Transceivers with decision feedback equalizers - Signal Processing and Optimization for Transceiver Systems - page 640

in these cases do not maximize mutual information. Indeed the formulas
for σ f,k were very different from the water-filling formula. For example,
the linear MMSE transceiver without zero forcing used a formula of the
form

⎧
⎨
⎩

A if this is non-negative

c
σ h,k

1
σ h,k

−

σ f,k =

(19 . 138)

0 otherwise.

The bracketted term

A

1
σ h,k

−

does indeed look like a water-filling formula, but it is multiplied by c/σ h,k ,
which makes it different. So Eq. (19.138) is only a “pseudo” water-filling
equation. For the optimal linear transceivers of Chaps. 12 and 13, the
reconstructed signal at the detector input can still be written as in Eq.
(19.134), but the covariance of the error is not proportional to identity as
in Eq. (19.135). We used an extra unitary matrix U to minimize the error
probability. This matrix equalized the mean square errors, so [ R ee ] kk were
identical for all k , but R ee was not diagonal.

4. Losslessness of mutual information. Since the MMSE DFE system with-
out ZF maximizes the mutual information, it will be called a mutual-
information lossless system. In the literature it has been referred to as
a capacity lossless system [Jiang et al ., 2005b]. Technically, since the ZF
constraint does not make the capacity smaller (because capacity cannot be
decreased by insertion of invertible precoders at the transmitter), we prefer
the term mutual-information lossless . Since the system from s (transmit-
ted symbol) to

s (detector input) is characterized by Eq. (19.134) we can
conceptually represent it as in Fig. 19.23. Here s k are statistically inde-
pendent with identical power σ s . The fact that the mutual information has
been maximized means that we can in principle use an identical channel
coding scheme for each individual symbol stream s k ( n ) and get arbitrarily
close to capacity [Gallager, 1968] (while making error probabilities arbitar-
ily small). By contrast, in any of the other systems (e.g., DFE with ZF, or
any linear transceiver), since there is loss of mutual information, we cannot
independently code s k ( n ) to get close to the capacity of the channel.

19.8 Other algorithms related to decision feedback

The use of space-domain decision feedback within the block has been prevalent
owing to its importance in wireless MIMO channels and in multiuser detection
theory see Zhang et al. [2005] and references therein). In most of the early
versions, the precoder was identity, that is, F = I (the so-called lazy precoder).
The focus of this chapter so far, on the other hand, has been to optimize a
general precoder matrix jointly with the receiver matrices (Secs. 19.3-19.5). In
this section we describe some of the earlier lazy-precoder DFE systems, in order