On the Fundamental Limits of InterweavedCognitive Radios

G. Chung, and S. VishwanathWireless Networking and Communication Group

University of Texas at AustinAustin, TX 78712, USA

Email: {gchung,sriram}@ece.utexas.edu

C. S. HwangCommunication Lab., SAIT

Samsung Electronics Co. Ltd.Yongin, Korea

Email: cshwang@samsung.com

AbstractThis paper considers the problem of channel sensingin cognitive radios. The system model considered is a set of Nparallel (dis-similar) channels, where each channel at any giventime is either available or occupied by a legitimate user. Thecognitive radio is permitted to sense channels to determine eachof their states as available or occupied. The end goal of thispaper is to select the best L channels to sense at any giventime. Using a convex relaxation approach, this paper formulatesand approximately solves this optimal selection problem. Finally,the solution obtained to the relaxed optimization problem istranslated 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 onthe wireless medium. Thus, it is essential that we determineefficient means of utilizing limited spectral resources avail-able to us. Currently, bandwidth resources are divided intofrequency bands and allocated to different users exclusivelyin order to insure the quality of service (QoS) of multiplewireless systems, and the FCCs frequency allocation chart[1] shows that almost all frequency bands are currently dividedand allocated to different groups for varying purposes. Also,according to recent surveys [2] and [3], most of this allocatedradio frequency spectrum is vastly under-utilized. Cognitiveradios are emerging as promising solutions to enable betterutilization of spectrum especially in bands that are currentlyunderutilized [4]. The classical example of a cognitive radio isone that employs interweave cognition [4]. These interweavecognitive radios are permitted to occupy a channel (frequencyband) only when it is not occupied by a user licensed touse that band. If the presence of other radios can be sensedaccurately and quickly, then such a policy can help ensure thatcognitive users cause little to no interference to the licensedradios in the system. A majority of existing literature oncognitive radios focuses on such interweaved radios. For ananalysis of other classes of cognitive radios, see [5], [6] and[7]. One of the main issues under study in the interweavedcognitive 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 ofTechnology.

as possible, if a given channel is occupied at any given time[8], [9], and [11]. For example, [8] describes a simple energydetection scheme for additive white Gaussian noise channel.In [9], the performance of energy detection schemes in amultipath environment is analyzed, and in [11], the impact ofadditional side information is considered in determining theperformance of cognitive sensing. Overall, channel sensing isone of the better established fields of research on cognitivecommunication. In this paper, our goal is significantly differentfrom that of channel-sensing literature. Given a fairly accuratesensing algorithm, we desire to determine which channelsshould be sensed when. In addition, we desire to perform aresource-allocation problem across multiple channels whichmay or may not be available to the cognitive radio. Overall,we ask the question Given that there are multiple dissimilarchannels available for you to sense, which channels shouldyou sense and, if they are available, what rate/power shouldyou assign to them?

The dissimilarity between different channels arises fromvarious factors. The properties of the propagation environmentdepend on frequency and thus can be significantly differentfrom channel to channel. Some channels may suffer fromextraneous interference from non-legitimate sources (suchas in the industrial, scientific and medical (ISM) bands) thatreduce the channel quality. Thus, just as any other multibandradio, the cognitive radio must allocate resources across dif-ferent bands it uses while simultaneously determining whichones it is permitted to exploit. Note that, in isolation, theproblem of channel selection for cognitive radios [12], [13] iswell studied. Also, by itself, the resource allocation problemfor muti-band radios is also well-understood [14]. However,bringing the two together is both important and challenging asthey are tightly coupled in the context of interweaved cognitiveradios. A simple explanation of this strong interdependencebetween sensing (channel selection) and resource allocation isas follows: Let us say that the system is such that noisy(poor) channels are less frequently used by licensed usersthan clear (good) channels. If the sensing mechanism wereto choose to sense the infrequently-used channels, it willpresent the cognitive radio with available channels that are allpoor resulting in a low rate. On the flip side, if the resourceallocation mechanism were to assign high rates to the good

arX

iv:0

910.

1639

v1 [

cs.IT

] 9 O

ct 20

09

Time (slot)

Le

gitim

ate

Ch

an

ne

l

: Unoccupied Time-Frequency Channel Block

: Occupied Time-Frequency Channel Block

12

. . .

1 . . .32

N

T

21

2N

22

Fig. 1. Channel Model

channels, the sensing mechanism may find that they are notavailable for use and then again sustain a very poor rate. Thus,designing channel selection and allocation jointly is essentialfor cognitive radios. Note that this papers focus is on thefundamental limits of joint selection and resource allocationin cognitive networks to provide a benchmark on performance.Thus, aspects such as sensing error, delay, device and networknon-linearities etc. are not incorporated into the analysis.

