mohatta2015.pdf

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