Physical Communication 2 (2009) 39

ab

ls

in

, LCognitive radioSpectrum holeCapacityAchievable rate

Cognitive radio is a new concept of reusing a licensed spectrum in an unlicensed manner.The motivation for cognitive radio is various measurements of spectrum utilization, thatgenerally show unused resources in frequency, time and space. These spectrum holescould be exploited by cognitive radios. Some studies suggest that the spectrum is extremelyunderutilized, and that these spectrum holes could provide ten times the capacity of allexistingwireless devices together. The spectrumcould be reused either during timeperiodswhere the primary system is not active, or in geographical positions where the primarysystem is not operating. In this paper, we deal primarily with the concept of geographicalreuse, in a frequency-planned primary network.We perform an analysis of the potential forcommunication in a geographical spectrum hole, and in particular the achievable sum-ratefor a secondary network, to some order of magnitude.Simulation results show that a substantial sum-rate could be achieved if the secondary

users communicate over small distances. For a small number of secondary links, the sum-rate increases linearlywith the number of links. However, the spectrumhole gets saturatedquite fast, due to interference caused by the secondary users. A spectrum hole may looklarge, but it disappears as soon as someone starts using it.

2009 Elsevier B.V. All rights reserved.

1. Introduction

A spectrum is a scarce resource, and operators havemade huge financial investments to buy licensed spec-trums. The licensed spectrum is intended for specific com-munication technologies, and no one but the spectrumowner is allowed to use it. Cognitive radio is a new conceptof reusing a licensed spectrum in an unlicensed manner

I Invited Paper.II The research leading to these results has received funding fromthe European Communitys Seventh Framework Programme (FP7/2007-2013) under grant agreement no. 216076. This work was also supportedin part by the Swedish Research Council (VR), the Swedish Foundation forStrategic Research (SSF), and the CENIIT foundation. E. Larsson is a RoyalSwedishAcademyof Sciences (KVA) Research Fellow supported by a grantfrom the Knut and Alice Wallenberg Foundation. Corresponding author. Tel.: +46 0 13 281350.E-mail addresses: axell@isy.liu.se (E. Axell), erik.larsson@isy.liu.se

(E.G. Larsson), danyo@isy.liu.se (D. Danev).

[13]. The motivation for cognitive radio is various mea-surements of spectrumutilization, that generally showun-used resources in frequency, time and space [4,5]. Thesespectrum holes could be exploited by cognitive radios.The spectrum could be reused either during time periodswhere the primary system is not active, or in geographicalpositions where the primary system is not operating. Thispaper deals primarily with geographical, or spatial, reuse.The introduction of cognitive radios, sometimes called

secondary users, in an existing primary system will createinterference and thus a quality degradation of the primarysystem. In order to reduce the impact on the primarysystem, cognitive radios have to sense the spectrum anddetect whether there are primary users in the vicinitythat are currently using the spectrum. The cognitive radiosneed to be positioned sufficiently far away from theprimary users and transmit at very low power levels.This has been analyzed in, e.g. [6] for a single cell, andin [7] for a frequency-planned network. It is unavoidable

1874-4907/$ see front matter 2009 Elsevier B.V. All rights reserved.doi:10.1016/j.phycom.2009.03.002Contents lists avail

Physical Com

journal homepage: www.e

Full length article

Capacity considerations for uncoordgeographical spectrum holesI,II

Erik Axell , Erik G. Larsson, Danyo DanevDivision of Communication Systems, Department of Electrical Engineering (ISY)

a r t i c l e i n f o

Keywords:

a b s t r a c tle at ScienceDirect

munication

evier.com/locate/phycom

ated communication in

inkping University, S-581 83 Linkping, Sweden

4 E. Axell et al. / Physical Communication 2 (2009) 39that there has to be some sort of compromise for theprimary system to allow secondary users. In [6,7] thiscompromise is a reduction of the primary cell radius.The cell radius is decreased by a small amount andit is required that the same signal-to-interference-plus-noise ratio (SINR) is experienced by the primary usersas without any secondary users. Hence, the aggregatedtransmit power of the secondary usersmust be constrainedto keep the interference level low. For example, if the cellradius is decreased by 5%, then the transmitter power forthe cognitive radios is constrained such that the primaryusers experience the same SINR as they did before, withoutany secondary users.A similar one-cell model for spatial frequency reuse

