On the Fundamental Limits of Interweaved Cognitive Radios

Abstract

This paper considers the problem of channel sensing in cognitive radios. The system model considered is a set of N parallel (dis-similar) channels, where each channel at any given time is either available or occupied by a legitimate user. The cognitive radio is permitted to sense channels to determine each of their states as available or occupied. The end goal of this paper is to select the best L channels to sense at any given time. Using a convex relaxation approach, this paper formulates and approximately solves this optimal selection problem. Finally, the solution obtained to the relaxed optimization problem is translated into a practical algorithm. Comment: 7 pages, 3 figures, IEEE Radio and Wireless Symposium, 2010

Full text

Page 1

On the Fundamental Limits of Interweaved
Cognitive Radios
G. Chung, and S. Vishwanath
Wireless Networking and Communication Group
University of Texas at Austin
Austin, TX 78712, USA
Email: {gchung,sriram}@ece.utexas.edu
C. S. Hwang
Communication Lab., SAIT
Samsung Electronics Co. Ltd.
Yongin, Korea
Email: cshwang@samsung.com
Abstract—This paper considers the problem of channel sensing
in cognitive radios. The system model considered is a set of N
parallel (dis-similar) channels, where each channel at any given
time is either available or occupied by a legitimate user. The
cognitive radio is permitted to sense channels to determine each
of their states as available or occupied. The end goal of this
paper is to select the best L channels to sense at any given
time. Using a convex relaxation approach, this paper formulates
and approximately solves this optimal selection problem. Finally,
the solution obtained to the relaxed optimization problem is
translated into a practical algorithm.
1
I. INTRODUCTION
As the number and types of wireless (multimedia) applica-
tions increase, so do the stringent requirements they impose on
the wireless medium. Thus, it is essential that we determine
efficient means of utilizing limited spectral resources avail-
able to us. Currently, bandwidth resources are divided into
frequency bands and allocated to different users exclusively
in order to insure the quality of service (QoS) of multiple
wireless systems, and the FCC’s frequency allocation chart
[1] shows that almost all frequency bands are currently divided
and allocated to different groups for varying purposes. Also,
according to recent surveys [2] and [3], most of this allocated
radio frequency spectrum is vastly under-utilized. Cognitive
radios are emerging as promising solutions to enable better
utilization of spectrum especially in bands that are currently
underutilized [4]. The classical example of a cognitive radio is
one that employs “interweave” cognition [4]. These interweave
cognitive radios are permitted to occupy a channel (frequency
band) only when it is not occupied by a user licensed to
use that band. If the presence of other radios can be sensed
accurately and quickly, then such a policy can help ensure that
cognitive users cause little to no interference to the licensed
radios in the system. A majority of existing literature on
cognitive radios focuses on such interweaved radios. For an
analysis of other classes of cognitive radios, see [5], [6] and
[7]. One of the main issues under study in the interweaved
cognitive radio domain is the so-called ”sensing problem”,
where we desire to determine, as accurately and efficiently
1This work is supported by a grant from Samsung Advanced Institute of
Technology.
as possible, if a given channel is occupied at any given time
[8], [9], and [11]. For example, [8] describes a simple energy
detection scheme for additive white Gaussian noise channel.
In [9], the performance of energy detection schemes in a
multipath environment is analyzed, and in [11], the impact of
additional side information is considered in determining the
performance of cognitive sensing. Overall, channel sensing is
one of the better established fields of research on cognitive
communication. In this paper, our goal is significantly different
from that of channel-sensing literature. Given a fairly accurate
sensing algorithm, we desire to determine which channels
should be sensed when. In addition, we desire to perform a
resource-allocation problem across multiple channels which
may or may not be available to the cognitive radio. Overall,
we ask the question ”Given that there are multiple dissimilar
channels available for you to sense, which channels should
you sense and, if they are available, what rate/power should
you assign to them?”
The dissimilarity between different channels arises from
various factors. The properties of the propagation environment
depend on frequency and thus can be significantly different
from channel to channel. Some channels may suffer from
“extraneous interference” from non-legitimate sources (such
as in the industrial, scientific and medical (ISM) bands) that
reduce the channel quality. Thus, just as any other multiband
radio, the cognitive radio must allocate resources across dif-
ferent bands it uses while simultaneously determining which
ones it is permitted to exploit. Note that, in isolation, the
problem of channel selection for cognitive radios [12], [13] is
well studied. Also, by itself, the resource allocation problem
for muti-band radios is also well-understood [14]. However,
bringing the two together is both important and challenging as
they are tightly coupled in the context of interweaved cognitive
radios. A simple explanation of this strong interdependence
between sensing (channel selection) and resource allocation is
as follows: Let us say that the system is such that “noisy”
(poor) channels are less frequently used by licensed users
than “clear” (good) channels. If the sensing mechanism were
to choose to sense the infrequently-used channels, it will
present the cognitive radio with available channels that are all
“poor” resulting in a low rate. On the flip side, if the resource
allocation mechanism were to assign high rates to the “good”
arXiv:0910.1639v1 [cs.IT] 9 Oct 2009

