Optimal Power Allocation For Cognitive Radio Based on aVirtual Noise Threshold

Majed Haddad , Aawatif Menouni HayarMobile Communications Group, Institut Eurecom

2229 Route des Cretes, B.P. 19306904 Sophia Antipolis, France

majed.haddad,aawatif.menouni@eurecom.fr

Merouane DebbahPlateau de Moulon, 3 rue Joliot-Curie91192 Gif sur Yvette Cedex, Francemerouane.debbah@supelec.fr

ABSTRACTIn this paper1, we investigate the idea of using cognitive ra-dio to reuse locally unused spectrum for communications.We consider a multiband/wideband system with two usersin which the primary (licensed) user and the secondary (cog-nitive) user wish to communicate to the base station, sub-ject to mutual interference. We introduce the notion of thevirtual noise-threshold which represents a proxy for the pri-mary user to allow cognitive communications. We deter-mine, under the assumption that the primary user is obliv-ious to the presence of the cognitive user the acceptableinterference level within a given quality of service. We alsogive an interesting way in the paper to acquire the primaryuser’s side information. The proposed strategy is proved tobe the optimal one that achieves the maximum rate for eachof the two users under the constraint that the secondary usermaintains a guarantee of service to the primary user whencognitive communication is considered.

KeywordsCognitive radio, interference temperature, power allocation,virtual noise threshold, capacity.

1. INTRODUCTIONThe recent boom in personal wireless technologies has ledto an increasing demand in terms of spectrum resources.To combat this overcrowding, the Federal CommunicationsCommission (FCC) has been investigating new ways to man-age RF resources involving progressive redefinition of rulesfor accessing to the radio spectrum and posing several tasksin the management and in the sharing strategies for such

1Eurecom’s research is partially supported by its indus-trial partners: Swisscom, Thales Communications, ST Mi-croelectronics, SFR, Cisco Systems, Sharp, France Tele-com, Bouygues Telecom, Hitachi Europe Ltd. and BMWGroup Research and Technology. The work reported hereinwas also partially supported by the French RNRT projectGRACE and Cruise

a precious resource. Within this setting, the FCC has re-cently recommended [1] that significantly greater spectralefficiency could be realized by considering cognitive radioSuch a scheme would define at least two classes of spectrumusers. The first would be primary users who already possessa license to use a particular frequency. The second wouldbe secondary (cognitive) users consisting of unlicensed users.Primary users would always have full access to the spectrumwhen they need it. Secondary users could use the spectrumwhen it would not interfere with the primary user. Cog-nitive radio systems offer the opportunity to improve spec-trum utilization by detecting unoccupied spectrum bandsand adapting the transmission to those bands while avoid-ing the interference to primary users. This novel approachto spectrum access is therefore based on reliable detection ofprimary users and adaptive transmission over a wide band-width. However, there are many challenges across all layersof a cognitive radio system design, from its application toits implementation.

In this work, we consider a TDD-uplink communication sce-nario in which the primary and the secondary user wish tocommunicate with the base station (BS), subject to mutualinterference in a heterogeneous network where devices op-erate in a wideband/multiband context. One property ofsuch systems is that, since the same frequency is used, thechannel characteristics are nearly the same in both links,provided the channel does not change too rapidly. Underthis scheme, we allow the secondary user to transmit si-multaneously with the primary user as long as the primaryuser has not his quality of service (QoS) affected. We im-pose that the secondary user can transmit simultaneouslywith the primary user as long as the level of interferencewith the primary user remains lower than a specified virtualnoise threshold. We derive the optimum power allocationfor each of the two users in terms of maximizing their owncapacity.

This work is motivated by the fact that in a cognitive ra-dio protocol as proposed by the Working Group on WirelessRegional Area Networks (WRANs), the primary user wouldalways have his QoS not affected when cognitive communi-cation is considered. The proposed strategy is proved to bethe optimal one that achieves the maximum rate for each ofthe two users under the constraint that the secondary usermaintains a guarantee of service to the primary user when

cognitive communication is considered.

The remainder of the paper is organized as follows: In sec-tion 2, we present the channel model system. Section 3describes the cognitive radio scenario as well as a simpleway to acquire the primary user’s side information. Section4 details the optimal power allocation policy for each user.Simulation results are provided in Section 5 and Section 6concludes the paper.

