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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THE MATH 81<br />
To obtain the MMSE solution, we substitute (4.66) into (4.69), which gives<br />
where<br />
F = E{(s(m 0 ) - z(m ( ]))(s(m 0 ) - z(m {) ))*}<br />
= E{|s(m 0 )| 2 - 2(mo)s*(m 0 ) - z(mo)*(m () )s(mo) + z(m ( |)z(mo)*}<br />
= 1 - w H E{vs*(m„)} - E{s(m 0 )v H }w + w H E{vv H }w<br />
= 1 - w H p - p H w - w H Rw, (4.70)<br />
p â E{v S *(mo)} (4.71)<br />
R â E{vv H }. (4.72)<br />
The vector p can be interpreted as the correlation of the data vector to the symbol<br />
of interest. The matrix R can be interpreted as a data correlation matrix, the<br />
correlation of v to itself.<br />
Notice that it is important that v have zero mean. Otherwise, p will depend<br />
on the true symbol value, which we do not know. Since v is zero mean, the data<br />
correlation matrix R is also the data covariance matrix C„. Note, if v had a known,<br />
nonzero mean, we could simply remove it first.<br />
Substituting (4.62) into (4.71) and (4.72) gives<br />
p = y/Ë~ h mo = y/Ë7 s h<br />
OO<br />
R — O v = E $ 2^ h m h m + -NoR-ni<br />
m= — oo<br />
(4.73)<br />
(4.74)<br />
where<br />
Rn(ji,J2) = R P (d h T s - d h T s ). (4.75)<br />
To determine the MMSE solution, we take the derivative of F with respect to<br />
the real and imaginary parts of each element in w and set the derivatives equal to<br />
zero. This can be written compactly as<br />
-2p + 2Rw = 0, (4.76)<br />
where 0 is a column vector of all zeros. From (4.76), we see that the MMSE weight<br />
vector can be obtained by solving the set of equations<br />
Substituting (4.73) and (4.74), we obtain<br />
Rw = p. (4.77)<br />
C„w = y/W a h. (4.78)<br />
which is independent of mo. Thus, the same weights can be used for all symbol<br />
periods. Also, from (4.68), A(ni(,) is also independent of mo- Keep in mind, we<br />
defined the processing delay dj as a relative delay, relative to m^T. Thus, the<br />
elements in v will change with different mo-