mohatta2015.pdf

100 MMSE AND ML DECISION FEEDBACK EQUALIZATION
In addition, we are also going to remove the influence of So on r 2 . Recall that Γ2
can be modeled as
r 2 = -10s 2 +9si +n 2 (5.4)
As there is no influence of So on r 2 , we simply get
which can be modeled as
y 2 = r 2 - 0, (5.5)
j/2 = -10s2+9si+n 2 . (5.6)
We now wish to estimate si using a weighted combination of y\ and ?/ 2 . Specifically,
Substituting (5.3) and (5.6) in (5.7) gives
z 1 =wiyi+ W2V2- (5.7)
z\ = u>i(—lOsi + ni) + ω 2 (—10s 2 + 9si + n 2 )
= — 10w 2 s 2 + (—lOwi + 9TO 2 )SI + W\n\ + ω 2 η 2 . (5.8)
Similar to the previous chapter, we can think of z\ as an estimate of s\. Consider
the error in the estimate, defined as
ei = z\- s 1
= — WW2S2 + (—10u>i + 9w2 — l)«i + wini + w 2 n 2 . (5.9)
To make this error small, we will minimize the average (mean) of the power (square)
of the error (ei). Hence the name minimum mean-square error (MMSE).
Recall from the previous chapter the following facts.
1. The average of asi is a? times the average of s\.
2. While symbols, such as s\, can be either +1 or —1, the square value is always
1. Thus, the average of the square value 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 in e 2 , denoted E2, is given by
E x = (w 2 (-10)) 2 (l) + (ω 2 (9) + wi(-10) - 1) 2 (1) + (wi9) 2 (l) + ω 2 (100) + w|(100).
(5.10)
As in the previous chapter, E\ (MSE) depends on w\ and w 2 . In Fig. 5.1, we plot
E2 vs. w\ for different values of w 2 . MSE is minimized to a value of 0.4158 when
wi = -0.04158 and w 2 = 0.01871. This is the MMSE solution for the feedforward
filter for s\. Notice that unlike the ZF DFE solution, w 2 is not zero in this case. A
block diagram is given in Fig. 5.2.
Another strategy which gives the same performance is the ML strategy. This
strategy is discussed more in later sections.