Sensing based interference and data queueback-pressure approach for cognitive radio

Kimmo Kansanen

Institute of Electronics and TelecommunicationsNorwegian University of Science and Technology

Trondheim NO-7491, NorwayEmail: kimmo.kansanen@iet.ntnu.no

AbstractThe Lyapunov drift based back-pressure frameworkis applied to the cognitive radio context. A group of spatiallydistributed sensing units are utilized to monitor re-used spectrum.The sensing results for secondary user received power are utilizedin virtual interference queues that, when stabilized, enforcespatially point-wise received power constraints, given in termsof a maximum allowed rate for exceeding an absolute receivedpower level. A system stabilizing algorithm consisting of packetadmission and transmit power control is applied to the problem.Numerical evaluations are utilized to characterize the systembehavior in terms of fulfilling interference constraints, requiredtransmitted energy per bit and queuing delay, when either areceived power or an interference outage type virtual queue, andeither water-filling type or binary ON-OFF power allocation isutilized.

I. INTRODUCTION

Cognitive Radio (CR) appears promising as a fundamen-tal approach to design and build future radio networks. Asignificant amount of work has been invested in finding thefundamental limits governing such communications (e.g.[1],[2], [3]) and finding practical solutions. One of the significantchallenges in cognitive radio networks is finding algorithmsthat tolerate dynamic system behavior and can deal withuncertainties of the environment. In this paper, an approachfor secondary use of licensed spectrum is developed, thatrelies on sensing feedback to control interference levels in thesystem. The back pressure framework is utilized to deal withuncertainty over instantaneous channel states towards primaryusers, and data communication needs. Related work can befound in e.g. [4] where the authors utilize the back-pressureframework to solve a similar problem.

In this paper, the joint problem of enforcing a receivedpower constraint at the sensor nodes and of stabilizing thedata queues at the secondary transmitters is considered. Onlinealgorithms that do not depend on explicit knowledge of radiochannel characteristics and secondary transmitter behavior aretargeted. Using the back-pressure framework, the approachtaken in [5] is closely followed, and extended to cover receivedpower constraints. A level crossing type interference constraintis formulated and virtual queues are utilized to model thedynamic received interference in order to define a stabiliz-ing algorithm that then fulfills the interference constraints.

Both optimal water-filling type power allocation and low-complexity ON-OFF binary allocation are considered.

The paper is organized as follows. The system model ispresented in Section II, and the cognitive radio related inter-ference constraints in Section III. The optimization frameworkis recapped for completeness and clarity in Section IV, andthe resulting power control schemes in Section V with someperformance bounds are given in Section VI. The interferenceoutage queue is introduced in Section VII, followed by nu-merical evaluations in Section VIII.

Vectors and matrices are denoted with bold lower case andbold capital letters, respectively. A scalar function with vectorargument operates element-wise, producing a vector result. Anall-ones vector is denoted by 1.

II. SYSTEM MODEL

An orthogonal multi access system is considered, withK secondary transmitter-receiver pairs that wish to oppor-tunistically re-use licensed spectrum. For interference controlpurposes, their transmissions are monitored by a network of Jspatially distributed sensors. This network can either consistof fixed sensors that have been deployed in the area for thatparticular purpose, or of other secondary user units. Eachsensor can provide an estimate of the instantaneous receivedamplitude on the desired frequency, and can furthermoreseparate primary and secondary user signals. In practice, thiscan be supported through the use of signal structures thataid signal separation and power estimation and correspondingmatched filtering at the sensor.

It is assumed the received primary user signal componentsafter matched filtering and the receiver noise are complexGaussian distributed. The matched filter output is then squaredand averaged over a time window, essentially performingenergy detection [6].

The system to be monitored has K transmitters. The trans-mission of user k experiences a propagation channel with asquared envelope ck between transmitter and the intended re-ceiver. The system uses time division multiple access (TDMA)between the users to let only one user transmit at a time.

The jth sensor (of J) receives at time t the transmissionof user k over a propagation channel with squared envelopedj,k, so that the received squared matched filter output vector,

containing the samples at all sensors, is given by

z(t) = D(t)p(t) + n(t), (1)

where p(t) denotes the transmit power vector at time t, andn(t) denotes the squared matched filter output noise term attime t, with unknown distribution but known average. Thelatter is due to the fact that the sensor does not attempt toestimate the statistics of the received primary signals. Thechannel dj,k has the expected value E [dk] = dk.

