Energy-Efficient Link Adaptation in
Frequency-Selective Channels
∗
Guowang Miao
†,[
, Student Member, IEEE, Nageen Himayat
††
, Member, IEEE, and Geoffrey Ye
Li
†
, Fellow, IEEE
†
School of ECE, Georgia Institute of Technology
††
Communications Technology Lab., Intel Corporation
Abstract—Energy efficiency is becoming increasingly impor-
tant for small form factor mobile devices, as battery technology
has not kept up with the growing requirements stemming
from ubiquitous multimedia applications. This paper addresses
link adaptive transmission for maximizing energy efficiency, as
measured by the “throughput per Joule” metric. In contrast to
the existing water-filling power allocation schemes that maximize
throughput subject to a fixed overall transmit power constraint,
our scheme maximizes energy efficiency by adapting both overall
transmit power and its allocation, according to the channel states
and the circuit power consumed. We demonstrate the existence
of a unique globally optimal link adaptation solution and develop
iterative algorithms to obtain it. We further consider the special
case of flat-fading channels to develop an upper bound on energy
efficiency and to characterize its variation with bandwidth,
channel gain and circuit power. Our results for OFDM systems
demonstrate improved energy savings with energy optimal link
adaptation as well as illustrate the fundamental tradeoff between
energy-efficient and spectrum-efficient transmission.
Index Terms– energy efficiency, link adaptation, frequency
selective channel, energy aware, OFDM
I. INTRODUCTION
The quality of wireless channel varies with time and fre-
quency. Therefore, link adaptation can be used to improve
transmission performance. With link adaptation, modulation
order, coding rate, and transmit power can be selected accord-
ing to channel state information (CSI).
Earlier research on link adaptation focuses on power allo-
cation to improve channel capacity subject to overall power
constraint. Optimal power allocation for frequency-selective
channels has been investigated in [1]. Here the highest data
rate on a band-limited channel is achieved when the total
received signal power at each frequency, consisting of channel
noise and desired signal component, is a constant. The termi-
nology, adaptive modulation, was first used in [2] even though
work on adaptive modulation [3] had been reported before.
In addition to throughput improvement, energy efficiency is
becoming increasingly important for mobile communications
because of the slow progress of battery technology [4] and
growing requirements of anytime and anywhere multimedia
∗
This work was supported by Intel Corp. and the U.S. Army Research
Laboratory under the Collaborative Technology Alliance Program, Coopera-
tive Agreement DAAD19-01-20-0011.
[
Corresponding author. Email: gmiao3@gatech.edu. Address: School of
Electrical and Computer Engineering Georgia Institute of Technology, Atlanta,
Georgia, 30332–0250
applications. With sufficient battery power, link adaptation
can be geared toward peak performance delivery. However,
with limited battery capacity, link adaptation could be adapted
toward energy conservation to minimize battery drain. Energy-
efficient communication also has the desirable benefit of reduc-
ing interference to other co-channel users as well as lessening
environmental impacts, e.g. heat dissipation and electronic pol-
lution. Hence, recent research has focused on energy-efficient
link adaptation techniques [5]–[7]. It is shown in [5] that when
the transmission bandwidth approaches infinity, the minimum
received signal energy per bit for reliable communication over
additive white Gaussian noise (AWGN) channels, approaches
−1.59 dB. For band-limited transmission, the lowest order
modulation should be used [6]. However, the investigation
in [5], [6] does not account for additional circuit power
consumed during transmission. Energy dissipation of both
transmitter circuits and radio-frequency output is investigated
in [8], where the modulation level is adapted to minimize the
energy consumption according to the simulation observations.
In [7], these ideas are extended to a detailed analysis of circuit
and transmit powers for both adaptive multiple quadrature
amplitude modulation (M-QAM) and multiple frequency shift
keying (MFSK) in AWGN channels for short range energy-
efficient communications.
Orthogonal frequency division multiplexing (OFDM) has
emerged as a primary modulation scheme for the next-
generation broadband wireless standards [9], [10]. The power
allocation and bit-loading algorithms for OFDM are summa-
rized in Chapter 3 of [11]. While extensive research has been
conducted to improve throughput [11], [12], limited work has
been done to address the energy-efficient communication for
OFDM systems. In this paper, we address the energy-efficient
link adaptation for frequency-selective fading channels. We
account for both circuit and transmit powers when designing
link adaptation schemes and emphasize energy efficiency
over peak rates or throughput. The proposed link adaptation
balances circuit power consumption and transmission power to
achieve the maximum energy efficiency, which is defined as
the number of bits transmitted per Joule of energy. In contrast
to the existing water-filling power allocation schemes that
maximize throughput subject to overall transmit power con-
straints, our scheme adapts both the overall transmit power and
its allocation according to the states of all subchannels and the
circuit power consumption to maximize the energy efficiency.
