Optimization of cyclic prefix transceivers - Signal Processing and Optimization for Transceiver Systems - page 547

require modification when the channel noise q ( n ) is colored, as is the case in
practical DSL systems [Starr et al., 1999]. In such situations the covariance
R qq of the noise vector q ( n ) is not diagonal. The general techniques described
in Chap. 12 should be used to obtain an equivalent system where the noise
covariance has the form σ q I . The final form of the transceiver will not be entirely
DFT-based, as it depends also on R qq .

17.6 The cyclic-prefix system with unitary precoder

In Chap. 15 we studied the special case of transceivers with unitary or orthonor-
mal precoders. In particular we considered in Sec. 15.2 orthonormal precoders
where all matrices ( F , H , and G ) are square. We showed that if such transceivers
are optimized to minimize the MSE under the power constraint (with or with-
out the zero-forcing constraint), then the precoder F can be assumed to be any
unitary matrix without loss of optimality.
In this section we consider the special case of cyclic-prefix systems with pre-
coder restricted to be unitary. Such systems will be referred to as orthonormal
cyclic-prefix systems , even though the receiver matrix is not unitary (it has to
perform equalization). 3 All the results of Sec. 15.2 can be applied, with the
additional feature that the channel matrix H is circulant. Since the precoder is
restricted to be unitary, all equalization in the optimized sytem will take place
at the receiver (instead of being split evenly between the transmitter and the
receiver).

17.6.1 Single-carrier and multicarrier cyclic-prefix systems

Since the unitary precoder F can be assumed to be any unitary matrix without
loss of optimality, we first consider the simplest example, where the precoder is
chosen to be identity:

F = I (17 . 52)

This is called the lazy precoder . In this case the transceiver is as in Fig. 17.12(a).
From Sec. 15.2 in Chap. 15 we then conclude that the optimal equalizer G (for
MMSE property) is given by

⎧
⎨

H − 1

ZF-MMSE case

H †

− 1

HH † + σ q

G = (17 . 53)

σ s I

pure-MMSE case.

⎩

3 In OFDM (orthogonal frequency division multiplex) systems the columns of the precoder
are mutually orthogonal but do not necessarily have unit norm. Thus, the O in OFDM signifies
orthogonal rather than orthonormal .