The notation z(t) = [z1(t), . . . zJ(t)]T, p(t) =[p1(t), . . . pK(t)]

T and [D(t)]j,k = dj,k is used. The ithcolumn of a matrix D is denoted by di. For simplicity, itis assumed that zk(t) < z0k,k, t, i.e. the interference hasbounded amplitude. Note, that the channel matrix D canmodel, unchanged, both orthogonal and non-orthogonal accessmethods provided that each user transmits once per timeinstant.

For convenience, a time slotted model is assumed, wherethe current system state is observed at time t, schedulingand resource allocation is performed, after which every usertransmits on their given time slot. The system model caneasily be modified to cover the case where system stateis continuously updated without major changes to the mainresults.

III. INTERFERENCE CONSTRAINTS

Two types of interference constraints are considered inthis paper. The soft interference constraint requires that theexpected received power at the primary receiver is below alimit such that the element-wise inequality

IE [z(t)] zav (2)

holds. The interference outage constraint requires that theinstantaneous received power at the primary receiver exceedsa pre-defined threshold with a small enough probability

P (zj(t) > zth) < Pth j, (3)

where typically the parameters zth and Pth are specific to theprimary system considered.

Note, that one can use the soft interference constraint tofulfill also the interference outage constraint. For a given Pthand zth, the admissible zav can be upper bounded by firstnoting that

Pthzth IE [z(t)]

always holds due to the Markov inequality. Then, if zav is setsuch that

zav = 1Pthzth,

and apply a strategy that fulfills the soft constraint (2), theinterference outage constraint (3) will be fulfilled regardlessof the distribution of z(t).

IV. LYAPUNOV BASED OPTIMIZATION

The soft interference constraint is considered first, and Jinterference queues are set up at the sensors as

x(t) = [x(t 1) zav]+ + z(t). (4)Intuitively, the interference queue represents the cumulativedeviation of the received power process z(t) from the desiredaverage power zth. If the interference queue can be stabilized,it will imply that the soft interference constraint is fulfilled.More strictly, if the channel matrix G denotes the channelsbetween the considered secondary transmitters and a set ofprimary receivers such that IE [G] < IE [D] holds element-wise, it follows that [5, Lemma 3]

limt sup

1

t

t=0

IE {G(t)p(t)}

< limt sup

1

t

t=0

IE {D(t)p(t)} zav. (5)

Or, in other words, if we find a set of receivers (sensors) whoseaverage channel state (IE {D}) dominates that of the primaryreceivers (IE {G}), we can fulfill the interference constraintsin the latter. Physically speaking sensors placed betweenthe secondary transmitters and primary receivers would bemost straightforward given the radio propagation environmentbehaves smoothly enough.

Each user holds a data queue, which has the related arrivalprocess ak(t) and transmissions rk(t), so that the vector ofdata queues behaves as

q(t) = [q(t 1) r(t)]+ + a(t). (6)The arrival process and the transmitted rate are assumed to beupper bounded by a0, and r0, respectively.

The system performance is optimized using the Lyapunovtechnique introduced in [5], where the quadratic Lyapunovfunction was shown to provide an asymptotically optimalenergy allocation. Since the focus is secondary spectrum usesystems that are best effort by nature, the suboptimal delay-energy behavior provided by the quadratic Lyapunov functionis of less importance. Thus, the quadratic Lyapunov functionwill be utilized as

L (x(t),q(t)) = qT(t)q(t) + xT(t)x(t). (7)

Next, the expected Lyapunov drift is computed, which is thenoptimized over all rate allocation strategies. The resultingalgorithm stabilizes the data and interference queues and seeksto maximize the weighted average expected throughput of theusers

Kk=1 kRk, where k > 0,k are arbitrary positive

weighting factors that reflect priorities between users, and

Rk = limt

t1=0

IE {ak()}

represents the long-term expected data rate admitted into thequeue.

