Multiuser Diversity in Delay-Limited Cellular Systems
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Multiuser Diversity in Delay-LimitedCellular Systems
Ralf R. Mü[email protected]
Department of Electronics & TelecommunicationsNorwegian University of Science & Technology, Trondheim
Giuseppe Caire & Raymond Knopp{caire,knopp}@eurecom.fr
Institut EurècomSophia-Antipolis, France
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Disclaimer
This is NOT a talk on fair scheduling of users.
Here:
• Fairness is not enough. • Each user has to get their message instantaneously with probability going to 1.
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Ergodic vs. Delay-Limited Capacity
1. The channel changes very often within a single codeword.
2. The channel is constant within a single codeword.
Two extreme cases of fading channels:
Ergodic capacity
Delay-Limited capacity
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Delay-Limited Capacity for a Single User
Let the fading d be known at both transmitter and receiver,
AWGN channel with energy per symbol E and noise
density N0:
C =d:
dEf (d)≤1inf log
21 + f(d)
|d |2 EN
0
⎛
⎝⎜
⎞
⎠⎟
if E |d|-2 exists, C > 0 and f (d) = |d|-2.
C = 0 for Rayleigh fading.
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Delay-Limited Capacity for Many Users
The result is given as a multi-dimensional optimization problem in
Stephen V. Hanly, David N.C. Tse:Multiaccess fading channels. Part II: Delay-limited capacitiesIEEE Trans. Inform. Theory, vol. 44, no. 7, pp. 2816 - 2831, Nov. 1998.
Can we obtain a gain by multiuser diversity without any drawback in quality of service, e.g. no additional delay?
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Gaussian Multiple-Access Channel without Fading
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Dual Representation of GMAC without Fading
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Transmit Power vs. Receive Power
Rescaling of axis in power region
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Transmit Power vs. Receive Power
2
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GMAC with Fading
Let the attenuations be random from codeword to codeword.
if E |d|-2 exists, C > 0
C = 0 for Rayleigh fading.
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%Eb
N0
= ln(2) 2Rxdx0
1
∫ ⋅dF
|d|2x( )
x∫
Single User vs. Infinite Users
Single User:
Infinite Users:
%Eb
N0
= ln(2) 2R F
|d|2x( )
⋅dF
|d|2x( )
x∫
jiR
R
j
i ,0 ∀∞<<for all rates
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Path Loss |di|2 ~ |ri|-4
no shadowing no Rayleigh fading
Signal Attenuation
r1
r2
r3
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Delay-Limited Capacity for Path Losssp
ec.
effic
ienc
y [b
its/s
ec/H
z]
infin
ite u
sers
[dB]
single
use
rin
f. or
th. u
sers
0
2
~
Nd
Eb
−
2
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There is a gain by multiuser diversity without any drawback in quality of service,
Without constraints to orthogonal separation of users, the gain is greater.
if E |d|-2 exists.
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Multiuser Diversity + Frequency Diversity
Let each user have M parallel channels.
Theorem 1:Subject to some technical conditions on the rates and the fading, each user uses only that channel of theirs which has the best propagation conditions, as the number of users approaches infinity.
For Rayleigh fading, DL-capacity is positive if M > 1.
Remark 1:Theorem 1 does not hold for a finite number of users, in general.
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Multiuser Diversity + Frequency Diversity
Corollary 1:Subject to some technical conditions on the rates and the fading, frequency diversity only re-shapes the fading distribution, as the number of users approaches infinity.
d = max{|d1|; |d2|; ··· ; |dM|}
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Path Loss |di|2 ~ |ri| -4
2nd order frequency diversity
no shadowing
Signal Attenuation
r1
r2
r3r1
r2
r3
Rayleigh fading
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0
2
~
Nd
Eb
−
Delay-Limited Capacitysp
ec.
effic
ienc
y [b
its/s
ec/H
z]
chan
nel c
apac
ity
[dB]
orth
ogon
al li
mit
without Rayleigh fadingwith Rayleigh fading
single u
ser
The mean fading changes as well.
2
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Uplink vs. Downlink
Theorem 2:Thanks to multiple-access broadcast duality on Gaussian channels, all results for multiple-access channels and transmit power, also hold for the Gaussian broadcast channel.
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Cellular Systems
Corollary 2:Subject to some technical conditions on the rates and the fading, minimizing interference power onto base stations of other cells instead of transmit power is equivalent to re-shaping the fading distribution as the number of users approaches infinity.
Cellular systems are interference limited.You want to minimize interference into other cells instead of transmit power.
Open question: Does MAC broadcast duality apply here?
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Cellular Systems (cont’d)
Consider a linear cellular system w.l.o.g.You want to minimize interference into other cells instead of transmit power.
I ∝
dn
2
n=−∞n≠0
+∞
∑
d0
2
Large number of users makes interference symmetric.
I ∝
d2n
2
n=−∞n≠0
+∞
∑
d0
2
Fd0
2(x) a Fd0
2
dn
2
n=−∞n≠0
+∞
∑
(x)
Fd0
2(x) a Fd0
2
d2n
2
n=−∞n≠0
+∞
∑
(x)Reuse factor 2:
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%Eb
N0
→ ∞
Delay-Limited Capacity of Cellular System
spec
. ef
ficie
ncy
[bits
/sec
/Hz]
path loss exponent
capacity
single user
Spectral effciency can be doubled.
reuse 2
reuse 2
reuse 3
reuse 3
reuse 4
reuse 4
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Coming Soon ...
Hexagonal cells ... Rayleigh fading
Just one more thing:
R =1
2Rxg(x)dx0
1
∫ g(x) =
π 2
4(1 −x)2 1 + tan2 π
2x( )( ) −1
path loss exponent 2:
with
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