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

12.3 Linear T-Spaced Equalization (TSE) 683
In other words, this equalizer will minimize the contribution of ISi to the mean square error
in d[nJ.
Finite Data Design
The MMSE (and ZF) design of Eqs. (12.39) and (12.43c) assumes statistical knowledge of
R z [mJ and S n -uZ* [n £J. In practice, such information is not always readily available and
may require real-time estimation. Instead, it is more common for the transmitter to send a short
sequence of training (or pilot) symbols that the receiver can use to determine the optimum
equalizer. We now describe how the previous design can be directly extended to cover this
scenario.
Suppose a training sequence {s n , n = n1 , n1 + l, ... , n2} is transmitted. To design an FIR
equalizer
we can minimize the average square error
F(z) = f [OJ + f[! Jz - 1
+ · · · + f [MJz - M
where
M
d[nJ = L f[iJz[n - iJ
i=O
To minimize J, we can take its gradient with respect to f UJ. By setting the gradient to zero,
we can derive the conditions required by the optimum equalizer parameters
M
l
u+n2
" f [iJ --- "
z[n - iJz* [n - jJ = ---
n2 - n1 + 1 n2 - n1 + 1
z=O
u+n2
n=u+n1
X
L S n -uz* [n - jJ j = 0, l, ..., M
n=u+n1
l
(12.44)
These M + l equations can be written more compactly as
R z [O, OJ
R 2 [0, lJ
[
R z [,MJ
R z [l, OJ
R z [l, I J
Rz [l:,MJ
z [M , OJ f [OJ
R z [M , lJ
] [ f[lJ
.
. .
. .
R z [M ,M] flM]
]
[
Rsz[-u]
lidu + ll
Rsz[-u + M]
]
(12.45)
where we denote the time average approximations of the correlation functions (for i, j =
0, 1, ... , M):
~ l
u+n2
R z [i,j] = ---- L z[n - iJz* [n - jJ
n2 - n1 + 1 n=u+n1
~ l
u+n2
Rsz[-u + j] = ---- L S n -uz* [n - j]
n2 - n1 + 1 n=u+n1