Page 2

Time (slot)
Legitimate Channel
: Unoccupied Time-Frequency Channel Block
: Occupied Time-Frequency Channel Block
1
2
. . .
1
. . .
32
N
T
2
1

2
N

2
2

Fig. 1. Channel Model
channels, the sensing mechanism may find that they are not
available for use and then again sustain a very poor rate. Thus,
designing channel selection and allocation jointly is essential
for cognitive radios. Note that this paper’s focus is on the
fundamental limits of joint selection and resource allocation
in cognitive networks to provide a benchmark on performance.
Thus, aspects such as sensing error, delay, device and network
non-linearities etc. are not incorporated into the analysis.
The rest of this is organized as follows. The next section
details the system model and notations used in the paper.
In Section III, we find the fundamental limit of the given
system model. In Section IV, we propose an algorithm for
joint channel selection and power allocation, and we conclude
with Section IV.
II. SYSTEM MODEL AND PROBLEM STATEMENT
The channel model is shown in Fig. 1. We consider N
parallel legitimate channels with equal bandwidth. In each time
slot, a channel n,1 ≤ n ≤ N, is occupied by a legitimate user
with probability qn. There are one cognitive transmitter and
one cognitive receiver. The cognitive transmitter is allowed to
transmit over channel n, if it is not occupied by any licensed
user. In legitimate channel n, cognitive radio’s channel is:
Yn= Xn+ Zn
where Znis additive Gaussian noise of variance σ2
this noise variance can be different from channel to channel, as
it represents the fading state of that particular channel. Before
the start of cognitive radio’s transmission using the legitimate
channels, the cognitive transmitter should know whether they
are occupied by the licensed users or not. Thus, at the start of
every time slot, the cognitive transmitter is allowed to sense
a subset of channels, and is allowed to exploit those channels
that are unoccupied; in this paper, we assume that the sensing
is performed perfectly. Also, the cognitive transmitter is not
allowed to transmit using the channel which is not sensed in
order to guarantee the transmission of the licensed users. As N
is assumed to be large, it is impractical to allow the cognitive
radio the ability to sense all of them at the start of every slot.
Instead, we require it to cleverly choose a subset of bands on
which to focus its efforts. The capacity of the cognitive radio
depends on which channels to sense from N parallel channels,
n. Note that
and power allocation among the available parallel channels.
Average total transmission power of cognitive transmitter is
constrained to P.
First, define the In(t) and IE,n(t) to be the indicator
function for selected channel to be sensed and an indicator
function for the unoccupied channel respectively, i.e.,
?
and
?
Denote the time average capacity of the cognitive radio with
the selection of the sensing channel In(t) and power allocation
Pn(t) in one time block as C (In(t),Pn(t)). Then,
C (In(t),Pn(t)) =1
T
n=1t=1
In(t) =
0
1
if channel n is not to be sensed
if channel n is to be sensed
(1)
IE,n(t) =
0
1
if channel n is occupied
if channel n is unoccupied
.
(2)
N
?
T
?
In(t)IE,n(t)
2
log
?
1 +Pn(t)
σ2
n
?
(3)
,
where T is the number of time slot in each time block.
In our model, we assume two constraints on the cognitive
radio:
a. An average power constraint on the cognitive transmitter of
P,
b. The number of channels that can be sensed by the cognitive
radio at any given time is L ≤ N.
Note that if the cognitive radio could sense all channels, L =
N, this problem has a fairly trivial solution. At the start of each
time slot, the cognitive radio would determine all available
channels and waterfill its power over them [10].
If the number of channels that cognitive radio can sense is
less than N, i.e. L < N, the resulting optimization problem
can be stated as follows:
max
Pn(t),In(t)C (In(t),Pn(t))
(4a)
such that
1
T
N
?
N
?
n=1
T
?
t=1
In(t)IE,n(t)Pn(t) ≤ P,
(4b)
n=1
In(t) ≤ L,
(4c)
and
Pn(t) ≥ 0,
In(t) ∈ {0,1},
IE,n(t) ∈ {0,1}.
(4d)
The optimization problem given by (4) determines the
maximum empirical average rate achieved by the cognitive
radio given constraints on the system. Note that it is an integer
programming (IP) due to the constraints in (4d), and multi-
dimensional due to its dependence on time t.
The next section studies the optimization problem given by
(4) in an ergodic policy setting.

