Optimal Downlink OFDMA Subcarrier, Rate, and ... AWS
Optimal Downlink OFDMA Subcarrier, Rate, and Power Allocation with Linear Complexity to Maximize Ergodic Weighted-Sum Rates Ian C. Wong and Brian L. Evans The University of Texas at Austin, Austin, Texas 78712 Email:
{iwong,
bevans}@ece.utexas.edu
Abstract In this paper, we propose a resource alloca-
Furthermore, previous research have assumed that
tion algorithm for ergodic weighted-sum rate maximization
algorithms to nd the optimal or near-optimal solution
in downlink OFDMA systems. In contrast to most previous
to the problem is too computationally complex for real-
research that focused on maximizing instantaneous rates using deterministic optimization techniques, we focus on maximizing ergodic rates using stochastic optimization techniques, which allow us to exploit the temporal dimension,
in
addition
to
the
frequency
and
multiuser
dimensions. Furthermore, in contrast to most previous
time implementation. Hence, the main focus of previous research efforts have been on developing heuristic approaches with typical complexities in the order of
O(M K 2 ).
Our approach, on the other hand, is based on
a dual optimization framework, which is less complex
algorithms that used greedy suboptimal heuristics with
(O(M K) per iteration, with less than 10 iterations) and
quadratic complexity, we use a dual optimization approach
achieves relative optimality gaps that are less than
that resulted in a simple subcarrier, rate, and power
(i.e. achieving
allocation algorithm that has complexity
M -user, K -subcarrier
O(M K)
for an
OFDMA system. Surprisingly, our
method is shown to result in duality gaps less than
10−4
in scenarios of practical interest, thereby allowing us to claim practical optimality. We present simulation results for a 3GPP-LTE system employing adaptive modulation.
99.9999%
10−4
of the optimal solution) in
typical scenarios, and thus actually allowing us to claim practical optimality. We focus on the discrete rate case in this paper. We also investigated the continuous rate case in [2]. Note that the dual optimization approach was also studied in [3] [4] [5], but their focus has been on instantaneous and continuous rate optimization.
I. I NTRODUCTION II. S YSTEM M ODEL
Next-generation broadband wireless system standards, e.g. 3GPP-Long Term Evolution (LTE) [1], consider Orthogonal Frequency Division Multiple Access (OFDMA) as the preferred physical layer multiple access scheme, esp. for the downlink. The problem of assigning the subcarriers, rates, and powers to the different users in an OFDMA system has been an area of active research, (see e.g. [2] [3] [4]). In most of the previous work, the formulation and algorithms only consider instantaneous
We consider a single OFDMA base station with
K -subcarriers and M -users indexed by the set K = {1, . . . , k, . . . , K} and M = {1, . . . , m, . . . , M } (typically K À M ) respectively. We assume an average ¯ > 0, bandwidth B , and noise transmit power of P density N0 . The received signal vector for the mth user at the nth OFDM symbol is given as
performance metrics. Thus, the temporal dimension is
ym [n] = Gm [n]Hm [n]xm [n] + wm [n]
(1)
not being exploited when the resource allocation is
ym [n]
and
xm [n]
are the
K -length
performed. Instead of considering only instantaneous
where
data rate, we formulate the problem considering user-
transmitted complex-valued signal vectors;
received and
weighted ergodic sum rate. This allows us to exploit all
diagonal gain allocation matrix with diagonal elements
three degrees of freedom in our system, namely time,
[Gm [n]]kk =
frequency, and multiuser dimensions. At the same time,
noise
we can enforce various notions of fairness through the
circular-symmetric, complex Gaussian (ZMCSCG) noise
user weights.
vector; and
Gm [n]
is the
p 2 I ) with pm,k [n]; wm [n] ∼ CN (0, σw K 2 variance σw = N0 B/K is the white zero-mean, Hm [n] = diag {hm,1 [n], . . . , hm,K [n]} is the
users, and that the resource allocation decisions are made
diagonal channel response matrix, where
hm,k [n] =
Nt X
known to the users through an error-free control channel.
gm,i [n]e
−j2πτi k∆f
.
