Capacity of Multi-Channel Cognitive RadioNetwork

Group 8: Tian Chu, Xinran Cai, Siqi Zhang, Song Liu

EE Dept. of Shanghai Jiaotong Universityhpday1987@sina.com

Abstract— With more and more research on the CognitiveRadio Network has been done, recently people began to combineKumar et al’s work on capacities of the ad-hoc network withthe CR network. With the help of the work done by the paper’Scaling Laws of the Cognitive Radio Network’, we extend theirmodel to multi-channel and/or multi-radio model to see whetherthe network capacity scales with the number of both primaryusers and the secondary users. Also, we suggested an algorithm inthe spectrum allocation and provided lower and upper bounds ofthe capacity if the network is organized with our algorithm. Oursimulation verified the theoretical calculations and we observedthat there exists an optimal value of the number of channels Lwith which the total capacity could be maximized.

I. INTRODUCTION

To increase the utilization of frequency, Dr Joseph Mitolainvented the idea of cognitive radio [3] in 1999. In cognitivenetworks, the unlicensed (secondary) users are allowed to usethe band allocated to but not being used by the licensed (pri-mary) users. Although this opportunistic policy for secondaryusers does not always work, it improves the capacity of thenetwork greatly.

The scaling laws of cognitive networks has been developedby the work of Mai Vu, Natasha Devroye, Masoud Sharif, andVahid Tarokh [2]. Their work is based on the paper of Guptaand Kumar [1], and different from previous work. Instead ofconsidering any homogeneous ad hoc wireless network, theystudied a cognitive network containing primary and secondaryusers. And they used a model of single-hop transmission, sothat each transmitter has a unique receiver. Figure 1 of [2] istheir network model.In the paper ”Scaling Laws of Cognitive Networks” [2],

several conclusions are drawn: firstly, with simultaneous one-hop cognitive transmissions, the sum-rate of cognitive usersscales linearly in the number of cognitive links n as n→ ∞,in presence of multiple primary users, when the cognitivetransmitters use constant power [2]; secondly, the sameresult holds in the presence of a single primary user, whenthe cognitive transmitters scale their power according to thedistance from the primary user [2]; finally, the authors derivedbounds on the radius of a primary exclusive region aroundeach primary transmitter [2]. Their work is limited to singlechannel model.

Our Main Contributions: In this paper, we extend theresearch of scaling laws of cognitive networks to multichannelnetworks. We focus on the derivation of capacity of multi-channel cognitive networks. Our work can be classified into

Fig. 1. Network model of ”Scaling Laws of Cognitive Networks”

two categories. One is the multi-channel multi-radio (MCMR)network, the other is the multi-channel single-radio (MCSR)network. Figure 2 introduces our models, and details areincluded in Section 2 and Section 3.

Fig. 2. (a) MCMR model (b) MCSR model

In both MCMR and MCSR networks, we suppose thatthere are several primary users scattered in a circular networkarea of radius R. Each primary user is surrounded by aPrimary Exclusive Region (PER) of radius Rp. And there area number of secondary users in each network. Each secondarytransmitter influences a distance of Rs. In the MCSR network,each primary user occupies the whole band, so their PERsshould not overlap. In the MCMR network, the PERs may

overlap if the primary users use different sub-channels. Weassume that both primary and secondary users are uniformlydistributed in the networks.

Within the MCMR network, we prove that no matter howmany other PERs a primary user is overlapped with, it willsubtract the same amount of capacity from the total capacity.Therefore we get the average capacity per user. We alsofind that the total capacity is linear with the number ofsub-channels and the number of primary users. When thenumber of subchannels L increases, the total capacity increaseslinearly. When the number of primary users m increases, thetotal capacity decreases linearly until zero. If the primary userskeep a constant density, the network is scalable with m→∞.