We demonstrate the existence of a unique globally optimal
link adaptation solution and provide iterative algorithms to
obtain this optimum. While the usefulness of our technique
is illustrated using frequency selective OFDM as an example
in this paper, the solution developed is applicable to more
general transmission scenarios where transmission occurs over
resources experiencing different channel conditions.
The rest of the paper is organized as follows. In Section III,
we investigate optimal conditions for energy-efficient trans-
mission and develop algorithms to obtain the globally optimal
solution. In Section III-B, we consider a special case when the
channel is with flat fading. We also consider energy-efficient
link adaptation when the user has either data rate requirement
or peak power limit in Section IV. As an example of energy-
efficient link adaptation, we apply the energy-efficient scheme
in OFDM systems and provide simulation results to demon-
strate energy efficiency improvement in Section VI. Finally,
we conclude the paper in Section VII.
II. PROBLEM FORMULATION
In this section, we formulate the problem of energy-efficient
link adaptation.
Assume that K subchannels are used for transmission, each
with a different channel gain. An example of this scenario
is OFDM transmission over frequency-selective channels. As-
sume block fading [13], [14], that is, the channel state remains
constant during each data frame and is independent from one
to another. Denote the data rate on Subchannel i as r
i
and the
data rate vector on all subchannels as
R = [r
1
, r
2
, · · · , r
K
]
T
, (1)
where []
T
is the transpose of a vector. The data rate vector,
R, depends on the channel state, coding, and power allocation.
Correspondingly, the overall data rate is
R =
K
X
i=1
r
i
. (2)
For a given channel state, the transmit power on each sub-
channel is determined by the requirement of reliable data
transmission. If we denote W as the subchannel bandwidth,
N
o
the power spectral density, g
i
the power gain, and P
T
i
the
allocated transmit power on Subchannel i, the channel output
signal-to-noise ratio (SNR) will be
η
i
=
P
T
i
g
i
N
o
W
(3)
and the achievable data transmission rate r
i
is determined by
[15]
r
i
= W log(1 +
η
i
Γ
), (4)
where Γ is the SNR gap that defines the gap between
the channel capacity and a practical coding and modulation
scheme. The SNR gap depends on the coding and modulation
scheme used and on the target probability of error. For a coded
quadrature amplitude modulation (QAM) system, the gap is
given by [15]
Γ = 9.8 + γ
m
− γ
c
(dB), (5)
where γ
m
is the system design margin and γ
c
is the coding
gain. For Shannon capacity [16], Γ = 0 dB. Denote the overall
transmit power as P
T
(R) and
P
T
(R) =
P
K
i=1
P
T
i
ζ
=
K
X
i=1
(e
r
i
W
− 1)
N
o
W Γ
g
i
ζ
, (6)
where ζ ∈ [0, 1] is the power amplifier efficiency and depends
on the design and implementation of the transmitter. P
T
(R) is
strictly convex and monotonically increasing in R. In fact, the
developed theory and approaches can be used for any P
T
(R)
that is strictly convex and monotonically increasing in R with
P
T
(0) = 0, where 0 = [0 , 0, · · · , 0]
T
.
In addition to transmit power, mobile devices also incur
additional circuit power during transmissions which is rela-
tively independent of the transmission rate [8], [17]. While
the transmit power models all the power used for reliable data
transmission, we let the circuit power represents the average
energy consumption of device electronics, such as mixers,
filters, and digital-to-analog converters, and this portion of
energy consumption excludes that of the power amplifier and is
independent of the transmission state. If we denote the circuit
power as P
C
, the overall power consumption given a data rate
vector will be
P (R) = P
C
+ P
T
(R). (7)
For energy-efficient communications, it is desirable to max-
imize the amount of data sent with a given amount of energy.
Hence, given any amount of energy 4e consumed in a
duration, 4t, i.e. 4e = 4t(P
C
+ P
T
(R)), the mobile wants
to send a maximum amount of data by choosing the data rate
vector to maximize
R 4 t
4e
, (8)
which is equivalent to maximizing
U(R) =
R
4e/ 4 t
=
R
P
C
+ P
T
(R)
. (9)
U(R) is called energy efficiency. The unit of the energy
efficiency is bits per Joule, which has been frequently used in
literature for energy-efficient communications [5], [6], [18]–
[20]. The optimal energy-efficient link adaptation achieves
maximum energy efficiency, i.e.