(2)
i=1 are the complex-valued frequency-domain wireless channel fading random processes, given as the discretetime Fourier transform of the
Nt time-domain multipath τi and subcarrier spacing
taps
gm,i [n]
∆f .
These taps are modeled as stationary and ergodic
with time-delay
2 , σm,i
discrete-time random processes with tap powers
which we assume to be independent across the fading paths
i
m. Since gm,i [n] is stationary hm,k [n]. Hence, the distribution of independent of n through stationarity, and
III. E RGODIC R ATE M AXIMIZATION
IN
OFDMA
A. Problem Formulation The data rate of the
k th
subcarrier for the
mth
user
can be given by the staircase function
η0 ≤ pm,k γm,k < η1 r0 , . . . . R(pm,k γm,k ) = ., . rL−1 , ηL−1 ≤ pm,k γm,k < ηL (4)
and across users
{ηl }l∈L , L = {0, . . . , L − 1},
and ergodic, so is
where
hm [n]
boundaries which dene a particular code-rate and mod-
is
are the SNR
rl
we can replace time averages with ensemble averages
ulation order pair combination that result in
in the problem formulations through ergodicity. In the
per transmission with a predened target bit error rate
subsequent discussion, we shall drop the index
taps
rl ≥ 0, rl+1 > rl , r0 = 0, η0 = 0, and ηL = ∞. Denote by p = [pT1 , · · · , pTK ]T the vector of T powers to be determined, where pk = [p1,k , · · · , pM,k ] .
gm,i ∼
Note that determining the power vector consequently
n
when
the context is clear for notational brevity. We
1
assume
that
the
time
domain
channel
are independent ZMCSCG random variables
2 ) with total power CN (0, σm,i
2 σm
=
PNt
2 i=1 σm,i . Then
(3)
Rhm = WΣm WH is the
e−j2πτi k∆f and
K × Nt DFT matrix with [W]k,i = 2 , . . . , σ2 Σm = diag{σm,1 m,Nt } is an
Nt ×Nt diagonal matrix of the time-domain path powers. Since we also assume that the fading for each user is independent, then the joint distribution of the stacked fading vector for all users
h = [hT1 , . . . , hTM ]T
is
likewise a ZMCSCG random vector with distribution
h ∼ CN (0KM , Rh ) where Rh
is the
diagonal covariance matrix with
KM ×KM
Rhm
block
as the diagonal
block elements. This is the distribution over which we shall take the weighted sum rate function in the problem formulations. We let
2 γm,k = |hm,k |2 /σw
γm = [γm,1 , . . . , γm,k ]T
where
γ¯m,k = subcarrier k
γm,k for a particular users m are independent but that
2 /σ 2 . Note σm w and different
not necessarily identically
distributed (INID) exponential random variables; and for
m
OFDMA can then be written as
and different subcarriers
k
p k ∈ P k ⊂ RM +,
0 P k ≡ {pk ∈ RM + |pm,k pm0 ,k = 0; ∀m 6= m } For notational convenience, we let
K · · · × P K ⊂ RM +
(5)
p ∈ P ≡ P1 ×
denote the space of allowable power
vectors for all subcarriers. Since we assumed perfect CSI, we can consider the power allocation vector a function of the realization of the fading CNR
p as γ =
T ]T . [γ1T , . . . , γM The ergodic discrete weighted sum rate maximization can then be formulated as
(
∗
f = max p∈P
denote the instantaneous channel-
to-noise ratio (CNR) with mean
a particular user
determines the subcarrier allocation (zero power means (4)). The exclusive subcarrier assignment restriction in
hm ∼ CN (0K , Rhm ) W
(BER), and where
the subcarrier is not allocated) and rate allocation (by
from (2), we have
where
data bits
s.t.
