Minimization of error probability in transceivers - Signal Processing and Optimization for Transceiver Systems - page 514

Proof. The mean square error can be expressed as

E e ⊥ = E [ e ⊥ e ∗ ⊥ x ) ∗ ]= E [ e ⊥ x ∗ ]

]= E [ e ⊥ ( x − − (using Eq. (16.70))

| 2

=(1 − α ) E | x (using Eq. (16.69))

so that

E e ⊥ =(1 − α ) E x , (16 . 74)

which proves Eq. (16.73). But in view of Eq. (16.68), we also have from
Eq. (16.69)

α ) 2 E x + E e ⊥ =(1 − E τ , (16 . 75)

where we have used the fact that α is real. Equating the right-hand sides
of Eqs. (16.74) and (16.75) we get Eq. (16.71). Dividing (16.74) by (16.71)
yields Eq. (16.72).

16.A.1 Bias-removed estimates

From Eq. (16.67) we see that the expected value of the estimate x, given the

transmitted symbol x, is given by

E [ x | x ]= αx + E [ τ | x ] . (16 . 76)

Usually τ is a zero-mean random variable,

E [ τ ]=0 , (16 . 77)

and it is statistically independent of x . In particular, therefore, E [ τ | x ]=0 , and

we see that

E [ x | x ]= αx = x. (16 . 78)

Since the conditional expectation of x is different from x , we say that the estimate

is biased . An obvious way to remove the bias is to just “kick it out.” That is,
define the bias-removed estimate

x br = x/α . This has the form

x br = x + τ
α

(16 . 79)

where “ br ” is a reminder for “bias-removed.” Clearly the mean squared error of
the bias-removed estimate is E τ /α 2 , which simplifies to E e br =

E e br = E e α (16 . 80)

using Eq. (16.72). Since α< 1 (unless the estimate is already unbiased), there
is an increase in the MSE due to bias removal.