Now let??™s suppose there is a frequency selective fading channel such as a two-ray model, then the
power equation can be written as follows (L 2).
(5.162) P [w1w*2
] # ch*
11 h*
12
h*
21 h*
22 d # ch11 h21
h12 h22 d # cw1
w2 d
w2 h1h*2
P
1
2 Zh1 Z2 2Re5h1h*2
w*2
6 Zh2 Z2 # Zw2 Z2
P [w1 w2] # c Zh1 Z2 h*1
h2
h*2
h1 Zh2 Z2 d # cw1
w2 d
P [w1 w2] # ch*1
h*2
d # [h1 h2] # cw1
w2 d
h1 [h11ch1L]T H [h1 h2] w [w1 w2 ]
P w* # (H*H) # w
h2 b # eju2 h1 a # eju1
r d # 1 22
# eju1 # 5a b6 r d # 1 22
# 5a # eju1 b # eju16 r d # 1 22
# a # eju1 d # 1 22
# ej(u1u2) # b # eju2
r d # w1 # a # eju1 d # w2 # b # eju2
w2
1 22
# h1 # h*2
Zh1 # h*2
Z
1 22
# ej(u1u2) w1
1 22
284 CHAPTER FIVE
After some simple manipulations we get the following equation that needs to be maximized.
(5.163)
Now this consists of two parts: The first part can be maximized when considering only the first multipath
while the second part can be maximized when considering only the second multipath. To jointly
optimize across both multipaths would possibly sacrifice the performance of each individual path for
the sake of the overall multipath channel.
Let us consider the following simplified problem of the channel matrix .
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