Radio Resource Allocation
Algorithms for the
Downlink of Multiuser OFDM
Communication Systems
Sanam Sadr, Alagan Anpalagan and
Kaamran Raahemifar
Ryerson University, Canada
IEEE COMMUNICATIONS SURVEYS &
TUTORIALS, 2009
1
Outline




Introduction
System Model
General Problem of Resource Allocation in Multiuser
OFDM Systems
Classes of Dynamic Resource Allocation in Multiuser
OFDM Systems




Rate Adaptive Algorithms
Margin Adaptive Algorithms
Current Research Areas
Conclusions
2
Orthogonal Frequency Division
Multiplexing (OFDM)

OFDM is based on the concept of multicarrier
transmission




Divide the broadband channel into N narrowband subchannels
Split high rate data stream into N substreams of lower rate data
and transmitted simultaneously on N orthogonal subcarriers
In a single user system, the user can use the total power to
transmit on all N subcarriers
In a multiuser OFDM system, there is a need for a
multiple access scheme to allocate the subcarriers and the
power to the users
3
Overview of the Problem of Resource
Allocation in OFDM Systems
(K users and N subcarriers)
4
System Model


Consider a multiuser OFDM system with K users and N subcarriers
 K = {1, 2, ...,K} and N={1, 2, ...,N}
The data rate of the kth user Rk in bits/s is given by:



B is the total bandwidth of the system
ck,n=1 only if subcarrier n is allocated to user k; otherwise it is 0
γk,n is the SNR of the nth subcarrier for the kth user and is given by:



pk,n is the power allocated for user k in subchannel n
hk,n and Hk,n denote the channel gain and channel to-noise ratio for user k in
subchannel n respectively
N0 as the power spectral density of additive white Gaussian noise (AWGN)
5
Modulation and Bit Error Rate

If r ≥ 2 and 0 ≤ γ ≤ 30 dB, the BER for channel with MQAM
modulation could be better approximated by [30]:
BER  0.2e 1.5 /(M 1)
(4)
where M = 2r and r denotes the number of bits
Using (4), the number of bits r is given by:


ln(5BER)  1.5 /( M  1)


where Γ is the SNR gap and a function of BER:
- ln(5BER)

 r
1.5
(2  1)
Knowing the modulation scheme, the effective SNR γk,n is adjusted
accordingly to meet the BER requirements
[30] A. J. Goldsmith and Soon-Ghee Chua, “Variable-rate variable-power MQAM for fading channels,” IEEE
Trans. Commun., October 1997.
6
General Form of Subcarrier and
Power Allocation Problem

From (1), the total data rate RT is given by:

The general form of the subcarrier and power allocation problem is
shown below:
(Rate Adaptive (RA))
(Margin Adaptive (MA))
(power constraint)
(fixed or variable rate requirements)
7
Classes of Dynamic Resource
Allocation Schemes

Two major classes of dynamic resource allocation
schemes

Rate Adaptive (RA) [8], [32], [33], [39–45]


Maximize the total data rate of the system with the
constraint on the total transmit power
Margin Adaptive (MA) [36–38]

Minimize the total transmit power while providing each
user with its required quality of service in terms of data rate
and BER
8
Rate Adaptive Algorithms

The dynamic resource allocation problem is formulated
as:


Sk is the set of subcarriers assigned to user k for which ck,n =1
is the union of all subcarrier
9
Water-filling Policy (1)

In a system with K users and N subcarriers,

each of N subcarriers is to be allocated to one of K users


Ideally, the subcarrier and power allocation should be
carried out jointly


assuming no subcarrier can be used by more than one user
which leads to high computational complexity necessitating
suboptimal algorithms
Suppose the subcarrier allocation is given, power
allocation problem can be formulated as:
Pi  hi2
RT   log 2 (1 
)
N
i 1
0
K
max
Pi
subject to :
K
P  P
i 1
i
total
10
Water-filling Policy (2)
K
Pi  hi2
max RT   log 2 (1 
)   ( Pi  Ptotal )
Pi
N0
i 1
i 1
K

hi2
RT

  0
Pi ( N 0  Pi  hi2 )

Pi 
*
1


N0
hi2
( Pi 
*
N0 1
 )
2
hi

K

Pi  Ptotal
Let 
i 1
K
K
1 N
Ptotal   Pi   (  20 )
hi
i 1
i 1 

*
In order to obtain the value of λ, we must rely on numerical methods to
solve the non-linear equation
11
Flat Power Allocation (1)

To solve the joint subcarrier and power allocation
problem, a very simple but highly efficient algorithm was
proposed in [32]


This algorithm is based on flat transmit power
It was suggested in [49] that in a single user water-filling
solution, the total data rate is close to capacity even with
flat transmit power spectral density (PSD)

as long as the energy is poured only into subchannels with good
channel gains
[32] W. Rhee and J. M. Cioffi, “Increase in capacity of multiuser OFDM system using dynamic subchannel
allocation,” in Proc. IEEE VTC, vol. 2, pp. 1085–1089, May 2000.
[49] P. S. Chow and J. M. Cioffi, “Bandwidth optimization for high speed data transmission over channels
with severe intersymbol interference,” in Proc. IEEE Globecom, vol. 1, pp. 59–63, December 1992.
12
Flat Power Allocation (2)

