The mathematical development begins with the superposition of
plane waves expressed in the low pass equivalent form ( ).
(3.86)
With N number of paths, fm maximum Doppler phase offset, n 2n/N.
H(t)
1 2N aN
n1
e j(2p# fm # t # cos(an)fn)
H(t) hI(t) jhQ(t)
Assuming N/2 is an odd number, the use of an omnidirectional antenna and the angle of arrivals
are uniformly distributed, the complex envelop of a Rayleigh multipath fading channel is given as
(with M denoting the number of frequency components)
(3.87)
(3.88)
where the following variables are defined.
; ; ; (3.89)
The Jakes Rayleigh fading channel model sums discrete sinusoids corresponding to discrete rays
with differing angle of arrivals or Doppler frequencies. The block diagram is shown in Fig. 3.55.
fm
v
l
fn
v
l
cosS2pn
M T bn
pn
M
M
1
2 aN
2
1b
hQ(t) 2aM
n1
sin[bn] # cosC2pfntD 22 sin [a] # cosC2pfmtD
hI(t) 2aM
n1
cos[bn] # cosC2pfntD 22 cos[a] # cosC2pfmtD
160 CHAPTER THREE
FIGURE 3.55 The classical Jakes multipath fading simulator generating the complex envelope of
the multipath fading channel.
2 sin[b1]
2 sin[bM]
sin[a]
2 cos[b1]
2 cos[bM]
cos[a]
~
cos[w1t]
~
~
2cos[?‰mt]
2cos[wMt]
.
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