has also been analyzed in [8], together with more generaldefinitions of a spectrumhole and somemetrics to quantifythe performance of a spectrum sensing algorithm. Thefocus of this work is more on the uncertainty of detectionversus the area that could actually be exploited.The concept of frequency reuse can be seen as a bucket

filled with rocks [8]. Although the bucket is filled thereis still plenty of room for sand. However this metaphorrequires a large-scale primary system, and a small-scalesecondary one. This problem has been stated in [9], andespecially the problem of coexistence of a secondarysystem with primary systems of different scales. The mainissue is the coexistence of a secondary system in thepresence of a small-scale primary network. The Part 74wireless microphone users that are a concern to the IEEE802.22 WRAN is given as an example. Relating to thebucket metaphor, if the bucket is filled with sand wecannot fit any more rocks or sand.In much of the literature, the main difficulty has been

perceived to be the detection of the primary users. Even ifthat could be solved, we need to know that the spectrumholes can really be exploited and provide some usefuldata rate. Some studies suggest that the spectrum isextremely underutilized, and that these spectrum holescould provide ten times the capacity of all existingwireless devices together [5]. The aim of this paper is toanalyze the potential for communication in a geographicalspectrum hole, and in particular the achievable sum-ratefor a secondary network, to some order of magnitude. InSection 2 we describe the system model, and Section 3shows some numerical results. Section 4 proposes someimprovements on the individual secondary links by usingmultiple devices. Section 5 concludes the work.

2. Model

We consider the downlink in a hexagonal frequency-planned network, shown in Fig. 1. We include the mainprimary base station and the first tier of co-channelinterferers. The cell radius is r , and the distance to the firsttier of interfering base stations is D = 3nr [10], where1/n is the frequency reuse factor of the primary system (nis the number of frequency groups). The positions of thebase stations are denoted by the vectors B0 = 0 and Bm =(D cos((m 1)pi3 ),D sin((m 1)pi3 )),m = 1, 2, . . . , 6.Following [6,7] we assume that the cognitive users

are permitted to operate only if they are located at adistance at least d from the nearest primary base station.Furthermore, we assume that N cognitive transmitters arespread out uniformly at random in the allowed region,i.e. in the area between the circles of radii D d andd respectively, as shown in Fig. 1. The positions of thecognitive transmitters are denoted by the vectors Ti =(xT ,i, yT ,i), i = 1, 2, . . . ,N . For each cognitive transmitterthere is an associated cognitive receiver at a distance d0from the transmitter and at an angle i. The angle i isuniformly distributed in the interval [pi, pi], but such thatthe receiver is also in the permitted area. The positionsof the receivers are denoted by the vectors Ri = (xT ,i +d0 cos(i), yT ,i + d0 sin(i)), i = 1, 2, . . . ,N .We consider a log-distance path loss model. Thus, we

define the channel gain function at distance x,

(x) =(xx0

)10/10, N(0, )

where is the path loss exponent, x0 is a normalizationconstant and is the standard deviation of the lognor-mal fading in dB. The distance, x, is in general a randomvariable since we consider random locations for the sec-ondary users. Hence, the received interference is random.However, the distance between a secondary transmit-ter/receiver pair d0 is fixed, and the only randomness ofthe received signal strength is the lognormal fading. Thebase stations transmit omnidirectionally with power P ,and each cognitive user transmits omnidirectionally withpower Pc . The transmit power is defined as the power re-ceived at the normalization distance x0.We assume that the secondary users are uncoordinated

and thus transmit simultaneously, using the same channel.Hence, the secondary users will interfere with each other.The interference power experienced by the ith cognitivereceiver, and caused by other secondary users can bewritten as Pc

k6=i (|Ri Tk|). The secondary users also

receive interfering signals from the primary base stations.The interference power from the primary system, for theith secondary user, can be written as P

6n=0 (|Ri Bn|).