This is a very important result since it completely
eliminates the major step of power allocation concentrating
mainly on subcarrier allocation
13
Comparison of the Achieved Data Rates of
Two Power Allocation Methods (1)

Suboptimal with Flat PSD: the best set of subcarriers is selected and the
total power is equally distributed among those subcarriers
14
Comparison of the Achieved Data Rates of
Two Power Allocation Methods (2)

The rate difference between the optimal and suboptimal
power allocation is negligible only when in the
suboptimal algorithm,



the optimal number and set of subcarriers are chosen to transmit
the data and
the total power is equally distributed among these subcarriers
while the rest of subchannels are allocated no power
There is no doubt that the dynamic power allocation
would increase the total throughput

The question however, is whether the achieved performance gain
is high enough to justify the significant additional computational
load
15
Flat vs. Dynamic Allocation (1)

[50] discussed the performance of four different cases in a
multiuser OFDM system with 48 subcarriers and 16 users
uniformly distributed in a single cell




Flat transmit power with fixed subcarrier assignment
Flat transmit power with adaptive subcarrier assignment
Dynamic power allocation with fixed subcarrier assignment
Dynamic power allocation with adaptive subcarrier assignment
[50] M. Bohge, J. Gross, and A. Wolisz, “The potential of dynamic power and sub-carrier assignments in
multi-user OFDM-FDMA cells,” in Proc. IEEE Globecom, vol. 5, pp. 2932–2936, December 2005.
16
Flat vs. Dynamic Allocation (2)

If the average attenuation among different users in the
cell is high (i.e., a large cell is considered)



The gain obtained from the dynamic power allocation has been
shown to be quite significant
Dynamic power allocation has a larger performance
improvement if it is applied with adaptive subcarrier
allocation rather than fixed subcarrier assignment
Therefore, in large cells with high channel gain
differences among the users, a fully dynamic approach
has been recommended
17
Fairness



The total throughput is maximized if each subchannel is
assigned to the user with the best channel gain and the
power is distributed using water-filling policy
 It was proved in [8] and [40]
When the path loss differences among users are large, the
users with higher channel gains will be allocated most of
the resources
Therefore, rate adaptive algorithms are divided into two
major groups based on the user rate constraints
 Constrained-fairness among the users
 A fixed rate requirement for each user
[8] J. Jang and K. B. Lee, “Transmit power adaptation for multiuser OFDM systems,” IEEE J. Select. Areas
Commun., vol. 21, pp. 171–178, February 2003.
[40] G. Song and Y. G. Li, “Utility-based joint physical-MAC layer optimization in OFDM,” in Proc. IEEE
Globecom, vol. 1, pp. 671–675, November 2002.
18
Rate Adaptive Algorithms with
Fairness



One way to accomplish both efficiency and fairness is to use utility
functions that are both increasing and marginally decreasing
 The slope of the utility curve decreases with an increase in the data rate
A logarithmic utility function U(R) = ln(R) is both increasing and
marginally decreasing
 A resource allocation policy using a logarithmic utility function is said
to be proportionally fair [46]
The problem of maximizing the total throughput with fairness was
formulated differently in [32] and [33]
 [32] studied the max-min problem (assured that all users achieve the
same data rate)
 [33] introduces proportional constraints among the users’ data rates
[32] W. Rhee and J. M. Cioffi, “Increase in capacity of multiuser OFDM system using dynamic subchannel allocation,” in
Proc. IEEE VTC, vol. 2, pp. 1085–1089, May 2000.
[33] Z. Shen, J. G. Andrews, and B. L. Evans, “Adaptive resource allocation in multiuser OFDM systems with proportional19
rate constraints,” IEEE Trans. Wireless Commun., November 2005.
[46] F. P. Kelly, “Charging and rate control for elastic traffic,” European Trans. on Telecommun., vol. 8, pp. 33–37, 1997.
Rate Adaptive Algorithms with
Fairness – Max-min Problem

Max-min problem [32]
20
Solutions for Max-min Problem
(1)

A flat transmit PSD was used in [32] indicating that the
power allocated to each subcarrier is constant and equal
to


The resource allocation then reduces to only subcarrier allocation
with N optimization parameters
The suboptimal algorithm proposed in [32] showed less
than 4% spectral efficiency loss compared to optimal
solution with adaptive power allocation

This algorithm achieves acceptable fairness as long as the
number of subcarriers is much larger than the number of users
 i.e., N >> K
21
Solutions for Max-min Problem
(2)

A very subtle but effective change in Rhee’s algorithm
[32] was made by Mohanram et al. [48]



The allocation procedure is the same as [32]
but the total power allocated to the user is distributed among the
assigned subcarriers with water-filling policy
The results are as follows:


