mohatta2015.pdf

THE MATH 105
Suppose we are detecting s(mg) using samples y(m.oT + djT s ),j = 0,..., J — 1.
The sample y(m n T + djT s ) is obtained from ν{πΐ(,Τ + djT s ) by removing ISI from
past blocks using detected symbol values. Specifically,
mu-1
y{ni()T + djTg) = v(m n T + djT s ) — \fE s VJ h(qT s — mT)s(m), (5.33)
m= — oo
which, assuming correct detections, can be modeled as
y{qT s ) h y/Ël Σ h(qT s - mT)s{m) + ñ(qT s ). (5.34)
m=mo
We can collect these samples into a vector y, which can be modeled as
where the rth row of h m is given by
oo
y |= VË~ S Σ h m s(m) + n, (5.35)
m=mc>
ftm(r) = h(d r T s + (mo — m)T). (5.36)
With MMSE DFE, we form the decision variable
which is then used to detect s(mo) using
z{m n ) = w H v, (5.37)
s(m () ) = detect(z(mo), A(mo)) (5.38)
A(mo) = w"h mo =w H h, (5.39)
where h is defined in (4.65). The weight vector w is designed to minimize the cost
function
F = E{|s(m 0 )-z(mo)| 2 }, (5.40)
where expectation if over the noise and symbol realizations.
The development is similar to that in Chapter 4, so that the weight solution ends
up being the solution to the set of equations
where
Using (5.34), it is straightforward to show that
Rw = p, (5.41)
p ^ E{y S *(m 0 )} (5.42)
R ^ E{yy H }. (5.43)
p = y/Ë~ s h mo = y/F s h (5.44)
R = C y = E s J2 h m h£+N () R n , (5.45)