Transceivers with decision feedback equalizers - Signal Processing and Optimization for Transceiver Systems - page 604

Thus the requirement (19.42) takes the form

V f [ Σ h ] − M V f

† = β 2 I M ,

I + B I + B (19 . 47)

where β 2 = cα 2 . The preceding equation simply says that

V f [ Σ h ] − M

I + B = β U (19 . 48)

for some unitary matrix U . This can be rewritten as
U † V f = β [ Σ h ] M . I + B (19 . 49)

Given the diagonal matrix [ Σ h ] M of the M dominant channel singular values
σ h,k , we can always find unitary matrices U and V f such that the preceding
equation is satisfied for an appropriately chosen, strictly upper triangular, B
matrix. This is a non-obvious result, a direct consequence of the QRS de-
composition or the GMD (geometric mean decomposition) introduced in the
pioneering papers of Jiang et al. [2005a, 2005b], and Zhang et al. [2005]. It is
stated in detail in Appendix 19.A and proved in Appendix 19.B (both at the
end of this chapter). In what follows we use this decomposition, and summarize
the results of this section.

19.3.4.A Summary: Optimal DFE transceiver with zero forcing

The optimal zero-forcing DFE transceiver which minimizes the mean square
error at the input of the detector in Fig. 19.3 can be constructed as follows. Let
the channel SVD be given by

V h
P×P

H = U h Σ h , (19 . 50)

J×J J×P

where the singular values are ordered as σ h, 0 ≥ M leading

principal matrix of Σ h containing the M dominant singular values be

σ h, 1 ≥ ... Let the M ×

⎡
⎣

⎤
⎦

σ h, 0 0 ... 0

0 σ h, 1 ... 0

[ Σ h ] M = . (19 . 51)

. . .

. . .

0 0 ... σ h,M− 1

The channel is assumed to have rank ≥ M so that

σ h, 0 ≥ σ h, 1 ≥ ... ≥ σ h,M− 1 > 0 . (19 . 52)

Compute the QRS decomposition (i.e., the GMD) of [ Σ h ] M (see Appendix 19.A
at the end of this chapter):

[ Σ h ] M = σ QRS † ,

(19 . 53)