2. THE CHANNEL MODELThe baseband discrete-frequency model for uplink channelwith two users as described in fig.(1) is:

yiBS = hi

1

qpi1s

i1 + hi

2

qpi2s

i1 + ni, (1)

where:

• hik is the block fading process of user k for k = 1, 2 on

the sub-band i for i = 1, ..., N ,

• sik is the symbol transmitted by user k on the sub-band

i,

• pik is the power control of user k on the sub-band i,

• ni is the additive gaussian noise at the ith sub-band.

We statistically model the channel h to be i.i.d distributed

over the two rayleigh fading coefficients and En��hi

k

��2o = 1.

The additive gaussian noise n at the receiver is i.i.d circularlysymmetric and n ∼ CN (0, σ2).

BS

. . . . . . N N 1 1

h1, p1, s1 h2 , p2, s2

Primary User Secondary User

(1)

(2)

(1)

Figure 1: Two-user cognitive radio uplink in a wide-band/multiband context.

3. THE COGNITIVE RADIO SCENARIOSpectrum utilization can be improved by making a secondaryuser to access a spectrum hole unoccupied by the primaryuser at the right location and the right time. In currentcognitive radio protocol proposals, the device listens to thewireless channel and determines, either in time or frequency,which part of the spectrum is unused. It then adapts its sig-nal to fill this void in the spectrum domain. Thus, a devicetransmits over a certain time or frequency band only whenno other user does like in [4]. In the same context, au-thors showed in [5] how we can improve the overall systemspectral efficiency on classical approaches by considering or-thogonal cognitive communications if there is ordering inthe transmission. The contribution of some recent studies[6] and [7] has extended the cognitive protocol to allow thecognitive users to transmit simultaneously with the primaryusers in the same frequency band. This is exactly the ques-tion tackled in this work where the cognitive radio behavioris generalized to allow the secondary user to transmit simul-taneously with the primary user as long as the level of inter-ference with the primary user remains within an acceptablerange.

As in [8], we consider a TDD-uplink communication sce-nario in which the primary and the cognitive users wish tocommunicate with the base station (BS), subject to mu-tual interference in a heterogeneous network where devicesoperate in a wideband/multiband context. We found outthat a cognitive user can vary its transmit power in order tomaximize its own capacity while maintaining a guarantee ofservice to the primary user. However, contrary to the workaddressed in [8], in this contribution, it is proposed that thecognitive user is able to gather the instantaneous channelstate information (CSI) of the primary user, while the pri-mary user is assumed to only know his proper channel gains.The acquisition of primary user CSI will be discussed later.

Specifically, we consider a communication scenario wheredevices operates in a wideband/multiband context. In tra-ditional systems, when the primary user considers only theambient noise level σ2, he will exploit all the resources bywater-filling [9] on this noise level and therefore, leaves noresources for the secondary user to transmit. In our case, theprimary user implicity over-estimates the noise level (whichhe considers as thermal noise plus interference) so as heleaves space for the other user. A key idea behind doingso is that, in any case, the primary user will not necessarilyneed all that rate. In fact, the primary user will experienceσ2

v, and as long as his capacity is maximized, he does notcare about what he leaves more.

As an example, imagine a primary user communicating withthe BS, subject to a virtual noise interference σ2

v and a sec-ondary user able to reliably sense the spectral environmentover a wide bandwidth, detect the presence/absence of thelicensed user and use the spectrum to communicate withthe BS only if the communication does not interfere withprimary users like in [10].

Moreover, one basic assumption throughout this paper isthat a cognitive user can vary its transmit power in orderto maximize the capacity while maintaining a guarantee ofservice to the primary user. This implies the following in-

equality:

log2

1 +

pi1

��hi1

��2σ2

v

!≤ log2

1 +

pi1

��hi1

��2pi2 | hi

2 |2 +σ2

!

Reliable communication can therefore be achieved when thevirtual noise threshold is higher than the cognitive interferercontributes, yielding:

σ2v ≥ pi

2 | hi2 |2 +σ2, i = 1, ..., N (2)

Accordingly, the virtual noise threshold has a double role:

(i) it allows cognitive user to profit from the primary user

resources in an opportunistic manner,

(ii) it maintains a guarantee of service to the primary userwhen cognitive communication is considered.