The rest of this is organized as follows. The next sectiondetails the system model and notations used in the paper.In Section III, we find the fundamental limit of the givensystem model. In Section IV, we propose an algorithm forjoint channel selection and power allocation, and we concludewith Section IV.

II. SYSTEM MODEL AND PROBLEM STATEMENT

The channel model is shown in Fig. 1. We consider Nparallel legitimate channels with equal bandwidth. In each timeslot, a channel n, 1 n N , is occupied by a legitimate userwith probability qn. There are one cognitive transmitter andone cognitive receiver. The cognitive transmitter is allowed totransmit over channel n, if it is not occupied by any licenseduser. In legitimate channel n, cognitive radios channel is:

Yn = Xn + Zn

where Zn is additive Gaussian noise of variance 2n. Note thatthis noise variance can be different from channel to channel, asit represents the fading state of that particular channel. Beforethe start of cognitive radios transmission using the legitimatechannels, the cognitive transmitter should know whether theyare occupied by the licensed users or not. Thus, at the start ofevery time slot, the cognitive transmitter is allowed to sensea subset of channels, and is allowed to exploit those channelsthat are unoccupied; in this paper, we assume that the sensingis performed perfectly. Also, the cognitive transmitter is notallowed to transmit using the channel which is not sensed inorder to guarantee the transmission of the licensed users. As Nis assumed to be large, it is impractical to allow the cognitiveradio the ability to sense all of them at the start of every slot.Instead, we require it to cleverly choose a subset of bands onwhich to focus its efforts. The capacity of the cognitive radiodepends on which channels to sense from N parallel channels,

and power allocation among the available parallel channels.Average total transmission power of cognitive transmitter isconstrained to P .

First, define the In(t) and IE,n(t) to be the indicatorfunction for selected channel to be sensed and an indicatorfunction for the unoccupied channel respectively, i.e.,

In(t) ={

0 if channel n is not to be sensed1 if channel n is to be sensed

(1)and

IE,n(t) ={

0 if channel n is occupied1 if channel n is unoccupied .

(2)Denote the time average capacity of the cognitive radio withthe selection of the sensing channel In(t) and power allocationPn(t) in one time block as C (In(t), Pn(t)). Then,

C (In(t), Pn(t)) = 1T

Nn=1

Tt=1

In(t)IE,n(t)2

log(1 +

Pn(t)2n

),

(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 ofP ,b. The number of channels that can be sensed by the cognitiveradio 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 eachtime slot, the cognitive radio would determine all availablechannels and waterfill its power over them [10].

If the number of channels that cognitive radio can sense isless than N , i.e. L < N , the resulting optimization problemcan be stated as follows:

maxPn(t),In(t)

C (In(t), Pn(t)) (4a)

such that

1T

Nn=1

Tt=1

In(t)IE,n(t)Pn(t) P, (4b)Nn=1

In(t) L, (4c)

andPn(t) 0,In(t) {0, 1},IE,n(t) {0, 1}.

(4d)

The optimization problem given by (4) determines themaximum empirical average rate achieved by the cognitiveradio given constraints on the system. Note that it is an integerprogramming (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.

III. OPTIMAL POWER ALLOCATION AND SELECTION OFSENSING CHANNEL

As a first step, we assume that our policy is ergodic andstatic , i.e., that our sensing and power allocation policies areonly functions of the channel statistics and do not evolve withtime. This results in the following (simplified) optimizationproblem:

maxPn,In

Nn=1

Inqn2

log(1 +

Pn2n

)(5a)

such thatNn=1

InqnPn P, (5b)

Nn=1

In L, (5c)

andPn 0,In {0, 1}. (5d)

Since Pn = 0 where In = 0, constraints (5b) can further berelaxed to

Nn=1

qnPn P. (5e)

Denoting the optimal selection of channels to be sensed andpower allocation for channel n as In and P

n respectively, the

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) ismaximized when:

In = argmaxIn

Nn=1

qnIn2

log

2n

+P n =

2n

+In,

whereNn=1

2n

+Inqn = P

Nn=1

In = L,

and dwe+ is maximum value of 0 and w.Proof: Note that (5a) is a concave function over Pn for aparticular choice of In. The following Lagrangian describesthe optimization of (5a) with respect to Pn for a given In:

L =Nn=1

Inqn2 log

(1 + Pn2n

)(1)

(Nn=1 qnPn P

)+Nn=1

(1)n Pn

(6)

