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Mutual Information analysis of NB-LDPC associated to high-order modulations and application of Signal Space Diversity techniques to improve performance

We consider the study of the mutual information of NB-LDPC codes associated to high-order modulations. To be specific, we consider Coded Modulation (CM) techniques where each NB-LDPC coded symbol is associated to a modulation signal and we compare its channel capacity to the one of the Bit Interleaved Coded Modulation (BICM). We also consider the use of Rotated Constellations (or Signal Space Diversity, SSD), a technique used in the DVB-T2 to improve performance in fading channels. We show here that the SSD technique applied to coded modulation (CM) schemes has multiple advantages, compared to Bit-Interleaved Coded Modulations (BICM). To be specific, we investigate the association of $q$-ary codes to high-order modulations, where the cardinality of the code equals the modulation order. This kind of scheme is called Coded Modulation (CM). We consider the optimization of the SSD technique through the analysis of the mutual information of the channel as a function of the rotation angle. This approach is applied to both BICM and CM schemes to show the interest and advantages of the CM solution. Note that for the BICM we consider a binary FEC code whereas for the CM scheme we adopt a Non-Binary (NB) code defined over a Galois Field GF($q$), which allows to directly map a coded symbol to a constellation point. <\p>

EVALUATING MUTUAL INFORMATION

Please refer to our paper for a detailed explanation of the mutual information evaluation for both Coded Modulation and Bit-Interleaved Coded Modulation schemes with NB codes. We provide by the following the MATLAB software to obtain:

The mutual information curves as a function of the SNR for the Gaussian channel

1. 16-QAM
2. 64-QAM
3. 256-QAM

The mutual information curves as a function of the SNR (with and without SSD) for the fast Rayleigh fading channel

We consider an optimised rotation angle for the SSD technique, for each modulation.
4. 16-QAM
5. 64-QAM
6. 256-QAM

The mutual information curves as a function of the rotation angle (SSD technique)

7. 16-QAM
8. 64-QAM
9. 256-QAM

MUTUAL INFORMATION AS A FUNCTION OF THE SNR

In the following figure, for the 16-QAM NB scheme, we observe that the CM with SSD outperforms the non-rotated scheme for all SNRs. This does not occur for BICM: at low SNR (up to approximately 12 dB for the 16-QAM), the rotation causes a mutual information loss compared to non-rotated BICM. This can also be observed in Figures 2 and 3 (see next Section), where the mutual information curve decreases with rotation. Therefore, the gains displayed by the rotated CM are more significant than for BICM.

CM and BICM mutual information curves for a 16-QAM modulation over fast flat fading Rayleigh channel (without erasures and with 10% erasures)

CM and BICM mutual information curves for a 256-QAM modulation over fast flat fading Rayleigh channel (without erasures and with 10% erasures)

EVALUATING MUTUAL INFORMATION TO OPTIMIZE SSD

In the following figures we show the mutual information curves as a function of the SSD rotation angle for both CM and BICM schemes. . Both figures show results for the flat fast Rayleigh fading channel with erasures ($p_e = 0.1$) and without erasure. From these curves we can conclude that, in all cases, the CM scheme presents higher mutual information than the BICM. Also, rotation always provides gain for the CM scheme (unlike for the BICM, where rate loss may appear with rotation). Another point is that the variations on the mutual information are more significant in the erasure channel. Thus, the SSD can potentially provide more gains in the erasure channel.

Mutual information as a function of the rotation angle for CM and BICM. SNR = 10, 15 and 25 dB. Rayleigh fading channel and 16-QAM modulation.

Mutual information as a function of the rotation angle for CM and BICM schemes. SNR = 15, 25 et 30 dB. Rayleigh fading channel and 256-QAM modulation.

SIMULATION RESULTS AND PERFORMANCE COMPARISON

We show some performance results of NB-LDPC codes associated to 256-QAM for coding rates 3/4 and 9/10.

FER simulation for 3/4-rate BICM-GF(2) and CM-GF(256) schemes over the the fast flat Rayleigh fading channel, with and without Rotated Constellation. The SSD technique introduces performance loss in the BICM scheme for spectral efficiencies lower than 7 bit/Hz/s. However, as announced by the theoretical study, a gain of about 0.3 dB is provided by the SSD technique for CM.

FER simulation for 9/10-rate BICM-GF(2) and CM-GF(256) schemes over the the fast flat Rayleigh fading channel, with and without Rotated Constellation. SSD introduces gain for both the BICM and CM schemes. This gain is more significant in the CM scheme (1.3 dB at FER = 10^2 compared to about 0.15 dB for BICM).