The conditional Lyapunov drift for time t is defined as

(x(t),q(t)) =

IE {L (x(t+ 1),q(t+ 1)) L (x(t),q(t)) |x(t),q(t)} . (8)For the sake of brevity, the time index will be dropped in thefollowing. By replacing the queue definitions for time instantst, t + 1 to (8), the development in [5] can be followed. Thealgorithm uses an arbitrary tuning parameter V > 0. First, thedrift expression is expanded and an upper bound is found as

(x(t),q(t))

IE{

rTr + aTa + zavTzav + (Dp)

TDpq,x}

2xTzav + V TIE {a|q,x}+ (2q V )T IE {a|q,x}

2IE{qTr xTDp|q,x} . (9)This can further be simplified as

(x(t),q(t))

rT0 r0 + aT0 a0 + zavTzav + IE{

pTDTDpq,x}

2xTzav + V TIE {a|q,x}+ (2q V )T IE {a|q,x}

2IE{qTr xTDp|q,x} . (10)Loosely speaking, a finite upper bound of the drift expression(10) guarantees stability of all queues [7]. To optimize thesystem, the last two lines of (10) can be minimized byadmission control and transmit power control, respectively.The remaining of the terms can be bounded or are negative,and can be neglected, even though this produces a possiblyworse performing algorithm than directly optimizing the fulldrift expression. The admission control admits new arrivalsinto the data queue if qk V k/2, and the transmit powercontrol seeks to find a power vector p minimising the last line,in other words finding

p = arg maxp

{IE{qTr xTDp|q,x}} . (11)

The constant V controls the maximum buffer length, andthe user priority constants k control the desired averagethroughputs between users. Next, the problem of finding thesolution to (11) in problem context is considered.

V. TRANSMIT POWER CONTROL

In the cognitive radio context, protection of the primaryusers from excessive interference is the ultimate objective,but one which is at the absence of channel state informationbetween secondary transmitters and primary receivers, infea-sible. The proposed solution utilizes third party observers,sensors, as a replacement for providing an estimate of theinterference experienced by the primary users. However, dueto the spatial decorrelation of radio channels, a sensor canonly provide an estimate of the received power that is basedon the macroscopic propagation phenomena that correlate over

larger distances. The resulting estimate corresponds, thus, tothe local received power averaged over short-term statisticalmultipath fading. Thus, the uncertainty within (11) is over thechannel short term fluctuation. The modified optimization forthe average channel state is then

p = arg maxp

{qTr xTDp} , (12)

where D = IE {D}. In the following the resulting allocationin the case where the transmitter has a peak power constraintwill be shown, and the secondary receiver is assumed to haveknowledge of instantaneous receiver noise and interferencedue to primary user transmissions. Note that the algorithmsprovide user power allocation while channel allocation isassumed fixed such that each user has an assigned time slotto transmit.

It is reasonable to assume that a relatively good estimate ofD is available, and the optimization (12) used as

p = arg maxp

{qTr yTp} , (13)

where y = DTx. The power-rate function of each user isapproximated with the Shannon capacity of their channels as

rk(pk) = log

(1 +

pkckN0 + Ik

), (14)

where N0 and Ik denote the power spectral density of re-ceiver noise and the received primary signal power densityat the secondary receiver, respectively.Since the secondaryusers utilize orthogonal channel access amongst themselves,the optimization problem (13) can be solved separately foreach user, and the optimal allocation for each user as per

pk = min

(Pmax,

[qkyk N0 + Ik

ck

]+), (15)

where Pmax is the peak power constraint. Note that while theoptimization problem separates in terms of power allocation,global knowledge of the interference queue vector x(t) isstill required to compute y. Also note, that if users wereto transmit simultaneously, the power-rate functions of userswould depend on the full vector of allocated powers p(t) (c.f.[8]).

A simplified allocation method can be found by approx-imating the power-rate function (14) for rk 0, whererk(pk) pkck/ (N0 + Ik), and the optimization becomes

pk = arg maxpk{qkckpk/ (N0 + Ik) ykpk} , (16)

subject to pk > 0. Then, the corresponding allocation becomesthe binary

pk =

{Pmax

ykqk< ckN0+Ik

0 ykqk >ck

N0+Ik

. (17)

This last algorithm allows a simple interpretation: when theinterference queue normalized with the data queue is smallerthan the channel SNR, positive power is allocated.

