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Mobile Communications(KECE425)
Lecture Note 144-16-2014
Prof. Young-Chai Ko
1
Prof. Y. -C. Ko
S(f)
f
Frequency Selective Fading Channels
• Frequency selective channel =) Inter-symbol interference (ISI)
s(t) r(t)H(f)
H(f)
f
R(f)
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Prof. Y. -C. Ko
Multi-Carrier Transmission
S(f)
fH(f)
• Each sub-block is transmitted so that it experiences the flat fading chan-
nels, that is, ISI-free channel.
3
Prof. Y. -C. Ko
fc + f0
• Orthogonality condition for the receiver
- If we set f0 =
1NT , then we have the orthogonality among each branch of
the transmitter in the multi-carrier transmission.
X
cos(2⇡(fc + f0)t)
X
cos(2⇡(fc + 2f0)t)
X
cos(2⇡(fc +Nf0)t)
m1(t)
m2(t)
mN (t)
+ s(t)
total bandwidth
1/NT 1/NT
2
NT
...
fc +Nf0
2
NT
fc +Nf0
· · ·
fc + f0
fc + 2f0
fc + 3f0fc + 2f0
2
NT
(N � 2)f0
(N � 2)f0 +2
NT=
N � 1
NT
4
Prof. Y. -C. Ko
• Transmit signal waveform
X
cos[2⇡(fc + kf0)t]
Passband filtermk(t)
s(t)
• Assume that there is no fading and no noise.
2
NT
X
cos[2⇡(fc + kf0)t]
Z NT
0(·) dts(t)
or equivalently
s(t) =NX
k=1
mk(t) cos[2⇡(fc + kf0)t]
0
1
2
Z NT
0mk(t) dt
5
Prof. Y. -C. Ko
X
cos[2⇡(fc + kf0)t]
Passband filtermk(t)
s(t) 2
NT
0
s(t) =NX
j=1
mj(t) cos(2⇡(fc + jf0)t)
s(t) cos(2⇡(fc + kf0)t) = cos(2⇡(fc + kf0)t)⇥NX
j=1
mj(t) cos(2⇡(fc + jf0)t)
=
NX
j=1
mj(t) cos(2⇡(fc + kf0)t)⇥ cos(2⇡(fc + jf0)t)
cos(2⇡(fc + kf0)t)⇥ cos(2⇡(fc + jf0)t) =
1
2
[cos (2⇡(2fc + (k + j)f0)t) + cos(2⇡(k � j)f0)t)]
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Prof. Y. -C. Ko
s(t) cos(2⇡(fc + kf0)t) =
1
2
NX
j=1
mj(t) [cos (2⇡(2fc + (k + j)f0)t) + cos(2⇡(k � j)f0)t)]
X
cos[2⇡(fc + kf0)t]
Passband filter
mk(t)s(t) 2
NT
0
=
mk(t)
2
+
mk(t)
2
cos(2⇡(2fc + 2kf0)t)
+
1
2
NX
j=1,j 6=k
mj(t) [cos (2⇡(2fc + (k + j)f0)t) + cos(2⇡(k � j)f0)t)]
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Prof. Y. -C. Ko
X
Passband filter
s(t)
2
NT
0
cos[2⇡(fc + f0)t]
m1(t)
X
Passband filter
2
NT
0
X
Passband filter
2
NT
0
cos[2⇡(fc + 2f0)t]
cos[2⇡(fc +Nf0)t]
m2(t)
mN (t)
......
8
Prof. Y. -C. Ko
• In reality, the information bearing signal mk(t) is a signal with a certain
pulse which satisfies the Nyquist criterion, for example, rectangular pulse,
root-raised cosine pulse, and etc.
• In that case, the amplitude spectrum of mk(t) is sinc type.
· · ·· · ·
9
Prof. Y. -C. Ko
⇥s(t)
h(t)
r(t)+
n(t)
h(t) = ↵(t)ej�(t)
• Distribution of the amplitude ↵(t):
– Rayleigh fading channel
– Ricean fading channel
– Nakagami-m fading channel
• Flat fading channel model
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Prof. Y. -C. Ko
• Received signal power
• Received signal
r(t) = ↵(t)ej�(t)s(t) + n(t)
0
Pr
PrdBm(d) = µdBm(d) + ✏dB
✏dB ⇠ N (0,�2✏ )
area mean
shadowing
local mean
↵2(t)Pr
tt
• Instantaneous power:
↵2(t)Pr or it is often written as ↵2(t)E[s2(t)]
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Prof. Y. -C. Ko
• (Instantaneous) Signal-to-Noise ratio, �:
� =↵2(t)Es
N0
• Average Signal-to-Noise ratio, �:
E[�] = E
↵2(t)Es
N0
�=
E[↵2(t)]Es
N0=
⌦tEs
N0
12
Prof. Y. -C. Ko
CDF of Exponential RV
• For Rayleigh channel, ↵2(t) follows the exponential distribution:
p
↵
2(x) =1
⌦p
e
�x/⌦p
• Then the CDF of ↵2(t) can e written as
P
↵
2(x) = Pr[↵2< x] =
1
⌦p
Zx
0e
�t/⌦pdt
= 1� e�x/⌦p
x > 0
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Prof. Y. -C. Ko
CDF of Instantaneous SNR
• (Instantaneous) Signal-to-Noise ratio, �:
� =↵2(t)Es
N0
p
�
(x) =1
�
e
�x/�
,
• PDF of �:
• CDF of �:
P
�
(x) = 1� e
�x/�
,
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Prof. Y. -C. Ko
0 10 20 30 40 50 60 70 80 90 1000
1
2
3
4
5
6
7
8
�
� = 1
• Example of exponential random samples for � = 1.
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Prof. Y. -C. Ko
0 10 20 30 40 50 60 70 80 90 100−25
−20
−15
−10
−5
0
5
10
� [dB]
� = 1 ! 0 [dB]
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Prof. Y. -C. Ko
Outage Probability
• Outage probability, Pout
:
Pout
= Pr[� �T ]
= 1� e��T /�
−30 −25 −20 −15 −10 −5 0 5 1010−3
10−2
10−1
100
Pout
ex. � = 0 dB
�T dB
17
Prof. Y. -C. Ko
−30 −25 −20 −15 −10 −5 0 5 1010−4
10−3
10−2
10−1
100
• Outage probability for � = 0, 2, 4, 6, 8, 10 dB over Rayleigh channel.
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