Page 3

III. OPTIMAL POWER ALLOCATION AND SELECTION OF
SENSING CHANNEL
As a first step, we assume that our policy is ergodic and
“static” , i.e., that our sensing and power allocation policies are
only functions of the channel statistics and do not evolve with
time. This results in the following (simplified) optimization
problem:
max
Pn,In
N
?
n=1
Inqn
2
log
?
1 +Pn
σ2
n
?
(5a)
such that
N
?
n=1
InqnPn≤ P,
(5b)
N
?
n=1
In≤ L,
(5c)
and
Pn≥ 0,
In∈ {0,1}.
(5d)
Since Pn= 0 where In= 0, constraints (5b) can further be
relaxed to
N
?
n=1
qnPn≤ P.
(5e)
Denoting the optimal selection of channels to be sensed and
power allocation for channel n as I∗
optimum solution for (5) is given by the following theorem:
Theorem 1: The joint channel selection & rate allocation
problem (characterized by the optimization problem in (5) is
maximized when:
nand P∗
nrespectively, the
I∗
n= argmax
In
N
?
?+I∗
n=1
qnIn
2
?
logλ
σ2
n
?+
P∗
n=?λ − σ2
n
n,
where
N
?
N
?
n=1
?λ − σ2
I∗
n
?+I∗
nqn= P
n=1
n= L,
and ?w?+is maximum value of 0 and w.
Proof: Note that (5a) is a concave function over Pn for a
particular choice of In. The following Lagrangian describes
the optimization of (5a) with respect to Pnfor a given In:
L =
?N
n=1
Inqn
2
log
?
1 +Pn
σ2
n
?
−λ(1)??N
n=1qnPn− P
?
+?N
n=1λ(1)
n Pn
(6)
Taking the derivative of (6) and setting it to zero, we get:
2
1

2
2

2
4

2
N
1


2
N

1 1
q I
2 2
q I
4 4
q I
11
NN
qI

N N
q I
2P
4P
1
N P
Fig. 2. power allocation with given In(I1= 1,I2= 1,I3= 0,...,IN= 1)
∂L
∂Pn
=
Inqnloge
2(P∗
n+ σ2
n)− λ(1)qn+ λ(1)
n = 0.
(7)
P∗
n=
?Inloge
2λ(1).
2λ(1)
− σ2
n
?+
=?λ − σ2
n
?+In,
(8)
where λ =
From (5e) we obtain,
log e
N
?
n=1
?λ − σ2
n
?+Inqn= P.
(9)
Fig. 2. provides a graphical representation for the power
allocation strategy in (8). Note that it is similar to the water-
filling solution, with the main difference that the each channel
has different width, qnIn. We refer to the policy in (9) as
modified water-filling throughout this paper. Given that we
understand the structure of the power allocation policy that
optimizes (5a), we now desire to determine I∗
the optimization problem in (5) with respect to In is an IP.
It can be found by an exhaustive search, but computationally
very hard to solve. Moreover, the power allocation strategy in
8, specifically, the water-level λ is tightly coupled with the
choice of In. In the next section, we present an algorithmic
framework that approximates I∗
using low-complexity iterative techniques.
n. Note again that
n(and thus the water-level λ)
IV. JOINT SELECTION AND POWER CONTROL
A typical integer program is non-polynomial in complexity.
Although multiple techniques exist for obtaining approximate
solutions to such a program (such as branch and bound [15],
relaxation), such techniques apply to any integer program and
do not take the structure of the problem into consideration.
Our focus is on developing an algorithm customized to this
problem setting. We perform this in two steps, which we call
“coarse” and “fine” optimization. The coarse optimization step
determines a set of L channels to be utilized by the cognitive
radio. It gives us the lowest possible waterlevel, λmin. The fine

Page 4

optimization step uses λminwhich we obtained from coarse
optimization to further optimize the choice of the L channels.
First, we describe the coarse optimization step:
Coarse Optimization: We iteratively find the channels to
sense along with modified water-filling which incur the lowest
water level. Let λmindenote the lowest water level, and Ic
Pc
which result in λmin. Detailed procedures to find λmin, Ic
and Pc
Step I: Start with L initial channels. We can choose L
channels with the largest qnas initial channels, for example.
nand
nindicate the selection of the channel and power allocation
n,
nis described in the following four steps.
In,0=
?
1
0
if qn is among L largests
otherwise
(10)
S0= {n ∈ [1,N]|In,0= 1}
j = 1
(11)
(12)
Step II: Perform the modified water-filling with In,j−1, j ≥ 1,
such that
N
?
n=1
?λj− σ2
n
?+In,j−1qn= P.
n), and select the largest L
(13)
Step III: Calculate qn(λj − σ2
channels.