R
∗
= arg max
R
U(R) = arg max
R
R
P
C
+ P
T
(R)
. (10)
Note that if we fix the overall transmit power, the objective
of Equation (10) is equivalent to maximizing the overall
throughput and the existing water-filling power allocation
approach [1] gives the solution. However, besides adapting the
power distributions on all subchannels, the overall transmit
power can also be adapted according to the states of all
subchannels to maximize the energy efficiency. Hence, the
solution to Equation (10) is in general different from existing
power allocation schemes that maximize throughput with
power constraints.
2
III. PRINCIPLES OF ENERGY-EFFICIENT LINK
ADAPTATION
In the following, we demonstrate that a unique globally
optimal data rate vector always exists and give the necessary
and sufficient conditions for a data rate vector to be globally
optimal.
A. Conditions of Optimality
The concept of quasiconcavity will be used in our discussion
and is defined as [21].
Definition 1. A function f, which maps from a convex set
of real n-dimensional vectors, D, to a real number, is called
strictly quasiconcave if for any x
1
, x
2
∈ D and x
1
6= x
2
,
f(λx
1
+ (1 − λ)x
2
) > min{f(x
1
), f(x
2
)}, (11)
for any 0 < λ < 1.
Any strictly monotonic function is quasiconcave. Besides,
any strictly concave function is also strictly quasiconcave but
the reverse is not generally true. An example is the Gaussian
function, which is strictly quasiconcave but not concave.
It is proved in Appendix I that U(R) has the following
properties.
Lemma 1. If P
T
(R) is strictly convex in R, U(R) is strictly
quasiconcave. Furthermore, U(R) is either strictly decreasing
or first strictly increasing and then strictly decreasing in any
r
i
of R, i.e. the local maximum of U(R) for each r
i
exists at
either 0 or a positive finite value.
For strictly quasiconcave functions, if a local maximum
exists, it is also globally optimal [21]. Hence, a unique
globally optimal transmission rate vector always exists and
its characteristics are summarized in Theorem 1 according to
the proofs in Appendix I.
Theorem 1. If P
T
(R) is strictly convex, there exists a
unique globally optimal transmission data rate vector R
∗
=
[r
∗
1
, r
∗
2
, · · · , r
∗
K
]
T
for (10), where r
∗
i
is given by
(i) when
P
C
+P
T
(R
(0)
i
)
R
(0)
i
≥
∂P
T
(R)
∂r
i
¯
¯
¯
R=R
(0)
i
,
∂U(R)
∂r
i
¯
¯
¯
R=R
∗
=
0, i.e.
1
∂P
T
(R
∗
)
∂r
∗
i
=
R
∗
P
C
+P
T
(R
∗
)
= U(R
∗
);
(ii) when
P
C
+P
T
(R
(0)
i
)
R
(0)
i
<
∂P
T
(R)
∂r
i
¯
¯
¯
R=R
(0)
i
, r
∗
i
= 0,
where R
(0)
i
= [r
∗
1
, r
∗
2
, · · · , r
∗
i−1
, 0, r
∗
i+1
, · · · , r
∗
K
] and R
(0)
i
=
P
j6=i
r
∗
j
, i.e. the overall data rate on all other subchannels
except i.
Theorem 1 has clear physical insights. P
C
+P
T
(R
(0)
i
) is the
power consumption of both circuit and all other subchannels
when Subchannel i is not used.
P
C
+P
T
(R
(0)
i
)
R
(0)
i
is the per-bit
energy consumption when Subchannel i is not used and the
overall per-bit energy consumption needs to be minimized
for energy-efficient communications.
∂P
T
(R)
∂r
i
¯
¯
¯
R=R
(0)
i
is the
per-bit energy consumption transmitting infinitely small data
rate on Subchannel i conditioned on the optimal status of all
other subchannels. Hence, Subchannel i should not transmit
anything when
P
C
+P
T
(R
(0)
i
)
R
(0)
i
<
∂P
T
(R)
∂r
i
¯
¯
¯
R=R
(0)
i
. Otherwise,
there should be a tradeoff between the desired data rate
on Subchannel i and the incurred power consumption. The
tradeoff closely depends on the power consumption of both
circuits and transmission on all other subchannels and can be
found through the unique zero derivative of U (R) with respect
to r
i
.
