# Golden Code Coset and Its Application in Cooperation System

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## Right processing : column permutation and lat- tice reduction

Let H denote the real channel matrix. The QR-decomposition on H may generate an \ill- conditioned » matrix R whose diagonal entries can be very small. For the MIMO decoders, an ill-conditioned matrix introduces either large complexity or a bad performance. To sovle this problem, it is suggested to perform the right-processing on the channel matrix before the lattice searching step.This processing can be considered to decomposite H as: H = HredU (2.19) where U is a unimodular marix such that ui;j 2 Z and UTU = I. The matrix U can be a permutation matrix which gives a decoding schedule. In the otherhand, it can be a lattice reduction matrix which simplies the lattice basis. In general, the lattice reduction matrix gives better performance than the permutation matrix. However, it is worthy to notice that it takes more time to calculate the lattice reduction matrix. Another inconvenience of the lattice reduction matrix is the shaping problem that the constellation set S are changed for the lattice searching step.
For the permutation matrix, the algorithm Greedy Ordering (GO) proposed in [17, 18] can be applied to nd a good decoding order. It is shown that the permutation matrix provides the optimal order for the Decision Feedback Equalizer (DFE) algorithm which will be presented in section 2.4.

### MIMO decoder performance

In this section, we will discuss the performance of MIMO decoders in terms of Word Error Rate (WER) and decoding complexity. We suppose the CSI is perfectly known at the receiver and the MMSE-GDFE is performed for the left-processing of channel matrix. We group the MIMO decoders into 2 categories : the xed-latency decoders and the variant-latency decoders.
The xed-latency decoders include ZF decoder, DFE decoder and KSE decoder. As shown in section 2.4, the xed-latency decoder uses the same structure in a recursive way. The decoding complexity is linear to the MIMO system’s dimension which means the decoder’s delay can be controlled. For the Wi-Fi system which demands a strict decoding latency2, we are inclined to apply the xed-latency decoders for hardware implementation [26]. The SE/SEF decoder, Fano decoder and stack decoder are considered as variant-latency decoders. The variant-latency decoder is not suitable for hardware level implementation since their processing delays are unpredictable. This type of decoders can be imple- mented in software level like the platforms with microprocessor support or for systems not sensitive to the latency. For the interest of complexity evaluation, we will evaluate the algorithm’s eciency by analysing the average number of visited points per level during the lattice searching.
For the right-processing mentioned in section 2.3, we will investigate 3 situations: without any right-processing (direct decoding), with permuation matrix using GO and with LLL reduction method.

#### Discussion on performance

The proposed Relay-SISO PHY layer illustrates a signicant cooperative diversity gain in at fading channel as well as in multipath channel. In both channel environments, the performances of Relay-SISO are better than legacy 802.11a system with a considerable gain especially for low data rate and HPG conguration. Application of cooperative diversity can enlarge the network coverage or alternatively alllow a lower transmit power by providing good protection. This solution can be very interesting to enhance the system quality for the applications that are very demanding in terms of Quality of Service (QoS). For example the video broadcasting service needs a FER about 10􀀀4 and Relay-SISO can be implemented to update the present WiFi networks with reasonable decoding complexity. The inconvenience of Relay-SISO is that there is no signicant gain in LPG conguration known as \crossing point » problem. This phenomenon is due to that in low SNR range the relay terminal amplies the noise term that harms the reception quality. This problem can be the optimization works in the future.

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1 Cooperative Diversity
1.1 Cooperative system
1.2 Cooperative diversity and STBC
1.3 Cyclic division algebra based STBC
2 MIMO Decoder
2.1 MIMO system description
2.2 Left processing: MMSE-GDFE
2.3 Right processing : column permutation and lattice reduction
2.4 Lattice decoding
2.4.1 ML decoding algorithm
2.4.2 Algorithm Zero-Forcing
2.4.3 Heuristic algorithms: DFE and KSE
2.4.4 Sequential decoder based algorithm : Fano decoder and stack decoder
2.4.5 Generalized Schnorr-Euchner decoder: SEF decoder
2.5 MIMO decoder performance
2.5.1 Direct decoding
2.5.2 Greedy ordering
2.5.3 LLL reduction
2.5.4 Conclusion
3 Relay-SISO
3.1 IEEE 802.11a PHY layer
3.2 Relay-SISO PHY layer
3.3 Relay-SISO preambles and cooperation procedure
3.3.1 Data eld
3.3.2 Carrier frequency oset estimation and correction
3.3.3 Amplify-and-forward procedure
3.3.4 Channel coecients estimation
3.3.5 Cooperation procedure
3.4 Relay-SISO Performance
3.4.2 Multipath channel
3.4.3 Discussion on performance
4 Hybrid Cooperation
4.1 Normalized minimum squared Euclidean distance
4.2 Equivalent metric for IEEE 802.11a and Relay-SISO
4.3 Hybrid mode and performance
5 Golden Code Coset and Its Application in Cooperation System
5.1 Golden code and partition
5.2 Golden code parition and coset bit mapping
5.3 Application and performance
5.3.1 New modulation and coding scheme for data rate 12Mbps in Relay- SISO
5.3.2 New modulation and coding scheme for data rate 24Mbps in Relay- SISO
5.3.3 Application in 2 2 MIMO system
6 Relay-MIMO
6.1 IEEE 802.11n PHY Layer
6.2 Relay-MIMO PHY Layer
6.3 Relay-MIMO preambles and cooperation procedure
6.3.1 HT data eld
6.3.2 Cyclic shift processing
6.3.3 AF procedure
6.3.4 Channel coecient estimation
6.4 Relay-MIMO Performance