Taking the derivative of (6) and setting it to zero, we get:

21

22

24

21N

2N

1 1q I 2 2q I 4 4q I 1 1N Nq I N Nq I

2P 4P 1NP

Fig. 2. power allocation with given In (I1 = 1, I2 = 1, I3 = 0, ..., IN = 1)

LPn

=Inqn log e2 (P n + 2n)

(1)qn + (1)n = 0. (7)

P n =In log e2(1)

2n+

= 2n

+In, (8)

where = log e2(1)

.From (5e) we obtain,

Nn=1

2n

+Inqn = P. (9)

Fig. 2. provides a graphical representation for the powerallocation strategy in (8). Note that it is similar to the water-filling solution, with the main difference that the each channelhas different width, qnIn. We refer to the policy in (9) asmodified water-filling throughout this paper. Given that weunderstand the structure of the power allocation policy thatoptimizes (5a), we now desire to determine In. Note again thatthe optimization problem in (5) with respect to In is an IP.It can be found by an exhaustive search, but computationallyvery hard to solve. Moreover, the power allocation strategy in8, specifically, the water-level is tightly coupled with thechoice of In. In the next section, we present an algorithmicframework that approximates In (and thus the water-level )using low-complexity iterative techniques.

IV. JOINT SELECTION AND POWER CONTROL

A typical integer program is non-polynomial in complexity.Although multiple techniques exist for obtaining approximatesolutions to such a program (such as branch and bound [15],relaxation), such techniques apply to any integer program anddo not take the structure of the problem into consideration.Our focus is on developing an algorithm customized to thisproblem setting. We perform this in two steps, which we callcoarse and fine optimization. The coarse optimization stepdetermines a set of L channels to be utilized by the cognitiveradio. It gives us the lowest possible waterlevel, min. The fine

optimization step uses min which we obtained from coarseoptimization to further optimize the choice of the L channels.First, we describe the coarse optimization step:

Coarse Optimization: We iteratively find the channels tosense along with modified water-filling which incur the lowestwater level. Let min denote the lowest water level, and Icn andP cn indicate the selection of the channel and power allocationwhich result in min. Detailed procedures to find min, Icn,and P cn is described in the following four steps.

Step I: Start with L initial channels. We can choose Lchannels with the largest qn as initial channels, for example.

In,0 ={

1 if qn is among L largests0 otherwise (10)

S0 = {n [1, N ]|In,0 = 1} (11)j = 1 (12)

Step II: Perform the modified water-filling with In,j1, j 1,such that

Nn=1

j 2n

+In,j1qn = P. (13)

Step III: Calculate qn(j 2n), and select the largest Lchannels.

In,j =

1 if qn(j 2n) > 0 &

qn(j 2n) is among L largests0 otherwise

(14)

Sj = {n [1, N ]|In,j = 1} (15)Step IV: If Sj = Sj1, terminate the iteration, and set thepower allocation and channel selection values.

min = n (16)Icn = In,j (17)

P cn =(j 2n

)In,j . (18)

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 closeto the optimum. This will be shown from the simulationresult in the next section. Here, the optimality of the coarseoptimization in one special case will be stated and proven.

Lemma 1: Define Sc to be the set of the channels whichare selected from coarse optimization;

Sc = {n [1, N ]|Icn = 1}.If the noise variances of all the channels which are not selectedin the coarse optimization are greater than the lowest waterlevel min, i.e.,

2n min, n [1, N ], n / Sc

then the coarse optimization is optimal.Proof: Define S to be the set of channels from optimalselection.

S = {n [1, N ]|In = 1}.From definition,

maxPn

nS

qn2log(1 +

Pn2n

) max

Pn

nSc

qn2log(1 +

Pn2n

).

(19)

Lets assume that there exist at least one legitimate channelwith noise variance higher than the lowest water level whichis included in the optimal channel selection.

n S, 2n min.Define S = S Sc, and allow the number of channel tosense to be M = |S Sc|, which are strictly larger than L.Then,

maxPn

nS

qn2log(1 +

Pn2n

) max

Pn

nS

qn2log(1 +

Pn2n

).

(20)

Note that Sc S, and 2n min for all n / Sc. Modifiedwater-filling of M channels in S will lead to Pn=0 for alln / Sc. Thus,

maxPn

nS

qn2log(1 +

Pn2n

)= max

Pn

nSc

qn2log(1 +

Pn2n

).

Combine the above result with (20), we obtain

maxPn

nSc

qn2log(1 +

Pn2n

) max

Pn

nS

qn2log(1 +

Pn2n

).

(21)

Above result contradict (19), unless Sc is the optimal. Thisconcludes the proof.

