Transceivers with decision feedback equalizers - Signal Processing and Optimization for Transceiver Systems - page 645

3. Generalizations. The relation between the VBLAST algorithm and gener-
alized DFE is also explained in Ginis and Cio [2001] and Zhang et al.
[2005]. The QRS method developed in Zhang et al. [2005] and Xu et al.
[2006], and the GMD based methods of Jiang et al. [2005a, 2005b] are
generalizations of the VBLAST in a major way. Namely, they show how
the transmitter (precoder) can be optimized jointly with the matrices G
and B in the DFE receiver. A knowledge of the channel H is therefore
required at the transmitter, unlike the VBLAST system which requires a
knowledge of the channel only at the receiver.

19.9 Concluding remarks

In this chapter we have shown that the use of a decision feedback matrix can
improve the performance of a MIMO transceiver significantly. We showed how
to design the jointly optimal DFE transceiver with and without zero forcing.
We also showed that these optimal transceivers have minimum average symbol
error probability. All derivations were under the assumption that there is no
error propagation in the feedback loop. The climax point in all derivations was
based on the algebraic tool called the GMD or the QRS decomposition.
It is often said that the DFE system minimizes the geometric mean rather
than the arithmetic mean of the mean square errors at the input of the decision
device. The fact is that in an optimized DFE system the AM and GM of the
errors are identical because the error covariance matrix at the detector input is
diagonal with identical diagonal elements. More important is the fact that in an
optimized DFE system the mean square error depends on the geometric mean
rather than the arithmetic mean of (reciprocals of the channel singular

values). This is what makes the performance of the DFE transceiver so much
better than that of a linear transceiver.
The feedback loop involved in DFE systems can be moved to the transmitter
side by using ideas similar to Tomlinson-Harashima precoding (Sec. 5.8). Op-
timal nonlinear transceivers can be developed based on this idea. For further
details the reader should study Jiang et al. [2005b] and Shenouda and David-
son [2008]. As in the case of linear transceivers with bit allocation (Chap. 14)
one can design transceivers with decision feedback equalizers to minimize power
under bit allocation. A generalized version of the GMD, called the GTD (gen-
eralized triangular decomposition), introduced by Jiang, Hager, and Li [2008]
(Appendix 19.A) plays a crucial role in such systems. It can be shown that with
bit allocation optimized, the linear transceiver performs as well as a DFE system
[Weng et al., 2010a]. The advantage of a DFE system using bit allocation is that
the GTD offers extra flexibility which often allows one to restrain the bits to be
integers and still obtain a global optimum. See Weng et al. [2010a] for details.
A one-page summary of the results of this chapter is included at the end of
the topic in Appendix I.

{ 1 /σ h,k }