MMSE transceivers with zero-forcing equalizers - Signal Processing and Optimization for Transceiver Systems - page 426

q ( n )

J

y ( n ) s ( n )

s ( n )

P

M

x ( n )

M

F G

H

precoder channel equalizer

Figure 12.11 . The transceiver with a rectangular channel.

To express the corresponding zero-forcing equalizer conveniently, consider again
the channel in SVD form:

V h

H = U h

K×K

Σ h
K×P

. (12 . 92)

P×P

Note the sizes of the matrices; in particular U h and V h can be of different sizes.
Proceeding as in Sec. 12.4.4 we have

[ Σ h ] M
0

HF = U h

K×K

Σ f
M×M

,

K×M

where [ Σ h ] M is the M × M leading principal submatrix of Σ h , containing the

dominant singular values

σ h, 0 ≥ σ h, 1 ≥ ... ≥ σ h,M− 1

Thus

( HF ) † HF = Σ f [ Σ h ] M .

The zero-forcing equalizer is therefore found to be

= ( HF ) † HF

− 1 ( HF ) †

G

0 ] U h

= Σ h ] − M Σ − 2 Σ f [[ Σ h ] M

f

[ U h ] M×K ,

= Σ h ] − M Σ − 1

f

which is exactly as in Eq. (12.61), with the exception that [ U h ] M×K appears
instead of [ U h ] M×P . Thus the optimal transceiver structure has the form shown
in Fig. 12.12, which is identical to the structure in Fig. 12.5 except for the
sizes of H and U h . The diagonal matrices Σ f and Σ g are exactly as described
in Theorem 12.1.