mohatta2015.pdf

THE IDEA 73
Notice that £2 (MSE) depends on two variables, w\ and wi. We can plot £2 versus
these two variables to find the values that minimize £2. Rather than forming a
three-dimensional plot, we can plot E 2 vs. w\ for different values of W2, as shown
in Fig. 4.3. MSE is minimized to a value of 0.603 when w\ = —0.0127 and
u¡2 = —0.0397. This is called the MMSE solution. While we have found the
solution with trial and error, there are mathematical techniques that allow us to
find the solution by solving a set of equations (see next section).
So far, we have used r\ and Γ2 to detect S2. Normally, we would use a sliding
window of data samples, so that r 2 and r 3 would be used to detect s.¡. In this case,
we would find that we could reuse the weights, weighting r 2 with —0.0127 and r-¿
with —0.0397. Similarly, we could use the same weights when detecting s\ using TQ
and rj. However, if we only have r\ and r 2 to work with, then we would need to
determine a new set of weights for detecting «i.
Returning to the detection problem, using MMSE linear equalization to detect
Si and «2 using only r\ and r 2 gives the decision variable values and detected values
in Table 4.1.
3
2.5
2
LU
ω 1.5
1
0.5
0
-0.15 -0.1 -0.05 0 0.05 0.1 0.15
Figure 4.3
w1
MSE vs. w\ for various values of W2 for LE.
Table 4.1
Example of MMSE LE decision variables
Decision Variable Value Detected Symbol
«i -0.18914 ^1
2 2 0.26488 +1