mohatta2015.pdf
signal processing from power amplifier operation control point of view
signal processing from power amplifier operation control point of view
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72 LINEAR EQUALIZATION<br />
4.1.1 Minimum mean-square error<br />
A better strategy is to minimize the sum of ISI and noise. This is referred to as the<br />
minimum mean-square error (MMSE) strategy. Consider using r\ and r 2 to detect<br />
«2- With the partial ZF strategy, we used r\ to fully cancel the ISI from s\. In<br />
doing so, we introduced ISI from s (l as well as an additional noise term, ri\. With<br />
the MMSE approach, we use r\ to partially cancel ISI from βχ. While this leaves<br />
some ISI from s\, it also reduces the ISI from SQ and the noise from n\.<br />
Let's look at the math. Suppose we are going to decide S2 using decision variable<br />
Z2 that is the weighted sum of r\ and r 2 . Specifically,<br />
To see what happens to the ISI and noise, recall that<br />
Substituting (4.13) in (4.12) gives<br />
«2 = win + w 2 r2- (4-12)<br />
rj = — lOsi + 9s() + n\<br />
r 2 = -10s 2 +9si +n 2 . (4.13)<br />
2 2 = wi(-10si+9s()+ ni) + w 2 (-10s 2 + 9si+n 2 )<br />
— —IOUI2S2 + (—lOwi + 9w2)si + 9wiS() + wini + W2"2- (4-14)<br />
We can think of z 2 in (4.14) as an estimate of s 2 . Consider the error in the<br />
estimate, defined as<br />
e 2 = 2 2 - s 2<br />
= (-10u>2 - l)s2 + (-ΙΟΐϋχ +9UJ 2 )SI +9u>iS() + w\Tii +ω 2 η 2 . (4.15)<br />
We would like this error to be as small as possible. However, there are trade-offs.<br />
To make the ,$2 term small, we want u> 2 close to —0.1. To make the si term small,<br />
we want — 10wi + 9w 2 close to zero. To make the rest of the terms small, we want<br />
w\ and W2 to be close to zero. We cannot make all of these things happen at the<br />
same time!<br />
What we can do is minimize the sum of all these terms in some way. For good<br />
performance, it turns out that it is good to minimize the average (mean) of the<br />
power (square) of the error (e 2 ). This gives it the name minimum mean-square<br />
error (MMSE).<br />
To do this, we need some additional facts.<br />
1. The average of asi is a 2 times the average of Sj.<br />
2. While symbols, such as si, can be either +1 or — 1, the square value is always<br />
1. Thus, the average of the sf is 1.<br />
3. While the noise terms, such as n-¡, are random, we were told that the average<br />
of their squared values is σ 2 = 100.<br />
With these facts, the average power of e 2 , denoted E2, is given by<br />
E 2 = (w 2 (-10) - 1) 2 (1) + (w 2 (9) + Wl(-10)) 2 (l) + (tB,9) 2 (l) + {w\ + w 2 )(100).<br />
(4.16)