mohatta2015.pdf

MORE DETAILS 103
Notice that this is a general expression, for any weight values. Substituting the
MMSE weights gives an SINR of 1.405. Observe that this is larger than the SINR
for MMSE LE (0.65714), showing that if the detected values are correct, it is better
to remove them than to linearly suppress them. Also, 1.405 is larger than the upper
bound for ZF DFE (1.0), showing that MMSE performs better than ZF. This is
because MMSE collects signal energy from both images rather than just one. It is
still less than the matched filter bound of 1.81, as expected.
For the general dispersive case, assuming s"o = «o, the estimation error on z\ can
be modeled as
ei = [cw\ + dwi — \)s\ + CW2S2 + [wiTi\ + ω2«2]· (5.19)
The average power in e\, denoted E%, is given by
Ei = (c»i + dw 2 - l) 2 + (cw 2 ) 2 + {w\ + w 2 )a 2 . (5.20)
As in the previous chapter, we can take the derivative of E\ (MSE) w.r.t. w\ and
W2 and set them to zero. This gives a set of equations of the form (4.36) where
R
c 2 + σ 2 cd , h=[ C cd c 2 +d 2 + σ d]. (5.21)
2
As before, R can be interpreted as a matrix of data correlations, only now it is the
data correlations for y\ and yi, which have so removed.
The solution to this set of equations is
„, = c(c 2 + d 2 + a 2 )-cd 2
1
(c 2 + d 2 + a 2 ){c 2 + σ 2 ) - c 2 d 2 { '
d{c 2 + σ 2 ) - c 2 d
W2
~ (c 2 + d 2 + a 2 )(c 2 + σ 2 ) - c 2 d 2 (
'
'
For the MIMO case, we need to rethink the triangularization process. Recall the
first step, when we eliminated S2 from r\ by forming X\ = r\ — (d/f)r2- What we
were really doing is detecting s\ using a form of ZF linear equalization with weights
w\ = 1 and v>2 = (d/f). We would do better if we used the MMSE linear equalizer
as described in the previous chapter.
Recall the second step, in which we formed X2 with noise uncorrelated with x\,
subtracted the influence of si on X2 used s\, then detected S2. We still need to
form X2 with noise uncorrelated with x\. Let w\ and »2 be the weights used in the
first step to form y\, i.e.,
x\ = w\T\ + ω 2 Γ 2 . (5-24)
Let's form X2 as before, using
x 2 = r 2 + hr 1 . (5.25)
As before, we determine h such that the noise on x\ and X2 are uncorrelated. This
gives
E{«i«2} = E{(w)iTii + W2Ti2)(hni + TI2)} = hw\a 2 + w 2 a 2 = 0, (5.26)