Eγ
X
wm
X
) R(pm,k γm,k )
k∈K (m∈M ) X X Eγ pm,k ≤ P¯
(6)
m∈M k∈K B. Dual Optimization Framework
are not
We begin our development by observing that the ob-
independent but identically distributed (NIID) exponen-
jective function in (6) is separable across the subcarriers,
tial random variables. Throughout the paper, we assume
and is tied together only by the power constraint. In these
that the transmitter has perfect knowledge of 1
γm
for all
Although the results of this paper are applicable to any fading
distribution, we shall prescribe a particular distribution for the fading channels for illustration purposes.
problems, it is useful to approach the problem using duality principles [6]. The dual problem is dened as
g ∗ = min Θ(λ) λ≥0
(7)
where the dual objective is given by
( Θ(λ) = max Eγ p∈P
X
wm
m∈M
X k∈K
Ã
= λP¯ + Eγ
O(L)
) (
X X
complexity. However, if we assume that the dis2
crete rate function (4) is concave , we can reduce the
R (pm,k γm,k )
+λ P¯ − Eγ (
A straightforward computation of (12) would require
complexity of nding the power allocation function by
)!
noticing that (12) is equivalent to
pm,k
∗ wm rlm,k −
k∈K m∈M
X k∈K
max
pk ∈P k
X (8)
m∈M
= λP¯ + Eγ
X k∈K
γm,k
≥ wm rl −
∗ ,. Thus, for all l > l∗ ∀l ∈ L \ lm,k m,k
ληl γm,k
(13)
and for all
∗ , l < lm,k
(13) is equivalent to
)
∗ rl − rlm,k
[wm R (pm,k γm,k ) − λpm,k ] (
∗ ληlm,k
⇔ max ∗
max
l>lm,k
m∈M
¾ max [wm R (pm,k γm,k ) − λpm,k ]
pm,k ≥0
where the second equality follows from the separabil-
∗ ηl − ηlm,k ∗ rl − rlm,k ∗ ηl − ηlm,k
Since the slope
rl∗ − rl λ < m,k ∗ wm γm,k ηlm,k − ηl ∗ rlm,k − rl λ ≤ < min ∗ ∗ l
{iwong,
bevans}@ece.utexas.edu
Abstract In this paper, we propose a resource alloca-
Furthermore, previous research have assumed that
tion algorithm for ergodic weighted-sum rate maximization
algorithms to nd the optimal or near-optimal solution
in downlink OFDMA systems. In contrast to most previous
to the problem is too computationally complex for real-
research that focused on maximizing instantaneous rates using deterministic optimization techniques, we focus on maximizing ergodic rates using stochastic optimization techniques, which allow us to exploit the temporal dimension,
in
addition
to
the
frequency
and
multiuser
dimensions. Furthermore, in contrast to most previous
time implementation. Hence, the main focus of previous research efforts have been on developing heuristic approaches with typical complexities in the order of
O(M K 2 ).
Our approach, on the other hand, is based on
a dual optimization framework, which is less complex
algorithms that used greedy suboptimal heuristics with
(O(M K) per iteration, with less than 10 iterations) and
quadratic complexity, we use a dual optimization approach
achieves relative optimality gaps that are less than
that resulted in a simple subcarrier, rate, and power
(i.e. achieving
allocation algorithm that has complexity
M -user, K -subcarrier
O(M K)
for an
OFDMA system. Surprisingly, our
method is shown to result in duality gaps less than
10−4
in scenarios of practical interest, thereby allowing us to claim practical optimality. We present simulation results for a 3GPP-LTE system employing adaptive modulation.
99.9999%
10−4
of the optimal solution) in
typical scenarios, and thus actually allowing us to claim practical optimality. We focus on the discrete rate case in this paper. We also investigated the continuous rate case in [2]. Note that the dual optimization approach was also studied in [3] [4] [5], but their focus has been on instantaneous and continuous rate optimization.