Now the signal-to-interference-plus-noise ratio (SINR) forthe ith cognitive receiver becomes

SINRi = Pc(d0) 2 + Pc

k6=i(|Ri Tk|)+ P

6n=0

(|Ri Bn|)

= (d0) 2

Pc+k6=i(|Ri Tk|)+ PPc

6n=0

(|Ri Bn|),

where 2 is the receiver noise floor.The primary system may be either interference limited

or noise limited, or somewhere in between. Following [7],we can quantify the operating point of the primary systemin terms of the expected interference-to-noise ratio at thecell border without secondary users:

, E

P6n=1

(|(r cos(), r sin()) Bn|) 2

= P 2 ,

E. Axell et al. / Physical Communication 2 (2009) 39 5

eand which is uniformly distributed over [pi, pi]. Inorder to make sure that the cognitive users do not causetoo much harmful interference to the primary users, thetransmit power Pc must be constrained. We will constrainthe aggregate cognitive radio transmit power, such thatNPc = P , for some > 0. The choice of will bediscussed later. Thus, we can rewrite the SINR and obtainthe following:

SINRi = (d0)N+k6=i(|Ri Tk|)+ N

6n=0

(|Ri Bn|).

The achievable rate for the ith secondary link ismodeledas

Ci = log2 (1+ SINRi)

= log2

1+ (d0)N+k6=i(|Ri Tk|)+ N

6n=0

(|Ri Bn|)

(bits/s/Hz).

Hence, the sum-rate offered by the network of allsecondary users, C , is

C =Ni=1Ci

A = (D d)2 d2D2

= 1 2 dD.

This is the fraction of the total system area in whichcognitive operation is permitted. Note that the allowedtransmit power Pc depends on the permitted area A and onthe primary interference-to-noise operating point . Therelationship between these parameters was investigatedin [7]. See also [6] for the special case of only asingle primary base station. The primary cell radiuswas decreased, and the secondary transmit power wasconstrained such that the SINR for the primary users wasnot decreased compared to the primary system withoutany secondary users. The allowed aggregated secondarytransmit power NPc = P was computed given a relativearea A and a primary interference-to-noise operating point . We will use the values of , and A obtained from thisanalysis for our simulations.

3. Simulation results

In this section we will show some numerical resultsfrom Monte-Carlo simulations. For each number of sec-ondary users N , we generated 5000 realizations of the sys-tem model. The achievable sum-rate was then calculatedas the mean of the sum-rate over all realizations. For eachrealization we placed N cognitive transmitters uniformlyFig. 1. Model for th

where

, E

[6n=1

(|(r cos(), r sin()) Bn|)],

and the expectation is taken over the lognormal fadingcognitive network.

=Ni=1log2

1+ (d0)N+k6=i(|Ri Tk|)+ N

6n=0

(|Ri Bn|)

.We define the relative area of cognitive operation

6 E. Axell et al. / Physical Communication 2 (2009) 39Table 1Parameter values used in the simulations for n = 7, 21 and = 10 dB,obtained from Fig. 3 in [7].

A (%) 1 25 50 (n = 21) (dB) 0 3 12 (n = 7) (dB) 0 5 20

at random in the allowed area between the circles of radiid and D d respectively. To obtain a uniform distributionover the circular area we created the transmitter positionsTi = (xT ,i, yT ,i), i = 1, 2, . . . ,N , in polar coordinates. Theangle was uniformly distributed over [pi, pi] and the ra-dius, R, was obtained by

R =((D d)2 d2)X + d2,

whereX wasuniformly distributed over [0, 1]. The receiverpositions were then created as

Ri = (xT ,i + d0 cos(i), yT ,i + d0 sin(i)) i = 1, 2, . . . ,N,where all i were uniformly distributed over [pi, pi]. Tomake sure that all users were inside the allowed region wesimply redrew the angle i whenever a receiver positionhappened to be outside of the allowed region.We used = 10 dB throughout all simulations.

This corresponds to a noise limited primary system. Weargue that a practical system where we eventually couldmake use of this kind of geographical spectrum reusewould typically be noise limited. It could, for example, be atelevision network with a very sparse frequency reuse andprimary transmitters located far away from each other. Inaddition, simulations have shown that the value of onlyhas a small impact on the results. The values of and Athat were used in the simulations are shown in Table 1,and obtained from Fig. 3 in [7] for n = 7, 21 frequencyreuses and = 10 dB. These values were obtainedassuming that the primary cell radius is decreased by5%, and the primary users experience the same SINR atthe 90%-percentile of the distribution, as without anysecondary users. Also, in accordance with [7], we use thepath loss and shadow fading parameters = 4 and = 6dB throughout the whole paper. We use frequency reusen = 21, except where otherwise stated.

