Transceivers with orthonormal precoders - Signal Processing and Optimization for Transceiver Systems - page 479

which is independent of the precoder. Thus, any unitary precoder is as good as
any other if we restrict F to be M M unitary. 1

× For any chosen F we simply

take the equalizer to be

G =( HF ) #

(15 . 7)

A # represents the

to satisfy the zero-forcing condition. Here the notation

(minimum-norm) left inverse of the matrix A (see Eq. (C.7b) in Appendix

C). The reconstruction error then simplifies to

M− 1

1
σ h,k

E mse = σ q Tr ( H † H ) − 1 = σ q (15 . 8)

k =0

where σ h,k are the M singular values of the channel H (all σ h,k > 0 because the
channelisassumedtohaverank M ). 2
We now present three examples of the choice of unitary F . Since any unitary
F is as good as any other, all of these are “optimal” solutions.

1. Lazy precoder .If F = I we have G = H # . This transceiver is shown in Fig.
15.1(a). This system has the advantage that all “collaboration” between
the different s k ( n )'s is done at the receiver. This is suitable for a multiuser
system in multiple access mode (Sec. 4.5).

2. SVD precoder . With the channel expressed in SVD form (15.4), if we decide
to choose the unitary precoder F = V h , then the equalizer is

G =( HF ) # =( U h Σ h ) # =[[ Σ h ] − M 0 ] U h ,

(15 . 9)

where [ Σ h ] M is the M M diagonal matrix with diagonal elements equal

to σ h,k . This transceiver is shown in Fig. 15.1(b).

×

3. Cyclic-prefix precoder . Recall that in the cyclic-prefix system the channel
is made to look like an M

× M circulant matrix. In this case

H = W − 1 Λ c W . (15 . 10)

Here Λ c is a diagonal matrix containing the DFT coe cients of the cha n-

nel, and W is the DFT matrix. Since W † W = M I ,thematrix W † / √ M is
unitary and we can take the unitary precoder and the zero-forcing equalizer
to be

F = W † G = Λ − c W

√ M , √ M . (15 . 11)

This results in the familiar OFDM system described in Sec. 7.3. See Fig.
15.1(c). Since any u nit ary F is as good as any other, we can say that the

IDFT matrix W † / √ M is also an “optimum unitary precoder” solution.

1 Note that we are only concerned about minimization of MSE. If we are interested in
minimizing error probability, as in Chap. 16, then the situation is different.
2 The second equality in Eq. (15.8) follows because (a) the trace is the sum of eignevalues,
(b) the eigenvalues of H † H are σ h,k , and (c) the eigenvalues of ( H † H ) − 1 are the reciprocals
of those of H † H .