I. I NTRODUCTION II. S YSTEM M ODEL
Next-generation broadband wireless system standards, e.g. 3GPP-Long Term Evolution (LTE) [1], consider Orthogonal Frequency Division Multiple Access (OFDMA) as the preferred physical layer multiple access scheme, esp. for the downlink. The problem of assigning the subcarriers, rates, and powers to the different users in an OFDMA system has been an area of active research, (see e.g. [2] [3] [4]). In most of the previous work, the formulation and algorithms only consider instantaneous
We consider a single OFDMA base station with
K -subcarriers and M -users indexed by the set K = {1, . . . , k, . . . , K} and M = {1, . . . , m, . . . , M } (typically K À M ) respectively. We assume an average ¯ > 0, bandwidth B , and noise transmit power of P density N0 . The received signal vector for the mth user at the nth OFDM symbol is given as
performance metrics. Thus, the temporal dimension is
ym [n] = Gm [n]Hm [n]xm [n] + wm [n]
(1)
not being exploited when the resource allocation is
ym [n]
and
xm [n]
are the
K -length
performed. Instead of considering only instantaneous
where
data rate, we formulate the problem considering user-
transmitted complex-valued signal vectors;
received and
weighted ergodic sum rate. This allows us to exploit all
diagonal gain allocation matrix with diagonal elements
three degrees of freedom in our system, namely time,
[Gm [n]]kk =
frequency, and multiuser dimensions. At the same time,
noise
we can enforce various notions of fairness through the
circular-symmetric, complex Gaussian (ZMCSCG) noise
user weights.
vector; and
Gm [n]
is the
p 2 I ) with pm,k [n]; wm [n] ∼ CN (0, σw K 2 variance σw = N0 B/K is the white zero-mean, Hm [n] = diag {hm,1 [n], . . . , hm,K [n]} is the
users, and that the resource allocation decisions are made
diagonal channel response matrix, where
hm,k [n] =
Nt X
known to the users through an error-free control channel.
gm,i [n]e
−j2πτi k∆f
.
(2)
i=1 are the complex-valued frequency-domain wireless channel fading random processes, given as the discretetime Fourier transform of the
Nt time-domain multipath τi and subcarrier spacing
taps
gm,i [n]
∆f .
These taps are modeled as stationary and ergodic
with time-delay
2 , σm,i
discrete-time random processes with tap powers
which we assume to be independent across the fading paths
i
m. Since gm,i [n] is stationary hm,k [n]. Hence, the distribution of independent of n through stationarity, and
III. E RGODIC R ATE M AXIMIZATION
IN
OFDMA
A. Problem Formulation The data rate of the
k th
subcarrier for the
mth
user
can be given by the staircase function
η0 ≤ pm,k γm,k < η1 r0 , . . . . R(pm,k γm,k ) = ., . rL−1 , ηL−1 ≤ pm,k γm,k < ηL (4)
and across users
{ηl }l∈L , L = {0, . . . , L − 1},
and ergodic, so is
where
hm [n]
boundaries which dene a particular code-rate and mod-
is
are the SNR
rl
we can replace time averages with ensemble averages
ulation order pair combination that result in
in the problem formulations through ergodicity. In the
per transmission with a predened target bit error rate
subsequent discussion, we shall drop the index
taps
rl ≥ 0, rl+1 > rl , r0 = 0, η0 = 0, and ηL = ∞. Denote by p = [pT1 , · · · , pTK ]T the vector of T powers to be determined, where pk = [p1,k , · · · , pM,k ] .