3.1. Achievable sum-rate

Fig. 2 shows the total system throughput C for A =50%, 25%, 1% and = 10 dB. When increasing N ,we observe an increase to some congestion limit. Abovethis congestion limit, adding more users only causes athroughput degradation due to increased interference.Thus, for a given d, the system throughput is maximizedfor a certain number of users N . Intuitively we wouldexpect the congestion level to be lower when the allowedregion is smaller. The operation region will be saturatedfor a smaller number of users since the area is smaller.This intuition is confirmed by Fig. 2: the number of usersmaximizing the throughput is higher when the allowedregion is larger. Note also that the total throughput ishigher for a 25% cognitive area than for 50% or 1%.Hence, the throughput is neither increasing nor decreasingin d. Rather there is some d that maximizes the total

10

20

30

40

50

60

70

80

sum

ra

te [b

its/s/

Hz]

0

90

50 100 150 200 250Number of secondary links

0 300

1 bit/s/Hz/link

A=50%, =12dBA=25%, =3dBA=1%, =0dB

Fig. 2. Total achievable sum-rate for the secondary system for d0 = 0.1r , = 4, = 6 dB, n = 21.

throughput. The interpretation of this is that the allowedtransmit power Pc is high and the expected interferencefrom the primary system is low for a small cognitive region,but the interference from the cognitive users increases fastas the number of users N increases. On the other hand,when the allowed region is large, the allowed transmitpower Pc is low and the expected interference from theprimary system is higher. The optimum seems to besomewhere in between these two extremes. These resultsare for frequency reuse n = 21, but the same behavior isseen also for other frequency reuse factors.As a reference, the solid line shown in Fig. 2 is a straight

line with sloping one, and corresponds to the case whereeach secondary user gets 1 bit/s/Hz. We consider ratesabove this line as acceptable whereas rates below theline are less acceptable. Arguably links with less spectralefficiency than 1 bit/s/Hz are not very useful. Note that fora large permitted area (A = 50%) many secondary userscan coexist and achieve a quite high sum-rate, but the rateper user is not acceptable.Fig. 3 shows the total achievable rate for different

transmitter-receiver distances d0, both for n = 7 andn = 21 frequency reuse. Due to the larger operatingarea for n = 21 than n = 7 frequency reuse, thesecondary users are allowed touse ahigher transmit powerwithout causing too much interference for the primaryusers. Also, since the operating area is larger, the distancebetween interfering cognitive users is larger on average.As expected, the performance is better for n = 21 thanfor n = 7 frequency reuse, both in terms of the totalachievable rate and in terms of the number of users thatcan be allowed before the congestion limit is hit. Wealso observe a large improvement from decreasing thecommunication distance. If we compare the maximumtotal achievable rates, we note that a distance decreaseby a factor 10 yields a throughput increase by a factor10 for n = 7 and a factor 20 for n = 21. Thus theorder of magnitude of the maximum sum-rate seems tostand in inverse proportion to the communication distancebetween the secondary users. It is also worth noting thatthe maximum throughput is attained for a larger number

E. Axell et al. / Physical Communication 2 (2009) 39 7

n=21n=7

50

100

150

200

250

300

350

400

450

sum

ra

te [b

its/s/

Hz]

0

500

50 100 150 200 250Number of secondary links

0 300

1 bit/s/Hz/link

Fig. 3. Comparison of the achievable sum-rates for d0 = 0.1r and d0 =0.01r , for A = 1%, = 4, = 6 dB.

of users when the distance is smaller. Hence, both the totalthroughput and the number of users can be larger for asmaller communication distance.

3.2. Geographic density

We have seen that the total achievable rate attains amaximum for a certain number of secondary users. If thereare more users they will get too close to each other andthe interference they generate to each other will increase.The question is then how the users should be distributed toachieve a maximum total system throughput. Is the userdistribution dense or sparse at the maximum sum-rateoperating point?A reasonable assumption is that the geographic area

filled up by each secondary link communicating over adistance d0 is equal to a circle of radius d0. Then Nsecondary links use an area Npid20. The total area ofsecondary operation is the area between the circles of radiiD d and d respectively, i.e. pi((D d)2 d2) = piAD2.We denote by Nmax the number of users for which themaximum total achievable rate is attained. We define theeffective area, AE , as the ratio of the area used by Nmaxsecondary users and the total area of secondary operation,i.e.

