mohatta2015.pdf

MORE DETAILS 61
where
«i = m - (d/f)n 2 . (3.20)
Can we simply keep r 2 as is? No. The reason is that noise on x\ (u\) is now
correlated with the noise on r 2 («2)· Specifically,
E{ Ul n 2 } = E{n x n 2 - (d/f)n 2 2} = -(d/f)a 2 φ 0. (3.21)
So let's consider a second combination of the form x 2 = r 2 + hr\, which can be
modeled as
x 2 = (e + hc)s 1 + (f + hd)s 2 + u 2 , (3.22)
where
u 2 =n 2 + hni. (3.23)
We want to pick h such that u\ and u 2 are uncorrelated. Setting the correlation to
zero gives
E{uiu 2 } = E{(m - (d/f)n 2 )(n 2 + hm)} = ha 2 - (d/f)a 2 = 0, (3.24)
which implies that we should set h = d/f.
Putting this together in matrix form, we obtain
1 -(d/f)
(d/f) 1
(3.25)
which can be modeled as
x \= Cs + u, (3.26)
where the elements of u are uncorrelated and have power (1 + (d/f) 2 )a 2 and
C = AH
c - de/f 0
e + dc/f f + d 2 /f
(3.27)
Success. Our channel matrix is now triangular. Note, we could have scaled the
elements in A by 1/^/(1 + (d/f) 2 ) to force the elements in u to have power σ 2 .
Instead, we allowed the signal and noise powers to increase, maintaining the same
SNR.
Now we detect si first, using y\. We can then detect s 2 using
As for SINR, the SINR for detecting si is
For detecting s 2 , an upper bound on SINR is given by
y 2 = X2 - (e + dc/f)s!. (3.28)
(c-de/f) 2
S I N R ! = , , . > : : ; „ , ■
(1 + (d//)> 2
( ^
SINR2 * (ΐ%//)> 2· ( 3·3°)