In,j=

1
if
qn(λj− σ2
qn(λj− σ2
otherwise
n) > 0 &
n) is among L largests
0
(14)
Sj= {n ∈ [1,N]|In,j= 1}
(15)
Step IV: If Sj = Sj−1, terminate the iteration, and set the
power allocation and channel selection values.
λmin= λn
Ic
Pc
(16)
(17)
(18)
n= In,j
n=?λj− σ2
n
?In,j.
Otherwise, j = j + 1 and repeat from step II.
The coarse optimization is performed for two reasons. One
is that the performance of coarse optimization is very close
to the optimum. This will be shown from the simulation
result in the next section. Here, the optimality of the coarse
optimization in one special case will be stated and proven.
Lemma 1: Define Sc to be the set of the channels which
are selected from coarse optimization;
Sc= {n ∈ [1,N]|Ic
n= 1}.
If the noise variances of all the channels which are not selected
in the coarse optimization are greater than the lowest water
level λmin, i.e.,
σ2
n≥ λmin, ∀n ∈ [1,N],n / ∈ Sc
then the coarse optimization is optimal.
Proof: Define S∗to be the set of channels from optimal
selection.
S∗= {n ∈ [1,N]|I∗
n= 1}.
From definition,
?
max
Pn
n∈S∗
qn
2log
?
1 +Pn
σ2
n
?
≥ max
Pn
?
n∈Sc
qn
2log
?
1 +Pn
σ2
n
(19)
?
.
Let’s assume that there exist at least one legitimate channel
with noise variance higher than the lowest water level which
is included in the optimal channel selection.
∃n ∈ S∗, σ2
n≥ λmin.
Define S?= S∗∪ Sc, and allow the number of channel to
sense to be M = |S∗∪ Sc|, which are strictly larger than L.
Then,
?
max
Pn
n∈S?
qn
2log
?
1 +Pn
σ2
n
?
≥ max
Pn
?
n∈S∗
qn
2log
?
1 +Pn
σ2
n
(20)
?
.
Note that Sc⊆ S?, and σ2
water-filling of M channels in S?will lead to Pn=0 for all
n / ∈ Sc. Thus,
max
Pn
n∈S?
Combine the above result with (20), we obtain
?
n≥ λminfor all n / ∈ Sc. Modified
?
qn
2log
?
1 +Pn
σ2
n
?
= max
Pn
?
n∈Sc
qn
2log
?
1 +Pn
σ2
n
?
.
max
Pn
n∈Sc
qn
2log
?
1 +Pn
σ2
n
?
≥ max
Pn
?
n∈S∗
qn
2log
?
1 +Pn
σ2
n
?
.
(21)
Above result contradict (19), unless Sc is the optimal. This
concludes the proof.
The other reason for performing the coarse optimization
is that it provides the essential information, λmin, which is
necessary for further fine optimization. Upon the following
assumption, fine optimization is optimal.
Conjecture 1: If the noise variance σ2
water level in the coarse optimization (σ2
channel n is not likely to be sensed in the optimal strategy, or
even if it is included it will not increase the capacity much;
Intuition:The following gives the intuition for the above
conjecture. Define S+to be the set of channels with noise
variance greater then or equal to the λminand S−to be the
set of channels with noise variance less then λmin but not
included in Sc;
nis greater than the
n> λmin), then the
S+= {n ∈ [1,N]|σ2
S−= {n ∈ [1,N]|σ2
n> λmin},
n≤ λmin,n / ∈ Sc}.
We have the set of channel Sc which incur the lowest
waterlevel. Lemma(1) shows that average capacity cannot
increase by exchanging elements in S+with elements in