AE = Nmaxpid20

piAD2= Nmaxd

20

3nr2A.

Note that this ratio might actually be greater than one,since the secondary links could overlap. Analyzing thesimulation results shown in Fig. 2, we obtain the value ofNmax in various cases. These values and the effective areasare then shown in Table 2. For all sizes of the allowedsecondary operation region we see that the effective areais between 5% and 20% (similar numbers have also beenobserved for n = 7). The conclusion is that the users shouldbe quite sparsely distributed to obtain the maximumsystem throughput. It is also worth noting that for largeoperating regions, the rate per link for N = Nmax usersis non-acceptable (in the sense defined in Section 3.1)although the sum-rate is maximized. In this case the usersTable 2Effective area for n = 21, d0 = 0.1 r.A (%) 1 25 50Nmax 12 86 220AE (%) 19 5 7

Fig. 4. Point-to-point communication from A to B, either directly or bymultihop with time-division multiplexing.

have to be even more sparsely distributed in order to get asatisfactory rate per link.

4. Point-to-point improvement

The simulations in Section 3 show that the achievablesum-rate is strongly dependent on the transmissiondistance d0. A small distance yields a larger receivedsignal strength for the secondary users and thus a largerachievable rate. This also leads us to another interestingquestion. Suppose that we want to communicate betweena point A and another point B separated by a distance ,and using power Ptot. We neglect shadow fading, i.e. thechannel gain at a distance x is simply (x) = ( xx0 ) . Theachievable rate would then be

C = log2(1+ Ptot (/x0)

J

), (1)

where J is the received noise plus interference power fromthe primary system.Assume further that we can alternatively use in total

M + 1 devices (M links) spread out uniformly on thestraight line between A and B, and transmit in a multihopfashion. Then the distance between two neighboringdevices is /M . We assume that the devices share thechannel by time-division multiplexing as shown in Fig. 4,i.e. we let the sub-nodes transmit one at a time, starting atA, to the next sub-node until the message reaches B. Sinceonly one device transmits at a time, each device is allowedto use power Ptot. The SINR on each link is in this case

SINR =Ptot

(Mx0

)J

.

The receivednoise plus interference power, J , is assumed tobe equal for all devices, since the inter-node distances are

8 E. Axell et al. / Physical Communication 2 (2009) 39

M=1M=2M=10

50 100 150 200 250Number of secondary links

0 300

50

100

150

200

sum

ra

te [b

its/s/

Hz]

0

250

1 bit/s/Hz/link

Fig. 5. Comparison of the achievable sum-rate using time-divisionmultiplexed repeater devices between the cognitive receiver andtransmitter, for d0 = 0.1r , = 4, = 6 dB, n = 21.

small relative to the primary cell radius r . The achievablerate from A to B is

C = 1Mlog2

1+ Ptot(

Mx0

)J

= 1Mlog2

(1+ Ptot(/x0)

JM)(bits/s/Hz). (2)

This is also in accordance with (1), for M = 1. Note that,in this case, we are not interested in the sum-rate of thenetwork consisting of the M links, but in the achievablerate from point A to point B. The information received bythe intermediate devices is not useful to them, since theyonly act as repeaters. The achievable rate for each of theMlinkswill be identical, by symmetry.Wehave to useM linksto communicate from A to B, hence the division by M . Weobserve from (2) that the achievable rate goes to zero asMgoes to infinity, but the optimal strategy actually dependson the SINR.Fig. 5 shows some simulation results of the time-

division multiplexing strategy. For simplicity, we approxi-mate the interference at all sub-nodes by the interferencein between the transmitter and the receiver, i.e. for all sub-nodes associated with link iwe use the interference expe-rienced at the position (Ti + Ri)/2. The simulations showthat for a small number of secondary users, it makes no bigdifference what strategy is used. The transmission can bemade directly between the transmitter and receiver, or wemay use a few intermediate devices. For a large number ofsecondary users we can achieve some improvement by us-ing intermediate nodes. For example, by simply using oneextra device, the rate can be doubled.More sophisticated strategies are also possible. For

example out ofM linkswe could let every other, every thirdetc., be active in each time slot. Assume that we spread theM links uniformly over T time slots. The transmit powerfor each link would then be shared between the M/Tlinks in each time slot, i.e. the transmit power is PtotM/T . Theachievable rate for the kth active link is then

