Transceivers with orthonormal precoders - Signal Processing and Optimization for Transceiver Systems - page 480

σ 2

q ( n )

covar . I

s ( n )

s ( n )

H # I H

2

covar . I

σ s

(a)

lazy precoder ZF equalizer

channel

σ 2

Keep M outputs

q ( n )

covar . I

s ( n )

s ( n )

V h

1

−

H U h

[ ] M Σ

h

2

covar . I

σ s

unitary
precoder

(b)

unitary matrix

channel

equalizer

σ 2 q ( n )

covar . I

s ( n )

s ( n )

Λ − 1

H W

W

α

α

2

covar . I

σ s

unitary matrix

(c)

unitary
precoder

channel

equalizer

Figure 15.1 . Examples of non-redundant ( P = M ) transceivers with orthonormal
precoder and zero-forcing equalizer. (a) Identity (or “lazy”) precoder, (b) precoder
which “diagonalizes” th e ch annel matrix, and (c) the IDFT precoder as used in cyclic
prefix systems ( α =1 / √ M ). All of these are optimal in the ZF-MMSE sense.

15.2.1.A Comparison with general nonunitary precoder

We now compare the unitary precoder with the general precoder, assuming the
channel input power to be the same for both systems. Recall that, when the
precoder is unitary, the transmitted power (trace of the covariance at the input of
the channel H )is p 0 = σ s M. If we wish to have more control on p 0 (independent
of σ s ), then we can insert a matrix α I at the transmitter and a matrix α − 1 I at
the receiver. This preserves the ZF condition. This is shown in Fig. 15.2 where
the unitary precoder is denoted as U f , and its ZF equalizer is denoted as G 1 .
The scalar variable α> 0 is used for power control. The transmitted power is
now

p 0 = σ s α 2 M. (15 . 12)

We already showed that when the precoder is unitary, the MMSE under the ZF
constraint is as in Eq. (15.8). This corresponds to α = 1. For arbitrary α we
therefore have

M− 1

σ q
α 2

1
σ h,k

E mmse =

k =0