We can extend the Jakes model to create up to M multipath
faded signals using the same sinusoids. Here the nth sinusoids is given an additional phase shift of
for ( j 1, . . ., M). These values can be determined by imposing the additional requirement
that the multipath faded signals are uncorrelated, (or nearly uncorrelated as possible). Additionally by
using two quadrature sinusoids per offset, the use of the phase shifters to perform the phase
shift can be eliminated. The extended Jakes model is shown in Fig. 3.56.
gnj bnj
gnj bnj
unk
pn
M 1
2p
M 1
# (k 1)
bn
pn
M 1 an
2pn
M
WIRELESS MULTIPATH CHANNEL 161
~
~
. . .
Summer Summer
. . . . . .
hI(t) hQ(t)
cos[qnj] 2 cos
bnj
2
cos[qnj] 2 sin
bnj
2
sin[qnj] 2 sin
bnj
2
sin[qnj] 2 cos
bnj
2
cos[wnt]
??“sin[wnt]
FIGURE 3.56 Extending the Jakes multipath fading model generating the complex envelope multipath
channel signal.
With the above figure, use these definitions ( j 1, . . ., M) and (n 1, . . ., M)
; (3.92) gnj
2p( j 1)
M 1
bnj
pn
M 1 unj bnj gnj
In the presentation of the Jakes model we discussed how it can reasonably model the Rayleigh multipath
fading time variation and the autocorrelation function.
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