mohatta2015.pdf

80 LINEAR EQUALIZATION
4.3.1 MMSE solution
Assuming partial matched filtering at the front end, recall from (2.58) and (2.59)
that the received samples can be modeled as
where
v{qT s )^y/E¡ J2 h(qT a -mT)s{m)+ñ(qT a ), (4.60)
m= — oo
L-l
h(t) = Y^g t R p {t-n). (4.61)
t=n
Suppose we are detecting s(mo) using samples v(moT 4- djT s ),j = 0,..., J — 1.
Notice that the delay dj is a relative delay, relative to m^T. The relative delays dj
are parameters to be optimized as part of the design. We can collect these samples
into a vector v, which can be modeled as
where the jth row of h m is given by
From (4.61), we have
oc
v h y/F s Σ h m s{m) + n, (4.62)
m= — oo
hm{j) = h(djT s + (mo - m)T). (4.63)
L-\
hmü) = Σ gtRp(djT s + (m n - m)T - r e ). (4.64)
«=»
Observe that h m¡ ,(j) — h(djT s ), which is independent of mo- Thus, we can replace
hm„ with h, where
h = \h(d„T s ) ... h{dj- 1 T,)) T . (4.65)
With MMSE linear equalization, we form a decision variable
which is then used to detect s(mo) using
z(m„) = w H v, (4.66)
s(rao) = detect(z(mo), A(m^)) (4-67)
A{ma) = w Ä h m „ = w"h. (4.68)
The weight vector w is designed to minimize the cost function
F = E{|2(m 0 )-s(m„)| 2 }, (4.69)
where expectation if over the noise and symbol realizations.