116)
where d is the desired signal. The solution to the above criterion leads to the following equation for
the antenna weights:
(5.117)
where the received signal??™s covariance matrix and cross-correlation vector are defined and given as below.
(5.118)
(5.119)
In practice we have to estimate these parameters using the received signal since the channel is time
varying. We can replace the expectation operator by a sample mean estimator, which gives us the following
estimate of the covariance matrix.
(5.120)
And the following estimate of the cross-correlation vector
(5.121)
We have used N to denote the number of time samples used in the estimate calculation. The use of
these estimates is sometimes referred to as Sample Matrix Inversion (SMI) or Direct Matrix Inversion
(DMI) [59, 64??“66]. Hence the estimated MMSE array weights are given as
(5.122) w^
MMSE R ^
1
xx # r^
xd
r^
xd
1
NaN
i1
x(t ti) # d*(t ti)
R ^
xx
1
NaN
i1
x(t ti) # x*(t ti)
rxd E5x # d*6 Rxx E5x # x*6
wMMSE R1
xx # rxd
min
w
Ee gw* # x d g2 f
wMMSE
276 CHAPTER FIVE
..
.
Weight
Calculation
X
X
X
Desired Signal, d
+
Decision
Device
x1
x2
xM
w1
w2
wM
y
FIGURE 5.65 Adaptive antenna array block diagram.
Pages:
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453