Page 5

Sc. Thus, for elements in S+to be included in S∗, optimal
channel selection, elements in S−should be included also.
By exchanging elements in S−with elements in Sc the
waterlevel rises up. Elements in S+can only be in optimal
selection if exchanging S+with elements in Sc lower the
waterlevel which increased due to the inclusion of channels in
S−effectively. Intuition is that channels in Scare the channels
which can lower the waterlevel effectively already. Thus, effect
of lowering the waterlevel with channels in S+will not affect
much in increasing the average capacity. The validity of this
conjecture is shown from the numerical analysis.
Fine Optimization: From lemma(1), if the number of chan-
nels that is selected to sense from the coarse optimization
is less than L, it is optimal, and no further optimization is
needed. Otherwise, further optimization will be required. From
Conjecture(1), we reconstruct the problem, so that we can
optimize the selection of the channel over the channels with
noise variance smaller than or equal to lambdamin only. we
rearrange the useful channels by indexing from 1 to M, where
M is the number of channels that has noise variance smaller
than λmin;
M = |Sc∪ S−|
σ2
(22)
(23)
n− λmin/le0 ∀n ∈ [1,M].
Then, the optimization problem can be rewritten as follows;
max
λ,InC (λ,In) = max
λ,In
M
?
n=1
qn
2log
?
1 +
?λ − σ2
n
?+In
σ2
n
?
(24)
= max
λ,In
M
?
M
?
M
?
n=1
qnIn
2
log
?
1 +
?λ − σ2
?+
n
?+
σ2
n
?
(25)
= max
λ,In
n=1
qnIn
2
?
logλ
logλ
σ2
n
,
(26)
(a)
= max
λ,In
n=1
qnIn
2
σ2
n
,
(27)
such that
M
?
n=1
?λ − σ2
n
?+Inqn
M
?
(b)
=?λ − σ2
In≤ L,
n
?Inqn= P,
(28)
n=1
(29)
λ ≥ λmin
In∈ {0,1},
(30)
(31)
where (a) and (b) result from constraining λ ≥ λmin. Then
the optimal channel selection and power allocation can be
determined by using the following theorem.
Theorem 2:
?M
In= 0
if λ > σ2
λ =
n=1qnInσ2
?M
n+ P
n=1qnIn
ne1−σ2
n
λ
Proof: Relax the constraint on In, such that the Incan take
the value in the region [0,1]. Construct the objective function
C (λ,f (In)) such that it is concave over the region of In
and λ, and f(0) = 0 and f(1) = 1. Consider the function
f (In) = Ik
n, then f(0) = 0, f(1) = 1, and
C (λ,f (In)) =
M
?
n=1
qnIk
2
n
logλ
σ2
n
.
(32)
Concavity of C (λ,f (In)) can be found as follows;
?
∂λ∂In
∂2λ
?
qnkIk−1
n
∂2C(λ,f(In))
∂I2
∂2C(λ,f(In))
n
∂2C(λ,f(In))
∂In∂λ
∂2C(λ,f(In))
?
(33)
=
qnk(k − 1)Ik−2
n
log
λ
σ2
n
qnkIk−1
j=1qjIk
n
1
λloge
1
λ2loge
1
λloge
−?M
j
?
(34)
.
Since the matrix is symmetric, if the determinant, (1,1) and
(2,2) components of the matrix take the negative values, the
matrix is negative semi-definite. Thus,
k(k − 1) ≤ 0
+ k2loge ≤ 0
(35)
k(k − 1)logλ
σ2
n
(36)
are the condition for C (λ,f (In)) to be a concave function.
We can find k such that the condition can be satisfied;
?
logλmin
k = min
σ2
n
logλmin
σ2
+ loge
n
σ2
n
?
.
(37)
Now that we verified the concavity of the objective function,
we can construct the according Lagrangian multiplier:
?M
−µ0
+?N
Solving the optimization,
L =
n=1
qnIk
2
n
log
?
λ
σ2
i(λ − σ2
n=1µ3,i(Ii− 1) + µ4(λ − λmin).
n
?
??M
n=1qnIk
n=1µ2,iIi−?N
n) − P
?
− µ1
??M
n=1Ik
i− L
?
(38)
∂L
∂In
=
kqnIk−1
+µ2,i− µ3,i
= 0
n
log
λ
n− µ0qnkIk−1
σ2
n
(λ − σ2
n) − µ1kIk−1
n
(39)
∂L
∂λ=
M
?
n=1
qnIk
n
loge
λ
− µ0
M
?
n=1
qnIk
n+ µ4= 0
(40)
µ0
?M
n=1
?
qnIk
n(λ − σ2
n) − P
?
= 0,
(41)
µ1
?M
n=1
?
Ik
n− L
?
= 0,
(42)
µ2,iIn= 0,
(43)