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