B. P. Lathi, Zhi Ding - Modern Digital and Analog Communication Systems-Oxford University Press (2009)

680 DIGITAL COMMUNICATIONS UNDER LINEARLY DISTORTIVE CHANNELS
Let us now proceed to find an equalizer filter that can minimize the mean square error of
Eq. (12.35). Once again, we will apply the principle of orthogonality in optimum estimation
(Sec. 8.5), that the error (difference) signal must be orthogonal to the signals used in the filter
input. Because d[n] = I: 0 f[i]z[n - i], we must have
In other words,
(d [n] - S n -u) .l z[n - ,£] ,e = 0, 1, ...
(d [n] - S n -u) z* [n - ,£] = 0 ,e = 0, 1, ... (12.36)
Therefore, the equalizer parameters {f [i]} must satisfy
(I)[i]z[n - i] - S n -u) z* [n - ,£] = 0
z=O
,e = 0, 1, ...
Note that the signal S n and the noise w[n] are independent. Moreover, {sn} are also i.i.d. with
zero mean and variance E s. Therefore, s n -uw* [n] = 0, and we have
00
S n - u z* [n - ,£] = S n-u (L h[j]*s -j -
£ + w[n - ,£]*)
}=0
00
= L h[j]* S n -u s - j- £ + 0
}=0
(12.37)
Let us also denote
R 2 [m] = z[n + m] z* [n]
(12.38)
Then the MMSE equalizer is the solution to linear equations
I)[i] R z [,£ - i] = { i s h [u - ,£]*
i=0
,e = 0, 1, ... , u
,e = u + I, u + 2, ... , oo
(12.39)
Based on the channel output signal model, we can show that
*
R 2 [m] = ( f hi S n+m-i + w[n + m]) (f hj S n -} + w[n])
l=0 J=0
(12.40)