Within the MCSR network, we derive an upper boundand a lower bound of capacity, and we also get an iterationformula for a specific case. We find that when there are alarge number of secondary users, some users will be deprivedof transmission opportunity due to other users’ interference;on the other hand, if we divide total bandwidth into a largenumber of sub-channels, some sub-channels will be wasted.Network capacity grows with the number of users almostlinearly at first, but when n→ ∞, capacity will saturate dueto decreasing transmission opportunity. On the other hand,average capacity of one user decreases with the number ofusers, and approaches zero at infinity.

The paper structure is as follows. In Section II, we givedetails of the the MCMR network model, and derive the scal-ing law between capacity and other determinants. In SectionIII, we provide details of MCSR network model, and derivean upper bound and a lower bound of capacity, and we alsoget an iteration formula for a specific case. In Section IV, wemake stimulations of our research, and pay attention to someuseful phenomena. In Section V, we make our conclusions.

II. IMPACT OF PRIMARY USERS ON THE CAPACITY OFMULTI-CHANNEL MULTI-RADIO MODEL

The first problem we are going to discuss is the impactof the number of primary users on the capacity of the totalnetwork. For simplicity, we only apply MCMR model here.

Fig. 3. Multi-Channel Multi-Radio Model

In the network above, the circular region with the radiusRp represents the primary exclusive region(PER) of a primaryuser, in which no other primary users with the same bandcan lie in. Therefore the PERs can overlap as long as theoverlapped regions’ owners do not share the same band.

First of all we assume that the network density of secondaryusers in n, and the total network area is S. i.e, the totalnumber of secondary users is nS. If each secondary userhas a constant capacity of WCsub per channel, then the totalcapacity available in the network will be nSWLCsub, whereL is the number of bands available.

In the MCMR model, we show that when we put a primaryuser surrounded by it’s primary exclusive region into the wholearea, now matter how many other PERs it is overlapped with,it will subtract the same amount of capacity from the totalaverage capacity available.

Therefore if we put a PER into the network area, the will benπR2

p secondary users affected and each of them will have oneless band to use. The total capacity loss will be nπR2

pWCsub.Since we have m PERs, the total capacity after subtractionwill be

WnSLCsub −mnπR2pWCsub (1)

Dividing the above by the whole number of users nS, andnote that S = πR2, we derive the average capacity as:

Caverage = W (L− mR20

R2)Csub (2)

Csub means the capacity of each pair of secondary usersusing one sub-band.

With the formula above, we can easily find that the totalcapacity is linearly correlated with the number of sub-channels(L) and the number of primary users (m). When L increases,the total capacity increases linearly. When m increases, thetotal capacity decreases linearly until it reaches zero, whichmeans that m should be limited with a constant LR2

R20

. Also,the size of PER will have a quadratic impact on the capacity.

We can also find from the formula that if the primary userhas a constant density m

πR2 = λ, the network is scalable withm approaching infinite, that is:

Caverage = W (L− πλR20

R2)Csub (3)

The per node capacity converges to WLCsub.

III. MULTI-CHANNEL SINGLE-RADIO

In this section, we first describe our network model and wegive both upper bound and lower bound for the capacity of theSecondary Network. At last we discuss on a tradeoff betweentransmission opportunity and transmission rate.

A. Network Model

As we have described in our introduction, in MCSR (Multi-Channel Multi-Radio) Network, all primary users use thesame channel with bandwidth W, so their exclusive regioncan never overlap. However, for the secondary network, this

channel (bandwidth=W) is divided into L sub-channels (band-width=W/L), and each secondary user is permitted to transmiton only one of them.Therefore, the more sub-channels, the lower transmission rate,but the greater transmission opportunity for each SU. So ifwe have too many sub-channels, a great amount of spectrumwill be wasted, but if too few sub-channels, a lot of SUwill lose their transmission opportunity. So L (the number ofsub-channels) is an important parameter for setup an efficientsecondary network, and its value may be cooperatively anddynamically determined by all SUs.

