6 New applications and conclusion.

In this last chapter, we briefly outline further research directions involving perfect STBCs, namely generalization to wireless networks and applications to coded modulations.

6.1 Coding for Wireless Networks

A lot of attention has been paid recently to wireless networks. Coding strategies for wireless networks proposed so far (for example see [34]) have been looking for methods to exploit spatial diversity using the antennas of different users in the network. The idea is to have the nodes forming a virtual antenna array, to obtain the diversity known to be achieved by point-to-point MIMO systems. Such coding strategies have been called cooperative diversity schemes.

Different families of coding strategies have been proposed. They are mainly classified between Amplify-and-Forward protocols, and Decode-and-Forward protocols. Both protocols comprise a two-step transmission: first a broadcast phase, where the transmitter broadcasts his message to the neighbor relay nodes. In the amplify-and-forward protocol, relay nodes receive the signal, just amplify it, and in a second phase, forward the amplified version to the receiver. In the decode-and-forward protocol, relay nodes try to decode the received signal, and those which manage then forward the decoded signal to the receiver. The second phase of those protocols is usually a phase of cooperation, since both these two protocols can be improved by having the nodes cooperating in doing some encoding before sending the signal to the receiver. In the decode-and-forward case, relays which decoded can cooperate in re-encoding a Space Time code [16, 34]. In the amplify-and-forward case, a way of getting cooperation is to use distributed Space-Time coding [35], as we detail below.

6.1.1 Distributed Space Time coding

The following two-step protocol, which can be seen as an improved amplify-and-forward protocol, has been introduced in [35]. We report here the basic idea of the protocol, ignoring on purpose normalization factors. All random variables for noise and fading are assumed to be complex Gaussian with zero mean and unit variance. The transmitter sends its signal s to each relay which can sense it, so that the ith relay gets

[r.sub.i] = [f.sub.i]s + [v.sub.i],

where [v.sub.i] is the noise vector and [f.sub.i] is the fading at the ith relay. Now each relay transmits

[t.sub.i] = [A.sub.i][r.sub.i],

where [A.sub.i] is a unitary matrix, so that the receiver gets

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

The matrix S is called a distributed Space-Time code since it has been generated in a distributed way by the relay nodes. It has been shown in [35] by analyzing the behavior of the pairwise error probability that the rank criterion holds similarly to the point-to-point case.

Thus knowledge acquired for building Space Time codes is useful for coding for wireless networks. Adaptation of perfect Space-Time codes have been used for wireless networks for example in [19, 37, 42]. Furthermore, since in wireless networks, the number of relay nodes correspond to the number of antennas, it is useful to have general code constructions, as given in [21].

6.1.2 MIMO Amplify-and-Forward Protocol

While the work discussed in the previous subsection focused on the analysis of the pairwise probability of error as design criterion, a lot of work has been done using as criterion the diversity-multiplexing gain trade-off (DMT) described in Chapter 3. In [2], the amplify-and-forward protocol has been analyzed with respect to the DMT. Note that the network model considered assume a direct link from the transmitter link to the receiver link, unlike the distributed Space-Time code model. It was shown that in order to reach the trade-off, the protocol has to be such that the transmitter node always transmits, which yields to so-called non-orthogonal amplify-and-forward protocol. In [2], the DMT has been shown to be achieved using random Gaussian codebooks. Since perfect Space-Time codes achieve the DMT in the point-to-point case, they seem natural candidates to generalize in order to reach the trade-off in the relay case. This has been proposed in [62], where the protocol has further been extended to the case of relays equipped with multiple antennas.

6.2 Trellis/Block Coded Modulations

Wireless networks for multimedia traffic demand high spectral efficiency coding schemes with low packet delay. Perfect Space-Time codes provide some very good tools to solve this challenging design problem. Wireless channels are commonly modeled as slow block fading, i.e., the channel coefficients are fixed over the duration of a frame. The careful concatenation of a Space-Time block code with an outer trellis code provides a robust solution for high rate transmission over a slow block fading channel.

In [32], a concatenated scheme is considered, where the inner code is the Golden code and the outer code is a trellis code. We can view this as a multidimensional trellis coded modulation (TCM), where the Golden code acts as a signal set to be partitioned. This Golden Space-Time Trellis Coded Modulation (GST-TCM) scheme is appropriate for high data rate systems thanks to the great flexibility in the choice of the modulation spectral efficiency. Moreover, the ML decoder complexity remains independent of the frame length.

A fist attempt to design such a scheme was made in [10]. However, the resulting ad hoc scheme suffered from a high trellis complexity. In [32], a systematic design approach for GST-TCM over slow block fading channels was based on lattice set partitioning combined with a trellis code is used to increase the minimum determinant between codewords. The Viterbi algorithm is used for trellis decoding, where the branch metrics are computed using a sphere decoder for the inner code.

The different GST-TCM codes designed in [32] were searched using the standard Ungerboeck's design rules for TCM. For example, it is shown that a 16 state TCM, with the spectral efficiency of 6 bits per channel use (bpcu), achieves a significant performance gain of 4.2 dB over the uncoded Golden code in slow and fast block fading channels, at an frame error rate (FER) of 10-3.

A natural research direction is to extend those techniques to other perfect Space Time codes.

6.3 Other Issues

There are other recent extensions and developments of the applications of cyclic division algebras to the area of wireless communications. One of the most promising extensions is by using maximal orders of the algebra in order to have a larger set of codewords with at least the same minimum determinant [301. There are also other applications for division algebras based codes than the MIMO or the Relay channel such as, for instance, the MIMO-ARQ channel [47].

6.4 Conclusion

Designing efficient Space Time codes for coherent MIMO systems involve more than fulfiling the known rank and determinant criteria. In this paper, we detailed several other parameters to take into account to optimize the efficiency of Space-Time codes, such as constellation shaping, diversity-multiplexing gain trade-off and the information lossless property. In order to actually construct codes satisfying those constraints, we heavily rely on the algebraic structure of cyclic division algebras based on number felds. In order to make those division algebra based codes accessible, we provide a self-contained introduction to the algebraic techniques involved. In some sense, those are a generalization of previous methods used for single antenna coding, and we believe that these algebraic approaches are now very promising for facing new coding problems coming from wireless networks.