM capacity under multiple transmission [14] showed that one single mapping can not be optimal for the whole interested SNR range. It was showed that constellation rearrangement (CoRe) outperforms the mapping obtained by the minimization of BER upper bound (MBER) at low SNR. However, at high SNR, MBER exhibits better performance than CoRe. Since there are different operating SNR region at different code rate, this paper suggests that mapping should be adaptive considering the targeted spectral efficiency (code rate). Although adaptive mapping scheme has been proposed, mappings that are optimized for each SNR is still an open problem. Hence we aim to find these optimal mappings.
When iterative decoding is applied (BICM-ID), mapping design is especial crucial for obtaining large iterative decoding gain even for single transmission. Various mapping design methods have been proposed for single transmission. However, very few have addressed the issue of multiple transmissions mapping design. In [21],
pairwise error probability and the uncoded ideal prior pair-wise error probability. Since the uncoded pair-wise error probability is independent of the underlying coding scheme, the design is not optimized for a particular code and may cause large performance degradation. By the analysis of the EXIT chart, the first intersection of the demapper transfer curve and the decoder transfer curve should be as high as possible.
Since different mapping and coding have different transfer curve, their first intersection will be different. Therefore, a mapping that is good for a particular code may not be good for another one as well. Guided by the EXIT chart, mapping design should be dependent on the outer code. Furthermore, the dependency of the demapper transfer function on SNR also suggests that different mappings should be designed for the same code on different SNR. Hence we propose a method jointly considering the outer code and the operating SNR to design the retransmission mappings based on the EXIT chart.