VI. BOUNDS

A. Minimum expected energy

The average rate achieved by the users is given by

Ck (pk) = IE

{log2

(1 +

pkckN0 + Ik

)}. (18)

Given that ck and Ik are ergodic random processes, there is aminimum average transmit power that guarantees stability ofthe queues, even without the admission protocol, such that

Ck (IE {pk}) , IE {Ck(pk)} > IE {ak} .In other words, the mean arrival rate vector is strictly interiorto the channel capacity region. The required average powerallocation results in a feasible solution for our system if, inaddition,

IE {Gp} < IE {Dp} = IE {z} < zth (19)0 < pk < Pmax k. (20)

From the above, it would be expected that when the systemtuning parameter V grows, the system energy should reducetowards the minimum. In practice, and since an admissionprotocol is utilized, one can measure the average energyrequired to transmit one bit of information, and thus removethe effect of packet dropping to the throughput.

B. Guarantees

A few simple upper bounds on the queue states q(t) andx(t) can be developed by using the following observations.Whenever

qkyk

(V

2+ a0

)sup(ck)

N0. (22)

VII. INTERFERENCE OUTAGE QUEUE

Note that

P (zj(t) > zth) < Pth

1 (zj(t) > zth) dP (zj < z) < Pth.

In other words, a counter queue can be set up for theinterference outage event. If this queue is then stabilized, theinterference outage constraint is fulfilled. The counter queues

o(t+ 1) = [o(t) Pth]+ + 1 (z > zth) (23)

are then set up. Here, the transmission is the constant valueof Pth, and the arrival is the event that the current measuredreceived power exceeds the threshold value. The indicatorfunction 1 (a) is one whenever condition a is true, zerootherwise. When stabilized, the configuration of the virtualqueue will guarantee that

limt sup

1

t

t=0

IE {1 (z(t) zth)} < Pth,

which, for an ergodic z(t), simply equals

P (z(t) zth) < Pth.It is possible then apply the interference outage approachin the above framework, essentially by replacing x with oin (12) (17), and the power arrival process z(t) with thecorresponding complementary outage event 1(z < zth). Sincethe modification to the algorithm is straightforward, the detailsare omitted. The resulting power optimization to be performedis given by

p = arg maxp

{qTr xTP (z > zth)

}, (24)

while the admission protocol utilized above can be used as is.The difficulty of directly maximizing (24) lies within ex-

pressing and optimizing the transmit power wrt. the probabil-ities P (z > zth). One possible approach for this would be toperform e.g. a maximum entropy derivation for the distributionof the channels dj,k given the known mean channel gain. Thiswould result the assumed distribution of the received powerat the sensors to be that of a sum of exponentially distributedrandom variables. The interference outage probabilities wouldthen be possible to express in closed form with the help ofcharacteristic functions and residue calculus. The resultingoptimization thereafter would, however, be non-trivial. Forthe sake of developing simple algorithms, the probabilitiesP (z > zth) are approximated using the Markov inequalityonce again. The resulting optimization is then given by

p = arg maxp

{qTr xTI1zth Dp

}, (25)

where Izth denotes a diagonal matrix with the vector zth on itsdiagonal. Denoting

yT = xTI1zth D,

the resulting power allocation corresponding to the rate func-tion (14) can be expressed as

pk = min

(Pmax,

[qkyk N0 + Ik

ck

]+). (26)

Equivalently, the binary allocation corresponding to the linearapproximation is given by

pk =

{Pmax

ykqk< ckN0+Ik

0 ykqk >ck

N0+Ik

. (27)

While the interference outage queue approach also usesan approximation for the outage probability, it has one clear

100 101 1020

2

4

6

8

10

12

14

16

18

V

Ener

gy

Power queue, real alloc.Power queue, binary alloc.Counter queue, real alloc.Counter queue, binary alloc.Constraint

Fig. 1. Transmitted energy per bit

advantage over the previous approach. The interference queueupdate at the fusion centre requires the communication of onebit, on the average rate corresponding to the outage probability.If this is combined with the binary power allocation strategy,the communication requirements between network units re-main low per scheduling cycle.

