Transceivers with decision feedback equalizers - Signal Processing and Optimization for Transceiver Systems - page 641

to place the main results of this chapter in proper historical context. Recall at
this point that systems with a lazy precoder do not have collaboration between
the components s k ( n ) at the transmitter, and are applicable to multiuser systems
operating in the multiple access mode (Sec. 4.5).

19.8.1 DFE based on QR decomposition of the channel

Consider M MIMO channel H with additive noise q . With precoder

F = I the received signal is given by

a J ×

y = Hs + q . (19 . 139)

With the channel H represented using QR decomposition (Appendix 19.A at
the end of this chapter), we have

H = QR , (19 . 140)

where Q has orthonormal columns, that is,

Q † Q = I M ,

and R is upper triangular. Thus Eq. (19.139) becomes

y = QRs + q ,

so that

v = Q † y = Rs + Q † q . (19 . 141)
Thus, if we design a receiver with Q † in the front, the Q part is cancelled, and
the receiver sees a triangular channel R . Writing Eq. (19.141) out explicitly, we
have
⎡
⎣

⎤
⎦

⎡
⎣

⎤
⎦

⎡
⎣

⎤
⎦

r 00 r 01 r 02 ... r 0 ,M− 1

v v
.
v M− 1

s s
.
s M− 1

0 r 11 r 12 ... r 1 ,M− 1

+ Q † q .

= (19 . 142)

. . . .

. . .

000 ... r M− 1 ,M− 1

A linear zero-forcing receiver would estimate s by inverting R and then using a
decision device. Instead of this approach, we can detect one symbol at a time
and use it in future decisions by proceeding as follows: from the last equation
in Eq. (19.142) we see that the received noisy signal v M− 1 depends only on one
transmitted symbol, namely s M− 1 . So we estimate this first:

v M− 1
r M− 1 ,M− 1

s M− 1 = (19 . 143)

This is similar to zero forcing. An MMSE version can also be readily developed.
The symbol s M− 1 is now estimated using the decision device D ( . ):

s M− 1 ,est = D ( s M− 1 ) .