The other reason for performing the coarse optimizationis that it provides the essential information, min, which isnecessary for further fine optimization. Upon the followingassumption, fine optimization is optimal.

Conjecture 1: If the noise variance 2n is greater than thewater level in the coarse optimization (2n > min), then thechannel n is not likely to be sensed in the optimal strategy, oreven if it is included it will not increase the capacity much;Intuition:The following gives the intuition for the aboveconjecture. Define S+ to be the set of channels with noisevariance greater then or equal to the min and S to be theset of channels with noise variance less then min but notincluded in Sc;

S+ = {n [1, N ]|2n > min},S = {n [1, N ]|2n min, n / Sc}.

We have the set of channel Sc which incur the lowestwaterlevel. Lemma(1) shows that average capacity cannotincrease by exchanging elements in S+ with elements in

Sc. Thus, for elements in S+ to be included in S, optimalchannel selection, elements in S should be included also.By exchanging elements in S with elements in Sc thewaterlevel rises up. Elements in S+ can only be in optimalselection if exchanging S+ with elements in Sc lower thewaterlevel which increased due to the inclusion of channels inS effectively. Intuition is that channels in Sc are the channelswhich can lower the waterlevel effectively already. Thus, effectof lowering the waterlevel with channels in S+ will not affectmuch in increasing the average capacity. The validity of thisconjecture 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 optimizationis less than L, it is optimal, and no further optimization isneeded. Otherwise, further optimization will be required. FromConjecture(1), we reconstruct the problem, so that we canoptimize the selection of the channel over the channels withnoise variance smaller than or equal to lambdamin only. werearrange the useful channels by indexing from 1 to M , whereM is the number of channels that has noise variance smallerthan min;

M = |Sc S| (22)2n min/le0 n [1,M ]. (23)

Then, the optimization problem can be rewritten as follows;

max,InC (, In) = max

,In

Mn=1

qn2log

(1 +

2n

+In

2n

)(24)

= max,In

Mn=1

qnIn2

log

(1 +

2n

+2n

)(25)

= max,In

Mn=1

qnIn2

log

2n

+, (26)

(a)= max

,In

Mn=1

qnIn2

log

2n, (27)

such thatMn=1

2n

+Inqn

(b)=( 2n

)Inqn = P, (28)

Mn=1

In L, (29)

min (30)In {0, 1}, (31)

where (a) and (b) result from constraining min. Thenthe optimal channel selection and power allocation can bedetermined by using the following theorem.

Theorem 2:

=Mn=1 qnIn

2n + PM

n=1 qnIn

In = 0 if > 2ne1

2n

Proof: Relax the constraint on In, such that the In can takethe value in the region [0, 1]. Construct the objective functionC (, f (In)) such that it is concave over the region of Inand , and f(0) = 0 and f(1) = 1. Consider the functionf (In) = Ikn , then f(0) = 0, f(1) = 1, and

C (, f (In)) =Mn=1

qnIkn

2log

2n. (32)

Concavity of C (, f (In)) can be found as follows;[2C(,f(In))

I2n

2C(,f(In))In

2C(,f(In))In

2C(,f(In))2

](33)

=

[qnk (k 1) Ik2n log 2n qnkI

k1n

1 log e

qnkIk1n

1 log e

Mj=1 qjI

kj

12 log e

].

(34)

Since the matrix is symmetric, if the determinant, (1, 1) and(2, 2) components of the matrix take the negative values, thematrix is negative semi-definite. Thus,

k(k 1) 0 (35)k(k 1) log

2n+ k2 log e 0 (36)

are the condition for C (, f (In)) to be a concave function.We can find k such that the condition can be satisfied;

k = min2n

(log min2n

log min2n + log e

). (37)

Now that we verified the concavity of the objective function,we can construct the according Lagrangian multiplier:

L =

Mn=1

qnIkn

2 log(2n

)0

(Mn=1 qnI

ki ( 2n) P

) 1

(Mn=1 I

ki L

)+Nn=1 2,iIi

Nn=1 3,i(Ii 1) + 4( min).

