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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MORE DETAILS 77<br />
To obtain the unity-gain max SINR solution directly, we want the coefficient in<br />
front of s 2 to be 1. This is achieved by setting w 2 = 1/c, so that<br />
The impairment power is given by<br />
«2 = s 2 + {d/c + wic)si + (ωιηι + (l/c)ri2). (4-31)<br />
Ñ 2 = (c 2 + a 2 )w\ + 2dw! + {d 2 + a 2 )/c 2 . (4.32)<br />
Setting the derivative w.r.t. w\ to zero and solving for wi gives<br />
Wl<br />
= 72-^2- ( 4·33 )<br />
For the MMSE solution, we start with (4.30). Substituting models for τ\ and<br />
r 2 , the error e 2 = z 2 — s 2 can be modeled as<br />
e 2 = {w 2 c - l)s 2 + {w 2 d + wic)si + (wini + w 2 n 2 ). (4-34)<br />
The MSE is the power in e 2 , which is<br />
E 2 = (w 2 c - l) 2 + (w 2 d + wie) 2 + (w 2 + w 2 2)a 2 . (4.35)<br />
To find the MMSE weights, we take the derivative of E 2 w.r.t. w\ and set it to<br />
zero. We do the same w.r.t. w 2 . This gives two equations in two unknowns, which<br />
can be written in matrix form as<br />
where<br />
o _ Γ c 2 + d 2 + σ 2 cd<br />
"■-[ cd c 2 +d 2 + a 2<br />
The solution to this set of equations is<br />
Rw = h, (4.36)<br />
[Z\], and h=[0]. (4.37)<br />
= -c 2 d cjc 2 + d 2 +a 2 )<br />
Wl<br />
(c 2 + d 2 + σ 2 ) 2 - c 2 d 2 W2<br />
' {c 2 +d 2 +σ 2 ) 2 -c 2 d 2, [ '<br />
and the resulting MSE is given by<br />
MSE = w^Rw - 2wr h + x = ! - ^ I f 2) t ^ ,<br />
(4.39)<br />
where superscript "'Ρ' denotes transpose (turns a column vector into a row vector).<br />
The minimum ISI solution can be obtained by setting σ 2 to zero.<br />
The elements in R have a special interpretation. The diagonal elements are the<br />
average received sample power, i.e., the average of r\ or r\. Specifically, the average<br />
received signal power is the desired signal power (c 2 + d 2 ) and the average noise<br />
power (σ 2 ). The off-diagonal elements are the average of the product of adjacent<br />
received values, i.e., the average of r\r 2 . Specifically, using the models,<br />
E{nr 2 } = E{(cs 2 +dsi+n 2 )(csi+dso + ni)} (4.40)<br />
= E{c s 2 s\ + cds 2 SQ + cs 2 ni + dcs x<br />
+ d 2 siSo + ds\ni + n 2 cs\ + n 2 ds n + n 2 n\). (4.41)