mohatta2015.pdf

789
LINEAR EQUALIZATION
We now need a couple properties of taking the average or expected value:
1. the average of the sum is the sum of the averages, and
2. the average of a product is the product of the averages if the two quantities
are unrelated or uncorrelated.
The first property allows us to sum the averages of the individual terms. The second
property makes most of those terms zero, as «i, s 2 > n i and «2 are unrelated to one
another. The only nonzero term is the fourth term, giving
Ε{Γ!Γ 2 } = cd. (4.42)
We refer to averages of products of received samples as received sample correlations.
In matrix form,
R = E{rr T }, (4.43)
where r = \r\ r 2 ] T . Thus, we can alternatively express the equalization weights as
a function of received sample correlations (used to form R) and channel coefficients
(used to form h).
The SINR for the MMSE solution has a nice form. It helps to use matrices and
vectors to derive this form. The decision variable can be written as
zi = w T r. (4.44)
The impairment (interference plus noise) can be written as the total received signal
minus the desired signal term, i.e.,
u 2 = r — h.S2- (4.45)
Using the property that x T y = y T x, the impairment power is then
I + N = E{(w T u) 2 } (4.46)
= E{w T uu T w}. (4.47)
Substituting (4.45) and using the fact that E{aa:} = aE{a;} for a nonrandom number
a, we obtain
I + N = E{w T (r-hs 2 )(r-hs 2 )' r w} (4.48)
= w T [E{rr T } - E{s 2 r}h T - hE{s 2 r r } + E{s 2 ,}hh T ] w. (4.49)
Using (4.43) and the fact that E{s 2 r} = h, this simplifies to
I + N = w T [R - hh T ] (4.50)
= w T Rw - [w' r h] 2 . (4.51)
Now, for the case of MMSE weights, (4.36) holds, so that
I + N = w T h-[w T h] 2 . (4.52)
As for the signal power, the signal term in r is hs 2 , so that
S = (w T h) 2 (4.53)