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

688 DIGITAL COMMUNICATIONS UNDER LINEARLY DISTORTIVE CHANNELS
MMSE FSE Design
We will apply a similar technique to provide the MMSE FSE design. The difference between
FSE and TSE lies in the output signal
m
M
d[n] = L L J; [k]z;[n - k]
i= l k=O
To minimize the MSE ld[n - s 11-u l 2 , the principle of orthogonality leads to
(d[n] - S n-u) zj [n - .C] = 0
.C = 0, 1, ... ,M, j= 1, ... , m
(12.54)
Therefore, the equalizer parameters {{ilk]} must satisfy
m l'vf
L L J; lk]Zi [n - k]zj [n £] = S n -uzj
i=I k=O
* [n - t] .C=0, 1, ... ,M, j =l,2, ... ,m
There are m(M + 1) equations for the m(M + 1) unknown parameters {{i[k]}, i = 1, ... , m,
k = 0, ... , M. The MMSE FSE can be found as a solution to this set of linear equations. In
terms of practical issues, we should also make the following observations:
· When we have only finite length data to estimate the necessary statistics,
S n-u zJ [n - .C]
and z; [n - k] z/ 1 n - .C]
can be replaced by their time averages from the limited data collection. This is similar to the
TSE design.
· Also similar to the MMSE TSE design, different values of delay u will lead to different mean
square errors. To find the optimum delay, we can evaluate the MSE for all possible delays
u = 0, 1, ... , M + L - 1 and choose the delay that results in the lowest MSE value.
Since their first appearance, 7 adaptive equalizers have often been implemented as FSE.
When training data can be had, FSE has the advantage of suppressing timing phase sensitivity. 7
Unlike the case in TSE, linear FSE does not necessarily amplify the channel noise. Indeed, the
noise amplification effect depends strongly on the coprime channel condition. In some cases,
the subchannels in a set do not strictly share any common zero. However, there is at least one
point Z a that is almost the root of all the subchannels, that is,
H;(Z a ) ,::::; 0
i = l, ... , m
then we say that the subchannels are close to being singular. When the subchannels are coprime
but are close to being singular, the noise amplification effect can still be quite severe.
12.5 CHANNEL ESTIMATION
Thus far, we have focused on the direct equalizer design approach in which the equalizer
filter parameters are directly estimated from the channel input signal S n and the channel output