mohatta2015.pdf

72 LINEAR EQUALIZATION
4.1.1 Minimum mean-square error
A better strategy is to minimize the sum of ISI and noise. This is referred to as the
minimum mean-square error (MMSE) strategy. Consider using r\ and r 2 to detect
«2- With the partial ZF strategy, we used r\ to fully cancel the ISI from s\. In
doing so, we introduced ISI from s (l as well as an additional noise term, ri\. With
the MMSE approach, we use r\ to partially cancel ISI from βχ. While this leaves
some ISI from s\, it also reduces the ISI from SQ and the noise from n\.
Let's look at the math. Suppose we are going to decide S2 using decision variable
Z2 that is the weighted sum of r\ and r 2 . Specifically,
To see what happens to the ISI and noise, recall that
Substituting (4.13) in (4.12) gives
«2 = win + w 2 r2- (4-12)
rj = — lOsi + 9s() + n\
r 2 = -10s 2 +9si +n 2 . (4.13)
2 2 = wi(-10si+9s()+ ni) + w 2 (-10s 2 + 9si+n 2 )
— —IOUI2S2 + (—lOwi + 9w2)si + 9wiS() + wini + W2"2- (4-14)
We can think of z 2 in (4.14) as an estimate of s 2 . Consider the error in the
estimate, defined as
e 2 = 2 2 - s 2
= (-10u>2 - l)s2 + (-ΙΟΐϋχ +9UJ 2 )SI +9u>iS() + w\Tii +ω 2 η 2 . (4.15)
We would like this error to be as small as possible. However, there are trade-offs.
To make the ,$2 term small, we want u> 2 close to —0.1. To make the si term small,
we want — 10wi + 9w 2 close to zero. To make the rest of the terms small, we want
w\ and W2 to be close to zero. We cannot make all of these things happen at the
same time!
What we can do is minimize the sum of all these terms in some way. For good
performance, it turns out that it is good to minimize the average (mean) of the
power (square) of the error (e 2 ). This gives it the name minimum mean-square
error (MMSE).
To do this, we need some additional facts.
1. The average of asi is a 2 times the average of Sj.
2. While symbols, such as si, can be either +1 or — 1, the square value is always
1. Thus, the average of the sf is 1.
3. While the noise terms, such as n-¡, are random, we were told that the average
of their squared values is σ 2 = 100.
With these facts, the average power of e 2 , denoted E2, is given by
E 2 = (w 2 (-10) - 1) 2 (1) + (w 2 (9) + Wl(-10)) 2 (l) + (tB,9) 2 (l) + {w\ + w 2 )(100).
(4.16)