To further understand Theorem 1, we consider an example
when each subchannel achieves the Shannon capacity and the
transmit power on each subchannel is given in (6) with Γ = 0
dB and ζ = 1. The overall transmit power is
P
T
(R) =
K
X
k=1
(e
r
k
W
− 1)
N
o
W
g
k
. (12)
According to Condition (i) of Theorem 1, when r
k
> 0, we
have
1
∂P
T
(R)
∂r
k
=
1
e
r
k
W
N
o
g
k
= U(R
∗
). (13)
Hence, the transmit power on Subchannel k is
P
T
n
= (e
r
k
W
− 1)
N
o
W
g
k
=
W
U(R
∗
)
−
N
o
W
g
k
, (14)
which is a water-filling to level
W
U(R
∗
)
. Since the water level
is determined by the optimal energy efficiency, we refer to
our scheme as dynamic energy-efficient water-filling. Note that
while the absolute value of power allocation is determined by
the maximum energy efficiency U (R
∗
), which relies on both
the circuit power and channel state, the relative differences of
power allocations on different subchannels depend only on the
channel gains on those subchannels.
B. A Special Case: When the Channel is Flat Fading
To facilitate the understanding of the fundamental depen-
dence of energy efficiency on the channel gain, circuit power,
and bandwidth, we consider a special case that the channel is
experiencing flat fading in this section. Hence, all subchannels
are with the same channel gain and the same link adaptation
is applied on all subchannels. The overall data rate is
R = Kr. (15)
According to Theorem 1, the optimal transmission data rate
follows immediately and is summarized by Theorem 2, where
the upper bound is proved in Appendix II.
Theorem 2. If P
T
(R) is monotonically increasing and strictly
convex in R, there exists a unique globally optimal transmis-
sion data rate to maximize energy efficiency and is given by
R
∗
=
P
C
+ P
T
(R
∗
)
P
0
T
(R
∗
)
, (16)
where P
0
T
(·) is the first order derivative of function P
T
(·).
Besides, energy efficiency is upper bounded by
1
P
0
T
(0)
.
When Shannon capacity is achieved in AWGN channels,
the upper bound is
g
N
o
.
In the following, we investigate some basic properties
of energy-efficient link adaptation. Propositions 1, 2, and 3
3
summarize the impact of channel gain, circuit power, and
the number of subchannels on the optimal energy-efficient
transmission, and are proved in Appendix III.
Proposition 1. Both the data rate and energy efficiency
increase with channel gain.
Proposition 2. The data rate increases with circuit power,
while the energy efficiency decreases with it. With zero circuit
power, the highest energy efficiency,
1
P
0
T
(0)
, is obtained by
transmitting with infinite small data rate.
From Proposition 2, when circuit power dominates power
consumption, which is usually true with short-range com-
munication, the highest data rate should be used to finish
transmission as soon as possible, which has been commonly
assumed by most MAC layer energy-efficient optimization
schemes as describe in the introduction of this paper. However,
when the circuit power is negligible, which is usually true
with long-range communication like satellite communications,
the lowest data rate should be used, which coincides with the
results in [6] and [22].
Proposition 3. The data rate on each subchannel decreases
with increasing number of subchannels while the energy effi-
ciency increases with it. With infinite number of subchannels,
the highest energy efficiency,
1
P
0
T
(0)
, is obtained by transmitting
with infinite small data rate.
Propositions 1, 2, and 3 discover three ways to improve
energy efficiency: increasing channel power gain, reducing
circuit power, and allocating more subchannels. The energy-
efficiency upper bound is achieved by transmitting with infinite
small data rate when either circuit power is zero or infinite
number of subchannels is assigned.
IV. CONSTRAINED ENERGY-EFFICIENT LINK ADAPTATION
In this section, we study energy-efficient link adaptation
when user has either a data rate requirement or a peak power
limit.
