Transceivers with decision feedback equalizers - Signal Processing and Optimization for Transceiver Systems - page 598

q ( n )

s ( n )
est

y ( n ) s ( n )

x ( n )

s ( n )

J

M P M

detectors

F G

+

H

+

−

v ( n ) precoder channel feedforward

matrix

M

feedback
matrix

B

M

s ( n )

Figure 19.5 . Equivalent schematic drawing of the DFE system under the assumption
of no error propagation.

This equation imposes certain rank conditions on the matrices. Since the matrix
I M + B is upper triangular with unit diagonal elements, it has full rank M which
shows that the P M

matrix HF . Similarly G is required to have rank M and the product GH has
to have rank M.

× M matrix F is required to have rank M ,andsoisthe J ×

19.3.1 Optimal feedforward matrix G for fixed F and B

Under the ZF condition the reconstruction error (19.10) is simply

e ( n )= Gq ( n ) , (19 . 14)

so that the error covariance is

R ee = σ q GG † .

(19 . 15)

The objective function to be minimized is the mean square reconstruction error
E mse = σ q Tr

GG †

(19 . 16)

under the usual power constraint
σ s Tr FF † = p 0 . (19 . 17)

With ( HF ) # denoting a left inverse of HF , if we choose
G =( I M + B )( HF ) # , (19 . 18)
then GHF =( I M + B )( HF ) # HF = I M + B as desired. Choosing the left inverse
to be the minimum-norm left inverse (Appendix C), that is, 1
( HF ) # = ( HF ) † HF

− 1 ( HF ) † ,

(19 . 19)

1 From Eq.

= σ q G 2 , where G is the

(19.15) we see that the total error is Tr R ee

G

(Frobenius) norm of (see Sec. 8.2 in Chap. 8). The choice of minimum-norm inverse

therefore minimizes the total error.