Transcript PPT - Signals and Systems, Uppsala University

Multiuser Diversity in Delay-Limited
Cellular Systems
Ralf R. Müller
[email protected]
Department of Electronics & Telecommunications
Norwegian University of Science & Technology, Trondheim
Giuseppe Caire & Raymond Knopp
{caire,knopp}@eurecom.fr
Institut Eurècom
Sophia-Antipolis, France
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.
Ergodic vs. Delay-Limited Capacity
Two extreme cases of fading channels:
1.
The channel changes very often within a single
codeword.
Ergodic capacity
2.
The channel is constant within a single codeword.
Delay-Limited capacity
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:
2

|d | E
C  inf log2  1  f (d)

N0 

d : f (d ) 1
Ed
if E |d|-2 exists, C > 0
and f (d) = |d|-2.
C = 0 for Rayleigh fading.
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 capacities
IEEE 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?
Gaussian Multiple-Access Channel without Fading
Dual Representation of GMAC without Fading
Transmit Power vs. Receive Power
Rescaling of axis in power region
Transmit Power vs. Receive Power
2
GMAC with Fading
Let the attenuations be random from codeword to codeword.
if E |d|-2 exists, C > 0
C = 0 for Rayleigh fading.
Single User vs. Infinite Users
Single User:
E%
b
N0
1
 ln(2)  2 dx  
Rx

dF|d|2 x
0
x
Infinite Users:
E%
b
N0
RF
 ln(2)  2
for all rates
|d|2

x  dF|d|2 x

x
0
Ri

Rj
i, j
Signal Attenuation
Path Loss
|di|2 ~
r3
|ri|-4
r1
r2
no shadowing
no Rayleigh fading
spec. efficiency [bits/sec/Hz]
Delay-Limited Capacity for Path Loss
2
~
Eb
d
2
N0
[dB]
There is a gain by multiuser diversity without
any drawback in quality of service,
if E |d|-2 exists.
Without constraints to orthogonal separation
of users, the gain is greater.
Multiuser Diversity + Frequency Diversity
Let each user have M parallel channels.
For Rayleigh fading, DL-capacity is positive if M > 1.
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.
Remark 1:
Theorem 1 does not hold for a finite number of users, in general.
Multiuser Diversity + Frequency Diversity
d = max{|d1|; |d2|; ··· ; |dM|}
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.
Signal Attenuation
Path Loss |di|2 ~ |ri| -4
r3
Rayleigh fading
2nd order
frequency
diversity
r1
r2
no shadowing
spec. efficiency [bits/sec/Hz]
Delay-Limited Capacity without
with Rayleigh
Rayleigh
fading
fading
2
The mean fading changes as well.
~
Eb
d
2
N0
[dB]
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.
Cellular Systems
Cellular systems are interference limited.
You want to minimize interference into other cells instead of transmit power.
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.
Open question: Does MAC broadcast duality apply here?
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 
Reuse
factor 2:
F
d0
2
(x) a F
dd00


n 
n 0
n  
n 0
0
(x)
(x)
22
22
dn2n
Large number of users makes interference symmetric.
dn2n
dd00
22
22
spec. efficiency [bits/sec/Hz]
Delay-Limited Capacity of Cellular System
E%
b
N0

reuse 2
reuse 3
reuse 4
reuse 2
reuse 3
reuse 4
path loss exponent
Spectral
effciency
can be
doubled.
Coming Soon ...
Hexagonal cells ... Rayleigh fading
Just one more thing:
path loss exponent 2:
R 
1
1
Rx
2
 g(x)dx
0
with
g(x) 
2
4

(1  x)2 1  tan 2
 x  1

2
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