Ck = 1T log2

1+ PtotM/T(

Mx0

)J + Ik

[bits/s/Hz],where Ik is the interference caused by all other sub-nodestransmitting in the same time slot. Due to symmetry, weonly need to consider one time slot. The interference Ikis also dependent on M and T . For example if T = 2,the interference I will contain one term that is identicalto the received signal strength plus other, smaller, terms.This means that the SINR will be smaller than 0 dB. ForT > 2 however, much greater SINRs can be achieved. Theachievable rate from A to B is the mean of the achievablerates of the active sub-links:

C = 1M/T

Mk:k1 (mod T )

1Tlog2

1+ PtotM/T(

Mx0

)J + Ik

= 1M

M/T1m=0

log2

1+ PtotM/T(

Mx0

)J + ImT+1

(bits/s/Hz). (3)We will give an example of this for M = 9 and T = 3.During the first time slot, the first, fourth and seventhdevices transmit, and we are interested in calculating I1, I4and I7. The first active link, where the first device transmitsto the second, will experience interference from the fourthand the seventh devices, which are on distance 2M and5M respectively from the receiver. Hence, the interferencepower experienced by the receiver of the first link (thesecond device) is:

I1 = PtotM/T(2Mx0

)+ PtotM/T

(5Mx0

).

Similarly

I4 = PtotM/T(4Mx0

)+ PtotM/T

(2Mx0

)and

I7 = PtotM/T(7Mx0

)+ PtotM/T

(4Mx0

).

Inserting this in (3) yields the achievable rate from A toB for this sub-system. The point-to-point achievable rateof this scheme (M = 9, T = 3) is shown in Fig. 6. It isalso compared to the direct transmission (M = 1), andthe time-multiplexed scheme with 9 transmitting devices(M = 9, T = 9). We note that for high SINR, thebest strategy is actually to transmit directly, without usingany intermediate devices. It is also clear that for someSINRs, themixed scheme yields the largest achievable rate.Especially, in accordance with Fig. 5, using intermediaterepeaters can provide an order of magnitude increase ofthe achievable rate for low SINR. In general, the gain thatcan be obtained by a repeater strategy depends much onthe Ptot/J operating point in Fig. 6. The location of thisoperating point depends on d0, Pc and N , among others.

E. Axell et al. / Physical Communication 2 (2009) 39 9

M=1M=9, T=9M=9, T=3

1

2

3

4

5

Achi

evab

le ra

te [b

its/s/

Hz/lin

k]

0

6

10 5 0 5 10Ptot/J [dB]

15 15

Fig. 6. Comparison of the point-to-point achievable rate using eitherdirect transmission, 9 time division multiplexed repeater devices, or 9repeater devices of which 3 transmit simultaneously.

For the example in Fig. 2, the interesting region is arguablyaround 1 bit/s/Hz/link (cf. the reference line in Fig. 2).This paper will not derive the general expression for theachievable rate or go into further detail regarding thisstrategy, but we note that it offers one possible way ofimproving the results in Fig. 5.

5. Conclusions

In this paper we have analyzed the achievable rateof a potential spectrum hole in a frequency plannedenvironment, using spatial frequency reuse. Simulationresults show that a substantial sum-rate could be achievedprovided that the secondary users communicate oversufficiently small distances. For a small number ofsecondary links, the sum-rate increases linearly withthe number of links. However, the spectrum hole getssaturated quite fast, due to interference caused by thesecondary users. A spectrum hole may look large, butit disappears as soon as someone starts using it. Wehave assumed that the cognitive users can perfectly judgewhether it is far enough away from the primary basestation, and utilize all of the spectrum holes. This is arather strong assumption [6,7], and it remains to analyzethe effect of imperfect detection of the primary system.It is hard to draw strong and general conclusions for the

potential capacity of spectrumholes.We have seen that forthis kind of a frequency-planned network, cognitive radiomay be a solution that could provide a reasonable rate. Wehave provided some initial reflections on the question asto whether the utilization of spectrum holes is realistic ornot, but further research appears necessary to answer thequestion conclusively.