Fig. 4. Example of an MCSR Cognitive Radio Network

B. Network Capacity

In our model, we assume all secondary users are uniformlydistributed in the network. So the mathematical expectation ofnetwork capacity can be described by the following equation:

E[Cn] =n∑

i=1

E[µi]W

L(4)

In the equation, Cn is network capacity, W/L is sub-channelbandwidth and µi is defined as an indicator variable:

µi ={

1 if Secondary user i can transmit on a sub-channel0 if Secondary user i can’t transmit on any sub-channel

In order to calculate E[µi], we introduce a concept of trans-mission probability matrix:

P =

p11 p12 . . . p1L

p21 p22 . . . p2L

......

. . ....

pn1 pn2 . . . pnL

Each element of this matrix, say pik, represents the probabilityof secondary user i transmit on sub-channel k. Then with thehelp of transmission probability matrix, we can express E[µi]like this:

E[µi] =L∑

k=1

pik (5)

If we assume each secondary user selects their sub-channel ina round-rubin order, and tries the sub-channel of smallest ID

first. This means if the first sub-channel is already occupied byother SU, then try the next sub-channel until one is available,if no sub-channel is available, then this SU fails to transmit.According to this simple strategy, we can get an iterativesolution of the transmission probability matrix:

pi1 = [1−i−1∑j=1

pj1R2

s

R2] ,for i∈[2,n] (6)

p11 = 1 and p1k = 0 ,for k∈[2,L] (7)

pik = [1−k−1∑j=1

pij ][1−i−1∑j=1

pjlR2

s

R2] ,for k∈[2,L], i∈[2,n] (8)

As we have described earlier, the network parameter L and nshould be balanced, otherwise spectrum waste or network con-gestion will makes the whole secondary network inefficient.

Fig. 5. Transmission probability matrix when n is too large

Fig. 6. Transmission probability matrix when L is too large

In these two figures, z axis represents the value of elements intransmission probability matrix. Comparing these two figures,we can get following ideas about the network parameter n andL:

• Some users are unable to transmit, when n is large• When L is too large, some sub-channels may be wasted• Dynamic balance between parameters L and n is needed

C. Upper-bound and Lower-bound of Capacity

In the paper [2], the capacity of per user in single channelCR network is mainly determined by the interference of pri-mary and secondary transmitters. However, our work mainlyfocus on the capacity of the whole secondary network. PrimaryExclusive Region makes sure that interference from otherprimary transmitters and secondary transmitters are limited,so that the transmission of primary user (at the center of thePER) to its receiver is always successful.On the next, we will give the upper bound of the secondarynetwork. When the number of secondary user is not too large,as we described in last subsection, some sub-channels will bewasted. So the more SU, the greater spectrum utilization. Sothe upper bound is determined by the number of primary userm, when the number of secondary user is saturated.Because transmissions on sub-channel k are independent ontransmissions on sub-channel j, if k6=j, the total network canbe viewed as L disks with radius R, each of which representsa sub-channel of the secondary network.

X

Y

Z

k=1

k=2

k=3

k=4

k=5

Fig. 7. Example of Secondary Network when L=5

Upper Bound of Capacity:n∑

i=1

E[µi] ≤ (1−mR2

p

R2)LR2

R2s

(9)

E[Cn] ≤ (1−mR2

p

R2)LR2

R2s

W

L(10)

The lower bound of the secondary network is both deter-mined by the number of primary user m and secondary usern.

n∑i=1

E[µi] ≥ (1−mR2

p

R2)n{1− [

(n− 1)R2s

LR2]} (11)

Substitute E[Cn] ≥∑n

i=1 E[µi]WL into the last equation, we

can get the final expression of lower bound of secondarynetwork:

E[Cn] ≥ (1−mR2

p

R2)n{1− [

(n− 1)R2s

LR2]}W

L(12)

As we can see, in this expression, the transmission rate ofsub-channel W/L will decrease as we increase the numberof sub-channels L. On the other hand, increasing L willincrease n{1 − [ (n−1)R2

s

LR2 ]}, which means that more SU willhave opportunity to transmit. So we expect that there will besome value of L will makes the capacity of the total networkgreatest.