VIII. NUMERICAL EXAMPLES

The behavior of the control strategy is tested in a smallsymmetric case with K = 2, J = 2, and the channels arenormalized such that IE {di,j} = IE {ci} = 1. The arrivalprocess is assumed to be a Bernoulli process with arrivalprobability p = 0.1, while each packet for both users has size0.5 bits. The interference outage threshold to is set to 5 forboth sensors and the outage probability to 0.01. It is assumedthe receiver noise at the sensor is low enough to be negligible,an eventual non-negligible noise variance can be taken intoaccount in the interference threshold. The total interferenceand noise at the secondary receiver is assumed to have unitvariance. The behavior of the system using all combinationsof the allocation strategies given by (15), denoted as realallocation, and (17), denoted as binary allocation, andinterference queuing using the interference queue (4), denotedas power queue, and interference outage queue (23), denotedas counter queue.

Figure 1 portraits the energy required per bit for the differentstrategies. As a rule, the binary allocation strategies require ahigher energy for small delays but behave more similarly toreal allocation when average delay is allowed to be higher.No allocation strategy matches, however, the performance ofthe real valued allocation with the interference power queue,which approaches the limiting performance of the system asexpected. These results suggest that quantized informationexchange may be needed to close the gap to optimal allocationwhile maintaining low control signaling overhead.

Figure 2 presents the delay behavior of the system interms of average queuing delay. The delay for real allocation

100 101 102

100

101

102

V

Delay

Power queue, real alloc.Power queue, binary alloc.Counter queue, real alloc.Counter queue, binary alloc.

Fig. 2. Average delay

saturates quickly, which shows the offered load is supportablewith the power allocation in question. The delay of the binaryallocation schemes, however, grows with the allowed buffersize. This means the binary strategies cannot support theload and need to operate at close to full buffers and droppackets occasionally. These observations are supported by thedropping statistics, which confirm that with the simulationparameters as above, the dropping rate of the real allocationschemes vanish while the dropping rates of the binary schemesremain high even at V = 100.

Figure 3 shows the interference outage occurrence ratesachieved by the strategies. All strategies are able to fulfillthe interference outage constraint. However, the strategiesusing the power queue (4) and use the Markov inequality tofulfill the outage constraint behave clearly sub-optimally. TheMarkov inequality is very loose and results in a pessimisticstrategy. The methods employing the counter queue providea tighter fit to the constraint, and interestingly enough, thebinary allocation performs tightly to the constraint.

To explain the constraint tightness, Figure 4 presents anevaluation of the real valued allocation with the counter queuewith several different average arrival rates, given in terms oftotal arrivals in the system. It is clear that all tested ratesbeyond 1 bit/s/Hz cannot be stabilized without the packetadmission protocol that limits the queue size, and the averagedelay of packets grow with the buffer size. Figure 5 shows howthis coincides with the algorithm reaching closer to the allowedinterference constraint. This behavior, where the interferenceconstraint is tight when the system is pushed to the capacitylimit appears identical to the binary allocation behavior shownin Figures 2 and 3.

IX. CONCLUSIONS

The back-pressure framework is able to guarantee the ful-fillment of interference constraints, and given optimal real-valued power allocation approach the minimum transmittedenergy per bit. While the simple ON-OFF binary power

100 101 1020

0.002

0.004

0.006

0.008

0.01

0.012

V

Inte

rfere

nce

Outa

ge

Power queue, real alloc.Power queue, binary alloc.Counter queue, real alloc.Counter queue, binary alloc.Constraint

Fig. 3. Interference outage rate

100 101 102

100

101

102

V

Delay

1 bits/s/Hz1.5 bits/s/Hz2 bits/s/Hz2.5 bits/s/Hz

Fig. 4. Delay behavior of real valued allocation

allocation schemes do not perform as well as their water-filling type counterparts, they show promise when higherdelays are allowed. The direct conversion of an interferenceoutage constraint to an average interference constraint withthe Markov inequality results in a loose design that can beimproved upon. The tested algorithms perform tightly with theconstraint only when arrivals are unsupportable on the channel.

100 101 1026

6.5

7

7.5

8

8.5

9

9.5

10

10.5x 103

V

Inte

rfere

nce

Outa

ge

1 bits/s/Hz1.5 bits/s/Hz2 bits/s/Hz2.5 bits/s/HzConstraint

Fig. 5. Interference outage behavior of real valued allocation

ACKNOWLEDGMENTS

The author would like to thank the anonymous reviewers fortheir thoughtful and precise comments that helped to improvethe manuscript and focus further work.

The research leading to these results has received fundingfrom the European Communitys Seventh Framework Pro-gramme (FP7/2007-2013) under grant agreement n 216076.

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