Minimization of error probability in transceivers - Signal Processing and Optimization for Transceiver Systems - page 501

16.3.1 Bias removal

Consider Fig. 16.3, which shows a transceiver with linear precoder and equalizer.
In general, the estimate s k ( n )of s k ( n ) is biased in the sense that the expected

value of s k ( n ) , given s k ( n ) , is different from s k ( n ):

s k ( n )

E [ s k ( n )] = s k ( n ) . (16 . 28)

If a biased estimate is used at the input of the symbol detector in the receiver,
then the detected symbol can suffer from serious errors (see Sec. 2.5.5). In prac-
tice therefore the bias is removed before symbol detection. In fact, bias removal
is automatically done when the threshold lines of the detector are appropriately
positioned with respect to the constellation symbols.
Let the transfer matrix from the input to the output of the transceiver before
bias removal be denoted as T . From Fig. 16.3 we have

T = GHF . (16 . 29)

Then the k th output is given by

s k = T kk s k +
m = k

T km s m + η k , (16 . 30)

where η k is the k th component of the transformed noise vector Gq . Here we
have omitted the arguments ( n ) for simplicity. Since q k are assumed to have
zero mean, the quantities η k also have zero mean. The symbols s k ( n ) come from
the constellation. We can regard s m ,m = k, as zero-mean random variables

statistically independent of s k . We then have

s k

E [ s k ]= T kk s k . (16 . 31)

This is not equal to s k unless T kk =1 . So the estimate s k is biased. Note that

for a zero-forcing system, since GHF = I ,wehave T kk =1,andthereisno
bias. But for MMSE systems without zero forcing, in general T kk =1,andthis

creates the bias.
We can remove the bias simply by dividing the estimate

s k by the number

T kk . Thus the bias-removed estimate has the form

s k,br = s k +
m = k

T km s m /T kk + η k /T kk . (16 . 32)

Defining the multipliers

1
T kk ,

γ k = (16 . 33)

the receiver which incorporates bias removal can therefore be schematically rep-
resented as in Fig. 16.4. Our interest is in optimizing G and H in this bias-free
system, for minimizing the symbol error rate.