The achieved total throughput [48] has up to 25% gain compared
to [32]’s algorithm
The algorithm shows higher achieved gain in total throughput
compared to [32] when the PSD of AWGN is higher
 This could be explained by the fact that applying water-filling
versus fixed power allocation yields larger gains at low SNRs
[48] C. Mohanram and S. Bhashyam, “A sub-optimal joint subcarrier and power allocation algorithm for multiuser
OFDM,” IEEE Commun. Lett., vol. 9, pp. 685–687, August 2005.
22
Rate Adaptive Algorithms with
Fairness – Proportional Constraint (1)

Proportional constraint [33]:

{α1, α2, ...αK} is the set of predetermined proportional constraints
23
Rate Adaptive Algorithms with
Fairness – Proportional Constraint (2)

Two special cases were analyzed in [33] to reduce the
complexity

High channel-to-noise ratio case
 Assume that the best subchannels were chosen for each user

small channel gain differences among them
H k ,1  H k , 2  ...  H k ,n

Assume that the BS can provide a large amount of power

the SNR is much larger than one
log( 1  SNR)  log( SNR)

Linear case
 Assume that the proportion of subcarriers assigned to each user is
approximately the same as the rate constraints
24
Rate Adaptive Algorithms with
Fixed Rate Requirements

When there is a fixed target data rate for each user, the
optimization is given by:

Rk and Rk,min are the achieved and the minimum required data rate
25
for the kth user respectively
Solution for Fixed Rate
Requirements (1)
In [39], the problem has been partitioned into three
steps:

1.
2.
3.
Determine how many subcarriers Nk and how much power pk are
needed for each user
Determine the particular set of subcarriers for each user
Determine the power allocation on each user’s assigned
subcarriers
The complexity of this problem arises from the fact that
the two variables Nk and pk are not independent


certain simplifying assumption have been considered in each
step
26
Solution for Fixed Rate
Requirements (2)

In [39], it is assumed that each user's channel across all
subcarriers is flat


Then, users are sorted according to their channel gain
 hk,1 > hk,2 > hk,3 >...
The frequency diversity is neglected
27
Margin Adaptive Algorithms (1)


In deriving the algorithms of this group, a given set of
user data rates is assumed with a fixed QoS requirement.
The optimization problem can then be formulated as:
28
Margin Adaptive Algorithms (2)


This problem was first addressed in [56] where the focus was only on
subcarrier allocation and further in [36] where adaptive power allocation
was also considered
To make the problem tractable, [36]
 relaxes the requirement ck,n∈{0,1} to allow ck,n to be a real number


i.e., subcarrier can be shared
Then, the problem is reformulated as a convex minimization problem
 The Lagrangian of the new problem can be obtained
 An iterative search algorithm is used to find the set of Lagrange multipliers
 The obtained set determines the optimal sharing factor of all the
subcarriers for all users; however, each subcarrier is assigned to only one
user that has the largest sharing factor on that subcarrier
[36] C. Y. Wong, R. S. Cheng, K. B. Letaief, and R. D. Murch, “Multiuser OFDM with adaptive subcarrier, bit and
power allocation,” IEEE J. Select. Areas Commun., vol. 17, pp. 1747–1758, October 1999.
29
[56] C. Y. Wong, C. Y. Tsui, R. S. Cheng, and K. B. Letaief, “A realtime subcarrier allocation scheme fore
multiple access downlink OFDM transmission,” in Proc. IEEE VTC, vol. 2, pp. 1124–1128, September 1999.
Margin Adaptive Algorithms (3)


The drawback of this approach is that the efficiency and the
convergence of the algorithm depend critically on the step
size and the initial point of the searching
It is prohibitively expensive and not suitable for real time
applications due to its high complexity


One solution to simplify the algorithm is to assume that the channel
is flat
[38] proposed a blockwise subcarrier allocation algorithm


Subcarriers must be assigned to users in blocks
This approach reduces the number of the optimization parameters
[38] L. Xiaowen and Z. Jinkang, “An adaptive subcarrier allocation algorithm for multiuser OFDM system,” in
Proc. IEEE VTC, vol. 3, pp. 1502–1506, October 2003
30
Current Research Areas

Multiple-Input-Multiple-Output (MIMO) OFDM


[62] takes Inter-antenna interference (IAI) into account
Resource Allocation in Multi-cell Systems


User can communicate with the neighboring BS if the SNR
received from its neighboring cell is relatively higher
[66] discussed adaptive cell selection to reduce the probability of
outage for those users with low average SNR
[62] F. S. Chu and K. C. Chen, “Radio resource allocation for mobile MIMOOFDMA,”in Proc. IEEE VTC, pp.
1876–1880, May 2008.
[66] Y. J. Zhang and K. B. Letaief, “Multiuser adaptive subcarrier-and-bit allocation with adaptive cell selection
for OFDM transmission,” IEEE Trans. Wireless Commun., vol. 3, pp. 1566–1575, September 2004.
31
Conclusions


We presented an overview of algorithms to adaptively
allocate resources in a multiuser OFDM system
General observations have been made from the reported
simulation results:


Increase in total throughput with higher number of users
 A result of multiuser diversity
Trade-off between performance and complexity
 Adaptive allocation: better performance, but higher
computational complexity
 Simplifying assumptions and methods:

flat transmit power, neglecting frequency diversity, blockwise allocation
or splitting the allocation procedure into separate steps
32