Throughout the rest of the paper, we will adopt this frame-work and derive the optimum power allocation policies ofeach user in terms of maximizing their proper capacity. Wealso determine what would be the average capacity rate ofthe primary and the secondary user in this setting.

4. OPTIMAL POWER ALLOCATION POLI-CIES

In this section, we will study the optimal power allocationpolicies that achieves the maximum capacity for each user insuch a scenario. For a given virtual noise-threshold σ2

v, themaximum achievable rate that the primary user can obtainover the N sub-bands is given by:

C1,N =1

N

NXi=1

log2

�1 +

pi1 | hi

1 |2

σ2v

�(bits/s/Hz) (3)

The optimal power allocation which maximizes the trans-mission rate at the primary user is solution of the optimiza-tion problem of:

maxp11,...,pN

1

C1,N ,

subject to: 8>>>>><>>>>>:

1

N

NXi=1

pi1 = 1,

pi1 ≥ 0,

(4)

In [9], authors looked at the problem of maximizing instan-taneous capacity subject to an average power constraint,and showed that the optimum power allocation follows fromShannon’s principle of water-filling, namely 2:

pi∗1 =

1

γ0− σ2

v

|hi1|

2

!+

, i = 1, ..., N (5)

Where γ0 is the Lagrange’s multiplier satisfying the averagepower in (4).

2(x)+ = max(0, x).

Let us now focus on the secondary user capacity. The ex-pression of the instantaneous capacity is given by:

C2,N =1

N

NXi=1

log2

�1 +

pi2 | hi

2 |2

pi1 | hi

1 |2 +σ2

�(6)

As far as channel estimation is concerned, this can be con-ducted in three steps: a) Each user estimates the pilot se-quence transmitted by the BS in order to determine hischannel gain via link (see link (1) in fig. 1). Notice herethat since we are in a TDD mode, when the users estimatethe channel in one way, they can also know it on the reverselink,

b) In the second frame, the primary user broadcasts a pilotsequence (see link (1) in fig. 1) so that the BS estimates thechannel h1,

c) In a third step, when the primary user sends his infor-mation with power p1, the secondary user can estimate thepower knowing the inter-user channel in link (2). In otherwords, assuming that the two users operate on a uniquestandard, the secondary user estimates the inter-user chan-nel via link (2) and can accordingly obtain the primary userpower p1.

Thus, the secondary user can gather the primary users’channel gains by reverse engineering. In fact, by only know-ing the virtual noise threshold (σ2

v) and the power allocationpolicy of the primary user (waterfilling), the cognitive usercan obtain the primary users’ channel gains following equa-tion in (5).

The secondary user offers the opportunity to improve thesum capacity over the system by reliably detecting primaryuser activity and adapting his transmission while avoidingthe interference to the primary user by satisfying constraintin (2). In fact, the spectrum utilization can be improvedby making a secondary user to access to the primary userspectrum at the right location and the sub-band in ques-tion. The optimal secondary user power allocation whichmaximizes the capacity of the system is solution of the op-timization problem:

maxp12,...,pN

2

C2,N (7)

subject to:

8>>>>>>>>><>>>>>>>>>:

1

N

NXi=1

pi2 = 1

pi2 ≥ 0; for i = 1, ..., N

σ2v ≥ pi

2 | hi2 |2 +σ2; for i = 1, ..., N

(8)

Theorem 1. The optimal secondary user power alloca-tion solution of the problem (7) under the constraints in (8)

is:

pi∗2 =

8>>><>>>:

1

λ− pi

1 | hi1 |2 +σ2

|hi2|

2

!+

; if λ >

��hi2

��2pi1 | hi

1 |2 +σ2v

σ2v − σ2

|hi2|

2 ; otherwise

(9)Where λ is the Lagrange’s multiplier satisfying the averagepower constraint in (8).

Proof. Due to the lack of space, we will not present theseconsistencies here and the reader is referred to the reference[11] for additional analysis results.