References

[1] R. Brodersen, A. Wolisz, D. Cabric, S. Mishra, D. Willkomm, CORVUS:A cognitive radio approach for usage of virtual unlicensed spectrum,White paper available at: http://bwrc.eecs.berkeley.edu/Research/MCMA/, 2004.[2] S. Haykin, Cognitive radio: Brain-empowered wireless communi-cations, IEEE Journal on Selected Areas in Communications 23 (2)(2005) 201220.

[3] I.J. Mitola, Software radios: Survey, critical evaluation and futuredirections, IEEE Aerospace and Electronic Systems Magazine 8 (4)(1993) 2536.

[4] FCC, Spectrum policy task force report, Tech. Rep. 02135, FederalCommunications Commission. Available at http://hraunfoss.fcc.gov/edocs_public/attachmatch/DOC-228542A1.pdf, 2002.

[5] M. McHenry, NSF spectrum occupancy measurements projectsummary, Tech. Rep., SSC. Available at http://www.sharedspectrum.com/, 2005.

[6] N. Hoven, A. Sahai, Power scaling for cognitive radio, in: Interna-tional Conference on Wireless Networks, Communications and Mo-bile Computing, vol.1, pp. 1316 June 2005, 250255.

[7] E.G. Larsson, M. Skoglund, Cognitive radio in a frequency plannedenvironment: Some basic limits, IEEE Transactions on WirelessCommunications 7 (2008) 48004806.

[8] R. Tandra, S.M. Mishra, A. Sahai, What is a spectrum hole and whatdoes it take to recognize one? Proceedings of the IEEE. (in press).

[9] S. Mishra, R. Tandra, A. Sahai, Coexistence with primary usersof different scales, 2nd IEEE International Symposium on NewFrontiers in Dynamic Spectrum Access Networks, DySPAN 2007,2007 pp. 158167.

[10] T. Rappaport, Wireless Communications: Principles and Practice,Prentice Hall, ISBN: 0130422320, 2001.

Erik Axell has been a doctoral student atthe Division for Communication Systems inthe Department of Electrical Engineering (ISY)at Linkping University (LiU) in Linkping,Sweden, sinceMarch 2006. He received hisM.Sc.degree from Linkping University in 2005. Hismain research interests are within the area ofwireless communication and especially systemaspects and spectrum sensing for cognitiveradio.

Erik G. Larsson is Professor and Head ofthe Division for Communication Systems inthe Department of Electrical Engineering (ISY)at Linkping University (LiU) in Linkping,Sweden. He joined LiU in September 2007.He has previously been Associate Professor(Docent) at the Royal Institute of Technology(KTH) in Stockholm, Sweden, and AssistantProfessor at the University of Florida and theGeorge Washington University, USA.His main professional interests are within

the areas of wireless communications and signal processing. He haspublished some 50 journal papers on these topics, he is co-author ofthe textbook Space-Time Block Coding for Wireless Communications(Cambridge Univ. Press, 2003) and he holds 10 patents on wirelesstechnology.He is Associate Editor for the IEEE Transactions on Signal Processing

and has been an editor for the the IEEE Signal Processing Letters and theIEEE Transactions on Vehicular Technology. He is a member of the IEEESignal Processing Society SAM and SPCOM technical committees.

Danyo Danev received his M.S. in mathematicsfrom Sofia University in 1996 and his Ph.D. inelectrical engineering fromLinkpingUniversityin 2001. In 2005 he obtained Docent title in DataTransmission. He is currently Associate Profes-sor at the Division of Communication Systemsin the Department of Electrical Engineering atLinkping University in Linkping, Sweden. Hisresearch interests are within the fields of cod-ing, information and communication theory aswell as wireless and optical communications. He

has authored or co-authored 2 book chapters, 12 journal papers andmorethan 20 conference papers on these topics.

Capacity considerations for uncoordinated communication in geographical spectrum holesIntroductionModelSimulation resultsAchievable sum-rateGeographic density

Point-to-point improvementConclusionsReferences