Fig. 8. The lower bound of the secondary network

IV. SIMULATION

In the simulation, because the result of MCMR cognitiveradio network is so simple that no simulation is needed. Tosimulate the MCSR cases, we use MatlabR2007a and setup anetwork with the number of secondary users n and primaryusers m. The following figure is just an example of the networkfor simulation. In this figure, the network radius R is 10, thenumber of SU (represented by blue crosses) is 1000 and thenumber of PU (represented by red crosses) m is 20.

Fig. 9. Example of MCSR Cognitive Radio Network

A. Total network capacity VS the number of SU

As we expect, simulation results show that total networkcapacity Cn grows with the number of users n almost linearlyat first, but when n approach infinite, capacity will saturatedue to decreasing transmission opportunity.

Fig. 10. Total network capacity VS the number of SU

B. Per user capacity VS the number of SUOn the other hand, average capacity of per secondary user

decreases with the number of users, and approaches zero atinfinity. Comparing these two figures, we can find anothertradeoff between the interests of the whole network andinterests of individual users. Increase the number of SU willdecrease the percentage of spectrum waste, which benefitsthe whole network, but it will also decrease the transmissionrate of SU, which damages the interests of individual users.Comparing these two figures, we can derive some following

Fig. 11. Per user capacity VS the number of SU

results:• Whole network’s capacity grows with n• Individual user’s capacity decrease with n• The secondary network can not be too densed

C. Network Capacity and Average Transmission ProbabilityVS number of sub-channels L• Transmission opportunity grows with L• Network capacity increase, because more users have

transmission opportunity• Network capacity decrease, because sub-channel band-

width decreases faster

V. CONCLUSION

In this project, in the first several weeks we mainly put ourefforts on searching papers and technical reports on Cognitive

Fig. 12. Network Capacity and Average Transmission Probability VS numberof sub-channels L

Radio (CR). After that, we try to use game theory and auctiontheory in the study of resource allocation of CR. However, wefinally founded that these areas have been rather intensivelyresearched. So we turn our focus on the capacity of CR, in theend. In this area, paper [2] has worked out a lot of useful resultabout the capacity of CR. Because [2] only considers thesingle channel cases, though it gives the relationship betweenthe per secondary user capacity and the number of secondaryusers based on the scaling law, its result is very limited inpractice. First because CR is based the capability of spectrumsensing, which helps CR node find the available bands and usethem opportunistically. So all SU and PU try to use the samechannel is unreasonable, because it does not increase spectrumefficiency by frequency reuse and spatial reuse. Therefore,the work we have finished tries to extend their work for theMulti-Channel cases. In the end, we put forward the capacityboundaries for both MCMR and MCSR cases. Under theMCSR situation, we also discussed on the tradeoff betweenthe per SU transmission rate and averaged SU transmissionopportunity.In the future, we will try to improve our work and extend theresult on Multi-Channel Multi-Hop Cognitive Radio Network.

REFERENCES

[1] Piyush Gupta, Student Member, IEEE, and P. R. Kumar, Fellow, IEEE:The Capacity of Wireless Networks

[2] Mai Vu, Natasha Devroye, Masoud Sharif, and Vahid Tarokh: ScalingLaws of Cognitive Networks

[3] J. Mitola, ”Cognitive Radio C An Integrated Agent Architecture for Soft-ware Defined Radio”, Ph.D. Dissertation, Royal Institute of Technology,Kista, Sweden, May 8, 2000.

Tian Chu,5060309889hpday1987@sina.com

Xinran Cai,5060309765lord vo7184@sina.com

Siqi Zhang,5060309762sceva624@sjtu.edu.cn

Song Liu,5060309851lsls137@163.com