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

12.6 Decision Feedback Equalizer 689
signals z; [n J. We should recognize that if MLSE receiver is implemented, the MLSE algorithm
requires the knowledge of channel parameters {h[k]}. When exact channel knowledge is not
available, the receiver must first complete the important first step of channel estimation.
In channel estimation, it is most common to consider FIR channels of finite order L.
Similar to the linear estimation of equalizer parameters introduced in the last section, channel
estimation should first consider the channel input-output relationship
L
z[nJ = L h[k ls n -k + w[nJ
k=O
(12.55)
If consecutive pilot symbols {s n , n = n 1 , n 1 + I, ... , 112} are transmitted, then because of the
finite channel order L, the following channel output samples
{z[nJ, n = n1 + L, n1 + L + 1, . . . , n2}
depend on these pilot data and noise only. We can apply the principle of MMSE to estimate
the channel coefficients {h[k]} to minimize the average estimation error:
n2
I
l
J (h[OJ, hl lJ, ..., h[L]) = ---- L I z[nJ - L h[kJs n -k
n2 111 - L + 1 111+r k=O
(12.56)
This MMSE estimation can be simplified by setting to zero the derivative of the
J (h[OJ, h[l l, ... , h[M]) with respect to each h[iJ. Removing redundant constants, we have
j = 0, 1, ... , L
Therefore, by defining
n2
1\,1/l #:. L z[nJs-j and R,. [j, kl #::. L s11-kS-j j = 0, 1, . .. , L
n1 +L
n2
n1 +L
we can simplify the MMSE channel estimation into a compact matrix expression:
j
R z [O, OJ R z [O, 1 J z [O, LJ h[OJ rs z [OJ
R z [l , OJ R z [l, lJ R [l,LJ h[lJ rsz [IJ
[
J [ J [
R z [L OJ R z [L l]
. . .
R z [L, LJ h[LJ rs z [MJ
(12.57)
Eq. (12.57) can be solved by matrix inversion to estimate the channel parameters h[iJ.
In the more general case of FSE, the same method can be used to estimate the ith subchannel
parameters by simply replacing z[n - kl with z;[n - k].
12.6 DECISION FEEDBACK EQUALIZER
The TSE and FSE we have discussed thus far are known as linear equalizers because the
equalization consists of a linear filter followed by a memoryless decision device. These linear