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