(38)Solving the optimization,

LIn

=kqnI

k1n log

2n 0qnkIk1n ( 2n) 1kIk1n

+2,i 3,i= 0 (39)

L

=Mn=1

qnIkn

log e 0

Mn=1

qnIkn + 4 = 0 (40)

0

(Mn=1

qnIkn( 2n) P

)= 0, (41)

1

(Mn=1

Ikn L)= 0, (42)

2,iIn = 0, (43)

3,i(In 1) = 0, (44)

4( min) = 0, (45)where 0, 1, 2,i, 3,i, and 4 are non-negative values. Fromthe condition (40),

=Mn=1 qnI

kn log e

0Mn=1 qnI

kn 4

. (46)

Note thatMn=1 qnI

kn log e,

Mn=1 qnI

kn , and 4 are the non-

negative value. Thus, 0 should be positive number in orderfor to be positive. Then, from the condition (41),

Mn=1

qnIkn( 2n) = P (47)

Thus,

=Mn=1 qnI

kn

2n + PM

n=1 qnIkn

, (48)

and from rearranging the equation (46),

0 =log e

+4M

n=1 qnIkn

. (49)

From the condition (39), we find that

In =

(qn log 2n 0qn(

2n) 1

)k

3,i 2,i

1

1k

(50)

If In is not either 0 or 1, from conditions (43) and (44),2,i and 3,i becomes 0, which will result in making Ininfinite number. Thus, In takes either 0,1 value, which givesthe desirable solution, such that the optimization of the re-laxed condition coincides with the condition for the originalproblem, and

=Mn=1 qnIn

2n + PM

n=1 qnIn, (51)

We set 4 to be zero, then from (49) and (50), we obtain

In =

(qn log

2ne1

2n

1)k

3,i 2,i

11k

. (52)

As a result, In can be 1 only if

> 2ne1

2n . (53)

This conclude the proof. With the Theorem(2), we can designiterative algorithm to find the optimal selection of channels tosensed iteratively.

Step I: Set the channels from coarse optimization to be theinitial channels.

In,0 ={

1 if n Sc0 otherwise (54)

S0 = {n [1, N ]|In,0 = 1} (55)j = 1 (56)

Step II: Calculate the waterlevel j from (51)

j =Mn=1 qnIn,j1

2n + PM

n=1 qnIn,j1. (57)

Step III: Calculate j 2ne12nj , and select the largest L

channels.

In,j =

{1 if j 2ne1

2nj is among L largests

0 otherwise(58)

Sj = {n [1, N ]|In,j = 1} (59)Step IV: If Sj = Sj1, terminate the iteration, and set thepower allocation and channel selection values. f , Ifn , andP fn are the waterlevel, channel selection, and power allocationfrom fine optimization, then

f = n (60)

Ifn = In,j (61)

P fn =(j 2n

)In,j . (62)

Otherwise, j = j + 1 and repeat from step II. Followingfrom theorem(??) this algorithm gives the optimal selectionof the channels to be sensed and power allocation, with theassumption that the channels with noise variance greater thanmin do not affect the optimization.

V. NUMERICAL ANALYSIS

In this section, we present numerical example of capacitiesfor coarse and fine optimization along with optimal solution. Inthis example, dissimilarity between channels is implementedby adapting the multi-path fading, which will incur frequencyselective channel. Also, occupation of the legitimate channelis modeled by having qn to be uniform in [0, 1] and identi-cally distributed. Sixteen legitimate channels are considered,N = 16, and cognitive radio is allowed to select and senseeight channels from all the legitimate channels, L = 8. Fig.3. compares the capacities of sub-optimal algorithms withthe optimal one, where the performance of optimal channelselection is obtained from the exhaustive search. The graphshows that performance of the fine optimization meet with thatof optimal one. Thus, it can be stated that the Conjecture (1) infine optimization is valid. Coarse optimization also performsoptimally in the low SNR region. In the low SNR region, itis likely that 2n min, n [1, N ], n / Sc, because thereare not much power to waterfill. From the Lemma (1), coarseoptimization is optimal in such case. It is also worthwhile tonote that coarse optimization perform as well as the optimalone.

0 5 10 15 20 25 300

2

4

6

8

10

12

SNR(dB)

Capa

city

Exhaustive SerachCourse OptimizationFine Optimization

Fig. 3. Perforamance Analysis

VI. CONCLUSION

In this paper, fundamental limits of interweaved cognitiveradio has been verified. In the case that there are large numberof legitimate channels and only limited number of them can besensed, the capacity has been analyzed. However, it requiresexhaustive search over the combination of all the legitimatechannels, which is not practical in terms of complexity. Thus,two steps of sub-optimal solutions have been developed.Coarse optimization is developed, and verified to be optimalin the low SNR cases, and further optimization is performedto ensure the performance in the high SNR region also.

ACKNOWLEDGMENT

The authors would like to thank Illsoo Sohn for usefuldiscussions and comments.

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IntroductionSystem Model and Problem StatementOptimal Power Allocation and Selection of Sensing ChannelJoint Selection and Power ControlNumerical AnalysisConclusionReferences