Accordingly, the optimal power allocation for this problemwas shown to be a mixture of channel inversion and water-filling allocation. Notice here that the proposed strategyprevents to obtain infinite power in extreme fading environ-ments, i.e. for bad fading states hi

2, the power allocationpolicy is the waterfilling.

Discussion: Assume σ2v and hi

1 fixed. For good hi2 the op-

timal power-control law is the channel inversion policy. Onthe contrary, for bad values of hi

2 the optimal power-controllaw is water-filling on the inverse of the channel gain. Thisstands in contrast to the traditional case of channel inver-sion policy where more power is allocated when the channelis bad than when the channel is good. Notice that in theevent of deep fades, i.e. hi

2 tends to be zero, the secondaryuser power allocation is zero.

5. SIMULATIONS AND RESULTSIn order to analyze the performance of the proposed policiesin terms of achievable rates, we model two rayleigh channelmodels, one for each user. In figure 2, the solid line standsfor the sum capacity of a system where the primary userdecides to maximize his rate selfishly. In other words, he willwater-fill over the ambient noise level σ2 and no resourceswill be left for cognitive users. The dashed line representsthe sum capacity of the proposed system where the primaryuser experiences the virtual noise threshold σ2

v. It is clearthat the cognitive system performs always better than forsystem where no cognition exists.

6. CONCLUSIONWe introduced the notion of the virtual noise-threshold inorder to maintain a guarantee of service to the primaryuser when cognitive communication is considered. We alsopresent a simple algorithm so that the secondary user cangather the side information about the primary user. Theproposed policies are proved to be the optimal ones thatachieve the maximum rate for each of the two users underthe constraint that the secondary user maintains a guaranteeof service to the primary user when cognitive communica-tion is considered. As a future work, it is of major interestto study the impact of the proposed strategy on the primaryuser in terms of outage capacity and to characterize this out-age with respect to classical communication systems. Therelated work can also be extended to the two-way channelcontext.

−5 0 5 10 150.5

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

SNR in dB

Sum

capaci

ty in

bits

/s/H

z

Cognitive Radio System: when the primary user experiences the virtual noiseNon Cognitive System: when the primary user experiences the ambient noise

Figure 2: Sum Capacity for N = 10.

7. REFERENCES[1] Federal Communications Commission, Cognitive Radio

Technologies Proceeding (CRTP),http://www.fcc.gov/oet/cognitiveradio/.

[2] The charter of IEEE 802.22, the Working Group onWireless Regional Area Networks (WRANs),http://www.ieee802.org/22/.

[3] D. Tse and P. Viswanath, Fundamentals of WirelessCommunication, Cambridge University Press, 2005.

[4] R. W. Brodersen, A. Wolisz, D. Cabric, S. M. Mishra,and D. Willkomm, Corvus, “ A Cognitive RadioApproach for Usage of Virtual Unlicensed Spectrum”,UC Berkeley White Paper, July 2004.

[5] M. Haddad, M. Debbah and A.M. Hayar , ”SpectralEfficiency for Cognitive Radio Systems”, submitted toIEEE GLOBECOM 2007, Washington, D.C, USA,2007.

[6] N. Devroye, P. Mitran, and V. Tarokh, “ AchievableRates in Cognitive Channels”, IEEE Trans. IT, vol. 52,no. 5, pp. 1813-1827, May 2006.

[7] A. Jovicic and P. Viswanath, “Cognitive Radio: AnInformation-Theoretic Perspective”, IEEEInternational Symposium on Information Theory,Seattle, USA, July 2006.

[8] M. Haddad, M. Debbah and A.M. Hayar,“Distributed Power Allocation for Cognitive Radio,” inproceeding of IEEE ISSPA, February 2007, Sharjah,United Arab Emirates.

[9] T. M. Cover and J. A. Thomas, Elements ofInformation Theory, Wiley, 1991.

[10] N. Hoven and A. Sahai, “Power Scaling for CognitiveRadio”, International Conference on WirelessNetworks, Communications and Mobile Computing ,June 2005.

[11] M. Haddad, M. Debbah and A.M. Hayar, “PowerAllocation Strategies For Cognitive Radio Systems”,In preparation. Available throughmajed.haddad@eurecom.fr.