gm,i ∼
Note that determining the power vector consequently
n
when
the context is clear for notational brevity. We
1
assume
that
the
time
domain
channel
are independent ZMCSCG random variables
2 ) with total power CN (0, σm,i
2 σm
=
PNt
2 i=1 σm,i . Then
(3)
Rhm = WΣm WH is the
e−j2πτi k∆f and
K × Nt DFT matrix with [W]k,i = 2 , . . . , σ2 Σm = diag{σm,1 m,Nt } is an
Nt ×Nt diagonal matrix of the time-domain path powers. Since we also assume that the fading for each user is independent, then the joint distribution of the stacked fading vector for all users
h = [hT1 , . . . , hTM ]T
is
likewise a ZMCSCG random vector with distribution
h ∼ CN (0KM , Rh ) where Rh
is the
diagonal covariance matrix with
KM ×KM
Rhm
block
as the diagonal
block elements. This is the distribution over which we shall take the weighted sum rate function in the problem formulations. We let
2 γm,k = |hm,k |2 /σw
γm = [γm,1 , . . . , γm,k ]T
where
γ¯m,k = subcarrier k
γm,k for a particular users m are independent but that
2 /σ 2 . Note σm w and different
not necessarily identically
distributed (INID) exponential random variables; and for
m
OFDMA can then be written as
and different subcarriers
k
p k ∈ P k ⊂ RM +,
0 P k ≡ {pk ∈ RM + |pm,k pm0 ,k = 0; ∀m 6= m } For notational convenience, we let
K · · · × P K ⊂ RM +
(5)
p ∈ P ≡ P1 ×
denote the space of allowable power
vectors for all subcarriers. Since we assumed perfect CSI, we can consider the power allocation vector a function of the realization of the fading CNR
p as γ =
T ]T . [γ1T , . . . , γM The ergodic discrete weighted sum rate maximization can then be formulated as
(
∗
f = max p∈P
denote the instantaneous channel-
to-noise ratio (CNR) with mean
a particular user
determines the subcarrier allocation (zero power means (4)). The exclusive subcarrier assignment restriction in
hm ∼ CN (0K , Rhm ) W
(BER), and where
the subcarrier is not allocated) and rate allocation (by
from (2), we have
where
data bits
s.t.
Eγ
X
wm
X
) R(pm,k γm,k )
k∈K (m∈M ) X X Eγ pm,k ≤ P¯
(6)
m∈M k∈K B. Dual Optimization Framework
are not
We begin our development by observing that the ob-
independent but identically distributed (NIID) exponen-
jective function in (6) is separable across the subcarriers,
tial random variables. Throughout the paper, we assume
and is tied together only by the power constraint. In these
that the transmitter has perfect knowledge of 1
γm
for all
Although the results of this paper are applicable to any fading
distribution, we shall prescribe a particular distribution for the fading channels for illustration purposes.
problems, it is useful to approach the problem using duality principles [6]. The dual problem is dened as
g ∗ = min Θ(λ) λ≥0
(7)
where the dual objective is given by
( Θ(λ) = max Eγ p∈P
X
wm
m∈M
X k∈K
Ã
= λP¯ + Eγ
O(L)
) (
X X
complexity. However, if we assume that the dis2
crete rate function (4) is concave , we can reduce the
R (pm,k γm,k )
+λ P¯ − Eγ (
A straightforward computation of (12) would require
complexity of nding the power allocation function by
)!
noticing that (12) is equivalent to
pm,k
∗ wm rlm,k −
k∈K m∈M
X k∈K
max
pk ∈P k
X (8)
m∈M
= λP¯ + Eγ
X k∈K
γm,k
≥ wm rl −
∗ ,. Thus, for all l > l∗ ∀l ∈ L \ lm,k m,k
ληl γm,k
(13)
and for all
∗ , l < lm,k
(13) is equivalent to
)
∗ rl − rlm,k
[wm R (pm,k γm,k ) − λpm,k ] (
∗ ληlm,k
⇔ max ∗
max
l>lm,k
m∈M
¾ max [wm R (pm,k γm,k ) − λpm,k ]
pm,k ≥0
where the second equality follows from the separabil-
∗ ηl − ηlm,k ∗ rl − rlm,k ∗ ηl − ηlm,k
Since the slope
rl∗ − rl λ < m,k ∗ wm γm,k ηlm,k − ηl ∗ rlm,k − rl λ ≤ < min ∗ ∗ l