With a data rate requirement Γ, the energy-efficient link
adaptation is given by
b
R
∗
= arg max
R
R
P
C
+ P
T
(R)
, (17a)
subject to
R ≥ Γ. (17b)
If the optimal data rate vector without constraint in (10)
satisfies R
∗
≥ Γ, it is also the solution to Problem (17), i.e.
b
R
∗
= R
∗
. Otherwise, Problem (17) is equivalent to
b
R
∗
= arg max
Γ
Γ
P
C
+ P
T
(R)
= arg min
R
P
T
(R), (18a)
subject to
R = Γ. (18b)
Since P
T
(R) is strictly convex, a unique globally optimal
b
R
∗
exists. Denote
f
k
(r
k
) =
∂P
T
(R)
∂r
k
(19)
and its inverse function to be f
−1
k
(). Then
b
R
∗
can be easily
obtained via the Lagrangian technique [23] and is
br
∗
k
= max
©
f
−1
k
(λ), 0
ª
(20)
for k = 1, · · · , K, where λ is determined by
K
X
k=1
br
∗
k
= Γ. (21)
When the channel capacity is achieved on each subchannel,
the corresponding optimal power allocation is a water-filling
allocation, which achieves the sum channel capacity Γ.
Similarly, with a maximum transmit power constraint, the
problem is to find
e
R
∗
= arg max
R
R
P
C
+ P
T
(R)
, (22a)
subject to
P
T
(R) ≤ P
m
. (22b)
If the optimal data rate vector without constraint in (10)
satisfies P
T
(R
∗
) ≤ P
m
, it is also the solution to Problem (22),
i.e.
e
R
∗
= R
∗
. Otherwise, via the the Lagrangian technique
again, we have the unique optimal solution as follows
er
∗
k
= max
©
f
−1
k
(λ), 0
ª
, k = 1, · · · , K, (23)
where λ is determined by
P
T
(
e
R
∗
) = P
m
. (24)
When channel capacity is achieved on each subchannel, the
power allocation is the classical water-filling where the water
level is determined by P
m
[1].
V. ALGORITHM DESIGN
Theorem 1 provides the necessary and sufficient conditions
for a rate vector to be the unique and globally optimum
one. However, it is usually difficult to directly solve the joint
nonlinear equations according to Theorem 1 to obtain the
optimal vector R
∗
. Therefore, we develop iterative methods
to search the optimal R for maximizing U (R). The global
optimality of the proposed methods is guaranteed by the strict
quasiconcavity of U (R). In the following, we describe our
low-complexity iterative algorithms.
A. Gradient Assisted Binary Search
When there is only one subchannel, Lemma 1 shows that
function U(r) has a unique r
∗
such that for any r < r
∗
,
dU(r)
dr
> 0, and for any r > r
∗
,
dU(r)
dr
< 0. Hence, we have
the following lemma to seek two points r
1
and r
2
such that
r
1
≤ r
∗
≤ r
2
.
Proposition 4.
Let the initial setting r
[0]
> 0 and set α > 1.
For any i ≥ 0, let
r
[i+1]
=
(
r
[i]
α
dU( r)
dr
¯
¯
¯
r
[0]
< 0
αr
[i]
otherwise
. (25)
4
Repeat (25) until r
[I]
such that
dU( r)
dr
¯
¯
¯
r
[I]
has a different sign
from
dU( r)
dr
¯
¯
¯
r
[0]
. Then r
∗
must be between r
[I]
and r
[I−1]
.
To locate r
∗
between r
1
and r
2
, let br =
r
1
+r
2
2
. If
dU( r)
dr
¯
¯
¯
br
=
0, r
∗
is found. If
dU( r)
dr
¯
¯
¯
br
< 0, r
1
< r
∗
< br and replace r
2
with br; otherwise, replace r
1
with br. This leads to the gradient
assisted binary search (GABS) for maximizing U (r), which
is summarized in Table I.
Algorithm GABS(r
o
)
(∗ algorithm for single-subchannel transmission. ∗)
Input: initial guess: r
o
> 0
Output: optimal transmission rate: r
∗
1. r
1
= r
o
, h
1
←
dU( r)
dr
¯
¯
¯
r
1
, initialize α > 1 (e.g.10)
2. if h
1
< 0
(∗ seek r
1
and r
2
such that r
1
< r
∗
< r
2
∗)
3. then r
2
←r
1
, r
1
←
r
1
α
, and h
1
←
dU( r)
dr
¯
¯
¯
r
1
4. while h
1
< 0
5. do r
2
←r
1
, r
1
←
r
1
α
, and h
1
←
dU( r)
dr
¯
¯
¯
r
1
6. else r
2
←r
1
∗ α and h
2
←
dU( r)
dr
¯
¯
¯
r
2
7. while h
2
> 0
8. do r
1
←r
2
, r
2
←r
2
∗ α, and h
2
←
dU( r)
dr
¯
¯
¯
r
2
9. while no convergence
(∗ seek r
∗
between r
1
and r
2
∗)
10. do br←
r
2
+r
1
2
;
b
h ←
dU( r)
dr
¯
¯
¯
br
11. if
b
h > 0
12. then r
1
= br;
13. else r
2
= br
14. return br
TABLE I: Gradient assisted binary search
B. Binary Search Assisted Ascent
To find the optimal data rate vector for the multiple sub-
channel case, we design a gradient ascent method to produce
a maximizing sequence R
[i]
, n = 0, 1, · · · , and
R
[i+1]
=
h
R
[i]
+ µ∇U (R
[i]
)
i
+
, (26)
where [R]
+
sets the negative part of the vector R to be zero,
µ > 0 is the search step size, and ∇U(R
[i]
) is the gradient
at iteration i. With sufficiently small step size, U (R
[i+1]
) will
be always bigger than U (R
[i]
) except when ∇U(R
[i]
) = 0
that indicates the optimality of R
[i]
[23]. However, small
step size leads to slow convergence. Besides, each element of
the gradient depends on the corresponding subchannel power
gain, which potentially differs from each other by orders of
magnitude. Hence, a line search of the optimal step size needs
to cover a large range to assure global convergence on all
subchannels, which is computationally expensive. Therefore,
at each R
[i]
, an efficient algorithm is needed to find the optimal
step size. Denote
f
i
(µ) = U(
h
R
[i]
+ µ∇U (R
[i]
)
i
+
). (27)
Similar to the proof of Lemma 1, it is easy to show that g
i
(µ)
is also strictly quasiconcave in µ and has a unique globally
maximum µ
∗
such that for any µ < µ
∗
,
df
i
(µ)
dµ
> 0, and for
any µ > µ
∗
,
df
i
(µ)
dµ
< 0. Let ∇U(R
[i]
) = [bg
1
, bg
2
, · · · , bg
K
].
Replace
dU( r)
dr
in GABS to be
df
i
(µ)
dµ
= [∇U(R
[i+1]
)]
T
e
G[i], (28)
where
e
G[i] =
d
[
R
[i]
+µ∇U(R
[i]
)
]
+
dµ
= [eg
1
, eg
2
, · · · , eg
K
], in which
eg
k
= bg
k
if the kth component of R
[i]
+µ∇U(R
[i]
) is positive
and eg
k
= 0 otherwise. Then GABS can be used for quick
location of the optimal step size. This leads to the binary
search assisted ascent (BSAA) algorithm in Table II.
Algorithm BSAA(R
o
)
(∗ algorithm for multi-subchannel transmission. ∗)
Input: initial guess: R
o
(default transmission rate can be used)
Output: optimal transmission rate vector: R
∗
1. R = R
o
,
2. while no convergence
3. do use GABS to find the optimal step size µ
∗
;
4. R = [R + µ
∗
∇U(R)]
+
5. return R
TABLE II: Binary search assisted ascent
C. The Rate of Convergence
While the global convergence of both GABS and BSAA
is guaranteed by the strict quasiconcavity of U(R) [24], we
further study the convergence rate in this section.
Theorem 3 characterizes the convergence of GABS and is
proved in Appendix IV.
Theorem 3. GABS converges to the globally optimal trans-
mission data rate r
∗
. A rate r, which satisfies |r − r
∗
| ≤ ²,
can be found within at most M iterations, where M is the
minimum integer such that M ≥ log
2
(
(α−1)r
∗
²
− 1).
It is difficult to theoretically analyze the global convergence
rate of BSAA because of the nonconcavity of U(R). Instead,
we run numerical simulations and observe the convergence.
Figure 1(a) illustrates the improvement of energy efficiency
with iterations. Here we assume the channel gain of each
subchannel has Rayleigh distribution with a unit average. The
circuit power is 5. The noise power on each subchannel is
0.01. The transmit power is given by Equation (6) with Γ = 0
dB. The energy efficiency is normalized by the optimal value
and the curves are the ensemble averages of 5000 channel
instances. Figure 1(b) shows the corresponding probability
distribution functions of the numbers of iterations necessary
for convergence. In both figures, we vary the number of
subchannels to verify its impact on the convergence rate. We
can see that BSAA converges very fast to the global optimum,
even with 1024 subchannels.
VI. SIMULATION RESULTS FOR OFDM
The proposed energy-efficient link adaptation is general
and can be applied to different kinds of OFDM, MIMO, and
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